Download Partial Differential Equations: An Introduction to Theory and

PartialDifferentialEquations
PartialDifferentialEquations
AnIntroductiontoTheoryandApplications
MichaelShearer
RachelLevy
PRINCETONUNIVERSITYPRESS
PrincetonandOxford
Copyright©2015byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,Woodstock,OxfordshireOX201TW
press.princeton.edu
CoverphotographcourtesyofMichaelShearerandRachelLevy.
CoverdesignbyLorraineBetzDoneker.
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Shearer,Michael.
Partialdifferentialequations:anintroductiontotheoryandapplications/MichaelShearer,RachelLevy.
Pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-691-16129-7(cloth:alk.paper)—ISBN0-691-16129-1(cloth:alk.paper)
1.Differentialequations,Partial.I.Levy,Rachel,1968–II.Title.
QA374.S452015
515′.353—dc23
2014034777
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinMinionProwithMyriadProandDINdisplayusingZzTEXbyPrinceton
EditorialAssociatesInc.,Scottsdale,Arizona
Printedonacid-freepaper.
PrintedintheUnitedStatesofAmerica
10987654321
Contents
Prefaceix
1.Introduction1
1.1.LinearPDE2
1.2.Solutions;InitialandBoundaryConditions3
1.3.NonlinearPDE4
1.4.BeginningExampleswithExplicitWave-likeSolutions6
Problems8
2.Beginnings11
2.1.FourFundamentalIssuesinPDETheory11
2.2.ClassificationofSecond-OrderPDE12
2.3.InitialValueProblemsandtheCauchy-KovalevskayaTheorem17
2.4.PDEfromBalanceLaws21
Problems26
3.First-OrderPDE29
3.1.TheMethodofCharacteristicsforInitialValueProblems29
3.2.TheMethodofCharacteristicsforCauchyProblemsinTwoVariables32
3.3.TheMethodofCharacteristicsinRn35
3.4.ScalarConservationLawsandtheFormationofShocks38
Problems40
4.TheWaveEquation43
4.1.TheWaveEquationinElasticity43
4.2.D’Alembert’sSolution48
4.3.TheEnergyE(t)andUniquenessofSolutions56
4.4.Duhamel’sPrinciplefortheInhomogeneousWaveEquation57
4.5.TheWaveEquationonR2andR359
Problems61
5.TheHeatEquation65
5.1.TheFundamentalSolution66
5.2.TheCauchyProblemfortheHeatEquation68
5.3.TheEnergyMethod73
5.4.TheMaximumPrinciple75
5.5.Duhamel’sPrinciplefortheInhomogeneousHeatEquation77
Problems78
6.SeparationofVariablesandFourierSeries81
6.1.FourierSeries81
6.2.SeparationofVariablesfortheHeatEquation82
6.3.SeparationofVariablesfortheWaveEquation91
6.4.SeparationofVariablesforaNonlinearHeatEquation93
6.5.TheBeamEquation94
Problems96
7.EigenfunctionsandConvergenceofFourierSeries99
7.1.EigenfunctionsforODE99
7.2.ConvergenceandCompleteness102
7.3.PointwiseConvergenceofFourierSeries105
7.4.UniformConvergenceofFourierSeries108
7.5.ConvergenceinL2110
7.6.FourierTransform114
Problems117
8.Laplace’sEquationandPoisson’sEquation119
8.1.TheFundamentalSolution119
8.2.SolvingPoisson’sEquationinRn120
8.3.PropertiesofHarmonicFunctions122
8.4.SeparationofVariablesforLaplace’sEquation125
Problems130
9.Green’sFunctionsandDistributions133
9.1.BoundaryValueProblems133
9.2.TestFunctionsandDistributions136
9.3.Green’sFunctions144
Problems149
10.FunctionSpaces153
10.1.BasicInequalitiesandDefinitions153
10.2.Multi-IndexNotation157
10.3.SobolevSpacesWk,p(U)158
Problems159
11.EllipticTheorywithSobolevSpaces161
11.1.Poisson’sEquation161
11.2.LinearSecond-OrderEllipticEquations167
Problems173
12.TravelingWaveSolutionsofPDE175
12.1.Burgers’Equation175
12.2.TheKorteweg-deVriesEquation176
12.3.Fisher’sEquation179
12.4.TheBistableEquation181
Problems186
13.ScalarConservationLaws189
13.1.TheInviscidBurgersEquation189
13.2.ScalarConservationLaws196
13.3.TheLaxEntropyConditionRevisited201
13.4.UndercompressiveShocks204
13.5.The(Viscous)BurgersEquation206
13.6.MultidimensionalConservationLaws208
Problems211
14.SystemsofFirst-OrderHyperbolicPDE215
14.1.LinearSystemsofFirst-OrderPDE215
14.2.SystemsofHyperbolicConservationLaws219
14.3.TheDam-BreakProblemUsingShallowWaterEquations239
14.4.Discussion241
Problems242
15.TheEquationsofFluidMechanics245
15.1.TheNavier-StokesandStokesEquations245
15.2.TheEulerEquations247
Problems250
AppendixA.MultivariableCalculus253
AppendixB.Analysis259
AppendixC.SystemsofOrdinaryDifferentialEquations263
References265
Index269
Preface
Thefieldofpartialdifferentialequations(PDEforshort)hasalonghistorygoing
backseveralhundredyears,beginningwiththedevelopmentofcalculus.Inthis
regard,thefieldisatraditionalareaofmathematics,althoughmorerecentthan
suchclassicalfieldsasnumbertheory,algebra,andgeometry.Asinmanyareas
ofmathematics,thetheoryofPDEhasundergonearadicaltransformationinthe
past hundred years, fueled by the development of powerful analytical tools,
notably, the theory of functional analysis and more specifically of function
spaces.Thedisciplinehasalsobeendrivenbyrapiddevelopmentsinscienceand
engineering, which present new challenges of modeling and simulation and
promotebroaderinvestigationsofpropertiesofPDEmodelsandtheirsolutions.
AsthetheoryandapplicationofPDEhavedeveloped,profoundunanswered
questions and unresolved problems have been identified. Arguably the most
visible is one of the Clay Mathematics Institute Millennium Prize problems1
concerning the Euler and Navier-Stokes systems of PDE that model fluid flow.
The Millennium problem has generated a vast amount of activity around the
world in an attempt to establish well-posedness, regularity and global existence
results, not only for the Navier-Stokes and Euler systems but also for related
systemsofPDEmodelingcomplexfluids(suchasfluidswithmemory,polymeric
fluids, and plasmas). This activity generates a substantial literature, much of it
highly specialized and technical. Meanwhile, mathematicians use analysis to
probenewapplicationsandtodevelopnumericalsimulationalgorithmsthatare
provably accurate and efficient. Such capability is of considerable importance,
given the explosion of experimental and observational data and the spectacular
accelerationofcomputingpower.
OurtextprovidesagatewaytothefieldofPDE.Weintroducethereadertoa
varietyofPDEandrelatedtechniquestogiveasenseofthebreadthanddepthof
thefield.Weassumethatstudentshavebeenexposedtoelementaryideasfrom
ordinarydifferentialequations(ODE)andanalysis;thus,thebookisappropriate
foradvancedundergraduateorbeginninggraduatemathematicsstudents.Forthe
studentpreparingforresearch,weprovideagentleintroductiontosomecurrent
theoretical approaches to PDE. For the applied mathematics student more
interested in specific applications and models, we present tools of applied
mathematicsinthesettingofPDE.Scienceandengineeringstudentswillfinda
rangeoftopicsinthemathematicsofPDE,withexamplesthatprovidephysical
intuition.
Our aim is to familiarize the reader with modern techniques of PDE,
introducing abstract ideas straightforwardly in special cases. For example,
strugglingwiththedetailsandsignificanceofSobolevembeddingtheoremsand
estimates is more easily appreciated after a first introduction to the utility of
specific spaces. Many students who will encounter PDE only in applications to
scienceandengineeringorwhowanttostudyPDEforjustayearwillappreciate
this focused, direct treatment of the subject. Finally, many students who are
interested in PDE have limited experience with analysis and ODE. For these
students, this text provides a means to delve into the analysis of PDE before or
while taking first courses in functional analysis, measure theory, or advanced
ODE.BasicbackgroundonfunctionsandODEisprovidedinAppendicesA–C.
To keep the text focused on the analysis of PDE, we have not attempted to
include an account of numerical methods. The formulation and analysis of
numerical algorithms is now a separate and mature field that includes major
developmentsintreatingnonlinearPDE.However,thetheoreticalunderstanding
gained from this text will provide a solid basis for confronting the issues and
challengesinnumericalsimulationofPDE.
A student who has completed a course organized around this text will be
prepared to study such advanced topics as the theory of elliptic PDE, including
regularity,spectralproperties,therigoroustreatmentofboundaryconditions;the
theoryofparabolicPDE,buildingonthesettingofelliptictheoryandmotivating
theabstractideasinlinearandnonlinearsemigrouptheory;existencetheoryfor
hyperbolicequationsandsystems;andtheanalysisoffullynonlinearPDE.
We hope that you, the reader, find that our text opens up this fascinating,
important, and challenging area of mathematics. It will inform you to a level
where you can appreciate general lectures on PDE research, and it will be a
foundationforfurtherstudyofPDEinwhateverdirectionyouwish.
Wearegratefultoourstudentsandcolleagueswhohavehelpedmakethisbook
possible,notablyDavidG.Schaeffer,DavidUminsky,andMarkHoeferfortheir
candid and insightful suggestions. We are grateful for the support we have
received from the fantastic staff at Princeton University Press, especially Vickie
Kern,whohasbelievedinthisprojectfromthestart.
Rachel Levy thanks her parents Jack and Dodi, husband Sam, and children
TulaandMimi,whohavelovinglyencouragedherwork.
Michael Shearer thanks the many students who provided feedback on the
coursenotesfromwhichthisbookisderived.
1.www.claymath.org/millennium/.
PartialDifferentialEquations
CHAPTERONE
Introduction
Partial differential equations (PDE) describe physical systems, such as solid and
fluid mechanics, the evolution of populations and disease, and mathematical
physics. The many different kinds of PDE each can exhibit different properties.
For example, the heat equation describes the spreading of heat in a conducting
medium,smoothingthespatialdistributionoftemperatureasitevolvesintime;
it also models the molecular diffusion of a solute in its solvent as the
concentrationvariesinbothspaceandtime.Thewaveequationisattheheartof
thedescriptionoftime-dependentdisplacementsinanelasticmaterial,withwave
solutions that propagate disturbances. It describes the propagation of p-waves
ands-wavesfromtheepicenterofanearthquake,theripplesonthesurfaceofa
pondfromthedropofastone,thevibrationsofaguitarstring,andtheresulting
sound waves. Laplace’s equation lies at the heart of potential theory, with
applicationstoelectrostaticsandfluidflowaswellasotherareasofmathematics,
suchasgeometryandthetheoryofharmonicfunctions.ThemathematicsofPDE
includes the formulation of techniques to find solutions, together with the
development of theoretical tools and results that address the properties of
solutions,suchasexistenceanduniqueness.
This text provides an introduction to a fascinating, intricate, and useful
branchofmathematics.Inadditiontocoveringspecificsolutiontechniquesthat
provideaninsightintohowPDEwork,thetextisagatewaytotheoreticalstudies
ofPDE,involvingthefullpowerofreal,complexandfunctionalanalysis.Often
we will refer to applications to provide further intuition into specific equations
andtheirsolutions,aswellastoshowthemodelingofrealproblemsbyPDE.
ThestudyofPDEtakesmanyforms.Verybroadly,wetaketwoapproachesin
thisbook.Oneapproachistodescribemethodsofconstructingsolutions,leading
to formulas. The second approach is more theoretical, involving aspects of
analysis,suchasthetheoryofdistributionsandthetheoryoffunctionspaces.
1.1.LinearPDE
TointroducePDE,webeginwithfourlinearequations.Theseequationsarebasic
to the study of PDE, and are prototypes of classes of equations, each with
different properties. The primary elementary methods of solution are related to
thetechniqueswedevelopforthesefourequations.
For each of the four equations, we consider an unknown (real-valued)
functionuonanopensetU⊂Rn.Werefertouasthedependentvariable,andx
=(x1,x2,…,xn)∈Uasthevectorofindependentvariables.Apartialdifferential
equation is an equation that involves x, u, and partial derivatives of u. Quite
often, x represents only spatial variables. However, many equations are
evolutionary, meaning that u=u(x,t) depends also on time t and the PDE has
time derivatives. The order of a PDE is defined as the order of the highest
derivativethatappearsintheequation.
TheLinearTransportEquation:
This simple first-order linear PDE describes the motion at constant speed c of a
quantityudependingonasinglespatialvariablexandtimet.Eachsolutionisa
travelingwavethatmoveswiththespeedc.Ifc>0,thewavemovestotheright;
ifc<0,thewavemovesleft.Thesolutionsareallgivenbyaformulau(x,t)=
f(x−ct).Thefunctionf=f(ξ),dependingonasinglevariableξ=x−ct,is
determinedfromsideconditions,suchasboundaryorinitialconditions.
Thenextthreeequationsareprototypicalsecond-orderlinearPDE.
TheHeatEquation:
In this equation, u(x, t) is the temperature in a homogeneous heat-conducting
material,theparameterk>0isconstant,andtheLaplacianΔisdefinedby
in Cartesian coordinates. The heat equation, also known as the diffusion
equation,modelsdiffusioninothercontexts,suchasthediffusionofadyeina
clear liquid. In such cases, u represents the concentration of the diffusing
quantity.
TheWaveEquation:
As the name suggests, the wave equation models wave propagation. The
parameter c is the wave speed. The dependent variable u = u(x, t) is a
displacement, such as the displacement at each point of a guitar string as the
string vibrates, if x ∈ R, or of a drum membrane, in which case x ∈ R2. The
acceleration utt, being a second time derivative, gives the wave equation quite
differentpropertiesfromthoseoftheheatequation.
Laplace’sEquation:
Laplace’s equation models equilibria or steady states in diffusion processes, in
whichu(x,t)isindependentoftimet,1andappearsinmanyothercontexts,such
asthemotionoffluids,andtheequilibriumdistributionofheat.
These three second-order equations arise often in applications, so it is very
usefultounderstandtheirproperties.Moreover,theirstudyturnsouttobeuseful
theoretically as well, since the three equations are prototypes of second-order
linearequations,namely,elliptic,parabolic,andhyperbolicPDE.
1.2.Solutions;InitialandBoundaryConditions
AsolutionofaPDEsuchasanyof(1.1)–(1.4)isareal-valuedfunctionusatisfying
theequation.OftenthismeansthatuisasdifferentiableasthePDErequires,and
the PDE is satisfied at each point of the domain of u. However, it can be
appropriate or even necessary to consider a more general notion of solution, in
whichuisnotrequiredtohaveallthederivativesappearingintheequation,at
leastnotintheusualsenseofcalculus.Wewillconsiderthiskindofweaksolution
later(seeChapter11).
As with ordinary differential equations (ODE), solutions of PDE are not
unique;identifyingauniquesolutionreliesonsideconditions,suchasinitialand
boundary conditions. For example, the heat equation typically comes with an
initialconditionoftheform
inwhichu0:U→Risagivenfunction.
Example1.(Simpleinitialcondition)Thefunctionsu(x,t)=ae−tsinx+be−4t
sin(2x) are solutions of the heat equation ut = uxx for any real numbers a, b.
However,a=3,b=−7wouldbeuniquelydeterminedbytheinitialcondition
u(x,0)=3sinx−7sin(2x).Thenu(x,t)=3e−tsinx−7e−4tsin(2x).
Boundary conditions are specified on the boundary ∂U of the (spatial)
domain.Dirichletboundaryconditionstakethefollowingform,foragivenfunction
f:∂U→R:
Neumannboundaryconditionsspecifythenormalderivativeofuontheboundary:
where ν(x) is the unit outward normal to the boundary at x. These boundary
conditions are called homogeneous if f ≡ 0. Similarly, a linear PDE is called
homogeneousifu=0isasolution.Ifitisnothomogeneous,thentheequationor
boundaryconditioniscalledinhomogeneous.
Equationsandboundaryconditionsthatarelinearandhomogeneoushavethe
propertythatanylinearcombinationu=av+bwofsolutionsv,w,witha,b∈
R, is also a solution. This special property, sometimes called the principle of
superposition,iscrucialtoconstructivemethodsofsolutionforlinearequations.
1.3.NonlinearPDE
We introduce a selection of nonlinear PDE that are significant by virtue of
specificproperties,specialsolutions,ortheirimportanceinapplications.
TheInviscidBurgersEquation:
is an example of a nonlinear first-order equation. Notice that this equation is
nonlinearduetotheuuxterm.Itisrelatedtothelineartransportequation(1.1),
but the wave speed c is now u and depends on the solution. We shall see in
Chapter 3 that this equation and other first-order equations can be solved
systematicallyusingaprocedurecalledthemethodofcharacteristics.However,the
method of characteristics only gets you so far; solutions typically develop a
singularity,inwhichthegraphofuasafunctionofxsteepensinplacesuntilat
some finite time the slope becomes infinite at some x. The solution then
continueswithashockwave.Thesolutionisnotevencontinuousattheshock,
but the solution still makes sense, because the PDE expresses a conservationlaw
andtheshockpreservesconservation.
For higher-order nonlinear equations, there are no methods of solution that
work in as much generality as the method of characteristics for first-order
equations.Hereisasampleofhigher-ordernonlinearequationswithinteresting
andaccessiblesolutions.
Fisher’sEquation:
withf(u)=u(1−u). This equation is a model for population dynamics when
the spatial distribution of the population is taken into account. Notice the
resemblance to the heat equation; also note that the ODE u′(t) = f(u(t)) is the
logisticequation,describingpopulationgrowthlimitedbyamaximumpopulation
normalized to u = 1. In Chapter12, we shall construct traveling waves, special
solutions in which the population distribution moves with a constant speed in
onedirection.Recallthatallsolutionsofthelineartransportequation(1.1)are
traveling waves, but they all have the same speed c. For Fisher’s equation, we
havetodeterminethespeedsoftravelingwavesaspartoftheproblem,andthe
travelingwavesarespecialsolutions,notthegeneralsolution.
ThePorousMediumEquation:
Inthisequation,m>0isconstant.Theporousmediumequationmodelsflowin
porousrockorcompactedsoil.Thevariableu(x,t)≥0measuresthedensityofa
compressiblegasinagivenlocationxattimet.Thevalueofmdependsonthe
equation of state relating pressure in the gas to its density. For m = 1, we
recovertheheatequation,butform≠1,theequationisnonlinear.Infact,m≥
2forgasflow.
TheKorteweg-deVries(KdV)Equation:
Thisthird-orderequationisamodelforwaterwavesinwhichtheheightofthe
wave is u(x, t). The KdV equation has particularly interesting traveling wave
solutions called solitarywaves, in which the height is symmetric about a single
crest.Theequationisamodelinthesensethatitreliesonanapproximationof
the equations of fluid mechanics in which the length of the wave is large
comparedtothedepthofthewater.
Burgers’Equation:
The parameter ν > 0 represents viscosity, hence the name inviscid Burgers
equationforthefirst-orderequation(1.6)havingν = 0. Burgers’ equation is a
combination of the heat equation with a nonlinear term that convects the
solutioninawaytypicaloffluidflow.(SeetheNavier-Stokessystemlaterinthis
list.)Thisimportantequationcanbereducedtotheheatequationwithaclever
change of dependent variable, called the Cole-Hopf transformation (see Chapter
13,Section13.5).
Finally,wementiontwosystemsofnonlinearPDE.
TheShallowWaterEquations:
in which g > 0 is the gravitational acceleration. The dependent variables h, v
representtheheightandvelocity,respectively,ofashallowlayerofwater.The
variablexisthehorizontalspatialvariable,alongaflatbottom,anditisassumed
that there is no dependence or motion in the orthogonal horizontal direction.
Moreover,thevelocityvistakentobeindependentofdepth.
TheNavier-StokesEquations:
describe the velocity u ∈ R3 and pressure p in the flow of an incompressible
viscous fluid. In this system of four equations, the parameter ν > 0 is the
viscosity,thefirstthreeequations(foru)representconservationofmomentum,
andthefinalequationisaconstraintthatexpressestheincompressibilityofthe
fluid. In an incompressible fluid, local volumes are unchanged in time as they
follow the flow. Apart from special types of flow (such as in a stratified fluid),
incompressibilityalsomeansthatthedensityisconstant(andisincorporatedinto
ν,thekinematicviscosity).
Interestingly,themomentumequation,regardedasanevolutionequationfor
u,resemblesBurgers’equationinstructure.Thepressurepdoesnothaveitsown
evolutionequation;itservesmerelytomaintainincompressibility.Inthelimitν
→0,werecovertheincompressibleEulerequationsforaninviscidfluid.Thisisa
singularlimitinthesensethattheorderofthemomentumequationisreduced.It
isalsoasingularlimitforBurgers’equation.
1.4.BeginningExampleswithExplicitWave-likeSolutions
ThelinearandnonlinearfirstorderequationsdescribedinSections1.1and1.3
nicely illustrate mathematical properties and representation of wave-like
solutions. We discuss these equations and their solutions as a starting point for
moregeneralconsiderations.
1.4.1.TheLinearTransportEquation
Solutionsofthelineartransportequation,
wherec∈Risaconstant(thewavespeed),aretravelingwavesu(x,t)=f(x−
ct). We can determine a unique solution by specifying the function f : R → R
fromaninitialcondition
Figure1.1.Lineartransportequation:travelingwavesolution.(a)t=0;(b)t>
0.
inwhichu0:R→Risagivenfunction.Thentheuniquesolutionoftheinitial
valueproblem (1.8), (1.9) is the traveling wave u(x, t) = u0(x − ct). A typical
travelingwaveisshowninFigure1.1.
Insteadofinitialconditions,wecanalsospecifyaboundaryconditionforthis
PDE.Hereisanexampleofhowthiswouldlook,forfunctionsϕ,ψgivenonthe
interval[0,∞):
Thesolutionuof(1.8),(1.10)willbeafunctiondefinedonthefirstquadrantQ1
={(x,t):x≥0,t≥0}inthex-tplane.ThegeneralsolutionofthePDEisu(x,
t)=f(x − ct); the initial condition specifies f(y) for y > 0, and the boundary
conditiongivesf(y)fory<0.Bothareneededtodeterminethesolutionu(x,t)
onQ1.
1.4.2.TheInviscidBurgersEquation
Thisequation,
haswavespeeduthatdependsonthesolution,incontrasttothelineartransport
equation(1.8)inwhichthewavespeedcisconstant.Ifweusethewavespeedto
track the solution, we can sketch its evolution. In Figure1.2 we show how an
initial condition (1.9) evolves for small t > 0. Points nearer the crest travel
faster,sinceuislargerthere,sothefrontofthewavetendstosteepen,whilethe
backspreadsout.NoticehowFigure1.2differsfromFigure1.1.Thesolutionu=
u(x,t)canbespecifiedimplicitlyinanequationwithoutderivatives:
Figure1.2.InviscidBurgersequation:nonlinearwavepropagation.(a)t=0;(b)
t>0.
Eventually,thegraphbecomesinfinitelysteep,andtheimplicitsolutionin(1.12)
isnolongervalid.Thesolutioniscontinuedtolargertimebyincludingashock
wave,definedinChapter13.
PROBLEMS
1.Showthatthetravelingwaveu(x,t)=f(x−3t)satisfiesthelineartransport
equationut+3ux=0foranydifferentiablefunctionf:R→R.
2.Findanequationrelatingtheparametersk,m,nsothatthefunctionu(x,t)=
emtsin(nx)satisfiestheheatequationut=kuxx.
3.Findanequationrelatingtheparametersc,m,nsothatthefunctionu(x,t)=
sin(mt)sin(nx)satisfiesthewaveequationutt=c2uxx.
4.Findallfunctionsa,b,c:R→Rsuchthatu(x,t)=a(t)e2x+b(t)ex+c(t)
satisfiestheheatequationut=uxxforallx,t.
5. For m > 1, define the conductivity k = k(u) so that the porous medium
equation(1.7)canbewrittenasthe(quasilinear)heatequation
6.Solvetheinitialvalueproblem
7.Solvetheinitialboundaryvalueproblem
ExplainwhythereisnosolutionifthePDEischangedtout−4ux=0.
8. Consider the linear transport equation (1.8) with initial and boundary
conditions(1.10).
(a)Supposethedataϕ,ψaredifferentiablefunctions.Showthatthefunction
u:Q1→Rgivenby
satisfies the PDE away from the line x = ct, the boundary condition, and
initial condition. To see where (1.13) comes from, start from the general
solutionu(x,t)=f(x−ct)ofthePDEandsubstituteintothesideconditions
(1.10).
(b)Insolution(1.13),thelinex=ct,whichemergesfromtheoriginx=t=
0, separates the quadrant Q1 into two regions. On the line, the solution has
one-sidedlimitsgivenbyϕ,ψ.Consequently,thesolutionwillingeneralhave
singularitiesontheline.
(i) Find conditions on the data ϕ, ψ so that the solution is continuous
acrossthelinex=ct.
(ii)Findconditionsonthedataϕ,ψsothatthesolutionisdifferentiable
acrossthelinex=ct.
9. Let f : R → R be differentiable. Verify that if u(x, t) is differentiable and
satisfies(1.12),thatis,u=f(x−ut),thenu(x,t)isasolutionoftheinitialvalue
problem
10.Letu0(x)=1−x2if−1≤x≤1,andu0(x)=0otherwise.
(a) Use (1.12) to find a formula for the solution u = u(x, t) of the inviscid
Burgersequation(1.11),(1.9)with−1<x<1,
.
(b)Verifythatu(1,t)=0,
.
(c) Differentiate your formula to find ux(1−,t), and deduce that ux(1−,t) →
−∞as
.
Note:ux(x,t) is discontinuous at x = ±1; the notation u(1−, t) means the
one-sidedlimit:
.Similarly,
means,
,with
.
1.However,therearetime-dependentsolutions,forexampleu(x,t)linearinxorindependentofx.
CHAPTERTWO
Beginnings
In the previous chapter we constructed solutions for example equations.
However, much of the study of PDE is theoretical, revolving around issues of
existenceanduniquenessofsolutions,andpropertiesofsolutionsderivedwithout
writingformulasforthesolutions.Ofcourse,existenceanduniquenessissuesare
resolvedifitispossibletoconstructallsolutionsofagivenPDE,butcommonly
thisconstructiveapproachisnotavailable,andmoreabstractmethodsofanalysis
arerequired.Inthischapterweoutlinetheoreticalconsiderationsthatwillcome
upfromtimetotime,giveasomewhatgeneralclassificationofsingleequations,
and then give a flavor of theoretical approaches by presenting the CauchyKovalevskayatheoremanddiscussingsomeofitsramifications.Finally,weshow
how PDE can be derived from balance laws (otherwise known as conservation
laws) that come from fundamental considerations underlying the modeling of
mostapplications.
2.1.FourFundamentalIssuesinPDETheory
Generally, the theoretical study of PDE focuses on four basic issues, three of
whicharelumpedtogetheraswell-posednessinthesenseofHadamard.1
1.Existence:IsthereasolutionofthePDEsatisfyingaspecificsetofboundary
andinitialconditions?
2.Uniqueness:Isthereonlyonesolutionforaspecificsetofboundaryandinitial
conditions?
3.Continuousdependenceondata:Dosmallchangesininitialconditions,
boundaryconditions,andparameterscreateonlysmallchangesinthe
solution?Wemightsaythesolutionisrobusttochangesinthedata.
Sometimes,thispropertyiscalledstructuralstability,ormoreloosely,stability.
Thefourthpropertyisgenerallyseparatedfromconsiderationsofwell-posedness:
4.Regularity:Howmanyderivativesdoesthesolutionhave?Wesometimesrefer
tothispropertyasthesmoothnessofthesolution.
Well-posedness is a desirable property if the goal is to model a repeatable
experiment, for example. Of the four properties, one could argue that the most
importantpropertyisexistence.Afterall,whatuseisaPDEmodelifitdoesnot
have a solution? In the theory of ODE, showing the existence of solutions is
generally straightforward, at least locally, based on the classical existence and
uniqueness theorem for initial value problems. In the previous chapter we
establishedexistencebyconstructingsolutions.However,ingeneralthetheoryof
existenceofsolutionsforPDEisacomplexandhighlytechnicalsubject.
Existence.Theapproachofthisbookistostudyexistenceissuesonlyforclasses
ofequations(andclassesofsolutions)forwhichthetheoryiselementary,suchas
classical (i.e., continuously differentiable) solutions of first-order equations. For
second-order equations, we begin by choosing problems for which we can
construct explicit solutions, thus avoiding the technicalities of proving general
existencetheorems.Towardtheendofthebook(seeChaps.9–11),weintroduce
some of the theoretical underpinnings of more general theories of PDE, such as
thetheoryofdistributions,theuseofSobolevspaces,andmaximumprinciples.
Uniqueness. Uniqueness is often the easiest property to establish. Moreover, it
doesnotrequiretheexistenceofsolutions,aswecanstate:“Thereexistsatmost
onesolution.”
Continuous dependence. Continuous dependence can be established using
techniques from analysis that estimate the closeness of distinct solutions with
differentdata,intermsoftheclosenessofthedata.Closenessofcourseinvolves
defining a suitable notion of distance—a metric—on both the space in which
solutions reside and on the space of data. These notions will be formally
introducedasneeded.
Regularity. Regularity is generally the hardest property to characterize,
requiring the most delicate analysis. In this text we make observations about
regularity from explicit solutions; regularity more generally and theoretically
involvesmoretechnicalmachinery.
2.2.ClassificationofSecond-OrderPDE
When studying ODE, it is convenient to be able to distinguish among different
kindsofequationsbasedonsuchcriteriaaslinearvs.nonlinearandseparablevs.
nonseparable. For PDE, there are also multiple ways to distinguish among
equations, some similar to the criteria for ODE. In the next chapter we discuss
first-orderPDEindetail,showingthatthetheoryislinkedcloselytosystemsof
first-orderODE.
For second-order equations, there are distinct families of equations,
distinguished by typical properties of their solutions. We identify the class of
hyperbolic equations, with wave-like solutions, and elliptic equations,
representing steady-state or equilibrium solutions. Between these two general
classes are the parabolic equations, which, like hyperbolic equations, have a
time-like independent variable but also have properties akin to those of elliptic
equations. The heat equation, the wave equation, and Laplace’s equation are
second-order linear constant-coefficient prototypes of parabolic, hyperbolic, and
elliptic PDE, respectively. Although this chapter is primarily about linear
equationsintwovariables,weincludesomeremarksaboutequationswithmore
independentvariablesandnonlinearequations.
2.2.1.ConstantCoefficients
Toexplainhowthetermshyperbolic,elliptic,andparaboliccometobeassociated
withPDE,itissimplesttoconsiderasecond-orderequationoftheform
wherethecoefficientsa,b,carerealnumbers,andtheright-handsidef=f(x,y,
u,ux,uy)isagivenfunctioncontaininganylower-orderderivativesofu.Thetype
oftheequationisdeterminedbythenatureofthequadraticformobtainedfrom
theleft-handsideof(2.1)byreplacingeachpartialderivativebyarealvariable.
More formally, we define the principal part of the PDE as the left-hand side of
(2.1).Thenthecorrespondingdifferentialoperatorwithprincipalindicatedbythe
superscript(p)is
Associated with this differential operator is the quadratic form, known as the
principalsymbol,
inwhichξ=(ξ1,ξ2)∈R2.Theconnectionbetweenprincipalpartandprincipal
symbolistheobservation
This conversion from differential operators ∂x,∂y to multiplication by iξ1, iξ2 is
typical of integral transforms; in this case, the connection is to Fourier
transforms.Thevector(ξ1,ξ2)istheFouriertransformvariable,orwavenumber.
FouriertransformsandtheirimportancefortheanalysisofPDEarediscussedin
Chapter7.
Thequadraticform(2.2)isassociatedwitheitherahyperbola(ifb2>ac),an
ellipse (if b2 < ac), or is degenerate (if b2 = ac). Correspondingly, we say the
PDE (2.1) is hyperbolic if b2 > ac, elliptic if b2 < ac, and parabolic if b2 = ac,
providedtheequationissecondorder(i.e.,notallofa,b,carezero).
Example1.(Classification)ThepartialdifferentialoperatorL=∂2x+α∂2y,is
ellipticforα>0,hyperbolicforα<0,andparabolicforα=0.
2.2.2.MoreGeneralSecond-OrderEquations
A similar classification applies to second-order equations in any number of
variables.Asusual,writex=(x1,x2,…,xn)∈Rn.Considertheequation
wheref=f(x,u,ux1,…,uxn).Weassumetherealcoefficientsaijintheprincipal
partL(p)u(givenbytheleft-handside)areconstantandsymmetricini,j:aij=aji.
(If they were not symmetric, we could rearrange the PDE using the equality of
mixedpartialderivativestoachievesymmetry.)Theprincipalsymbolisthen
The type of the PDE depends on the nature of this quadratic expression, which
wecanwriteinmatrixform:
where A = (aij) is a real symmetric n × n matrix. If we change independent
variableswithaninvertiblelineartransformationB,
then the chain rule changes the PDE (2.3). It is instructive (see Problem 2) to
work out that the principal symbol now has coefficient matrix BABT. If B is an
orthogonal matrix, then B−1 = BT, so that the linear change of independent
variablescorrespondstoasimilaritytransformationofA.Nowlet’schooseBto
diagonalizeA,sothatBABThastheneigenvaluesofAonthediagonalandzeroes
elsewhere. This is achieved by letting the columns of B be the orthonormal
eigenvectorsofA. The effect on the PDE is to convert the principal part into a
linearcombinationofpuresecond-orderderivatives,inwhichthecoefficientsare
theeigenvaluesofA.
WesaythePDEisellipticiftheeigenvaluesofAareallnonzero,andallhave
thesamesign.ThePDEiscalledhyperbolicifalleigenvaluesarenonzero,andall
butoneofthemhavethesamesign.(Thereisthethirdpossibilitythat,forn≥
4, all but k eigenvalues, with 2 ≤ k ≤ n/2, have the same sign. This case is
called ultrahyperbolic, but it does not occur much, so we ignore it.) Finally, if
there is at least one zero eigenvalue, then we could consider the PDE to be
parabolic. In practice, parabolic equations occur most commonly as time-
dependent PDE like the heat equation, with a single zero eigenvalue. Such
parabolicequationstypicallyhavetheform
where u = u(x, t), L is a linear elliptic operator with respect to the spatial
variables,andf=f(x,t,u,ux1,…,uxn).Inthisequation,onlyoneeigenvalueof
thecoefficientmatrixAiszero.
For each type of linear second-order PDE, we can find a change of
independentvariablestotransformtheequationintoacanonicalform,inwhich
the corresponding matrix A is diagonal, so that only pure second-order
derivativesoccur(i.e.,nocrossderivatives).Infact,thechangeofvariablescan
be done in general by observing how a linear change of independent variables
correspondstoasimilaritytransformationofA.Thenwecanreversetheprocess
tofindtheappropriatechangeofvariablesfromadiagonalizationofA.
Letx ∈ Rn be the independent variable, and suppose we introduce a linear
changeofvariablestoy,throughtheorthogonalmatrixBdefinedabove,sothat
BABTisdiagonal:
In coordinates, this reads
. If u = u(x), we define w(y) = u(Cy),
whereC=B−1.Thenacarefulcalculationgives
whereλ1,…,λnaretheeigenvaluesofA.
Example2.(SamplePDEoperators)Let’sadoptthenotation∂jinterchangeably
with∂/∂xj.HerewedisplayaPDEoperator,thecorrespondingmatrixA,andthe
typeoftheoperator:
1.
2.
;elliptic.
;x1=t,x2=x,x3=y;
;hyperbolic.
Noticethatforahyperbolicequation,theoneeigenvaluewithadifferentsign
suggestsatime-likedirection(associatedwiththecorrespondingeigenvalue).
AfterdiagonalizingA,wecanscaleeachindependentvariablesothatinthe
newvariables,wehave
Variable coefficients and nonlinear equations. When the coefficients aij in
(2.3)arefunctionsofx,u,ux1,…,uxn,thentheclassificationcanvarywithxand
canalsodependonthesolution.Herearesomeexamples:
1.TheTricomiEquation(relatedtosteadytransonicflow):uyy=yuxx.Thislinear
equationishyperbolicfory>0,ellipticfory<0,andthex-axisy=0is
calledtheparabolicline.Wesaytheequationchangestype.
2.TheNonlinearSmallDisturbanceEquation:
.Thisequation
changestypeonthelineφx=1.
3.TheQuasilinearWaveEquation:utt=F(ux)xishyperbolicwhenF′(ux)>0.To
seethatitishyperbolic,wewritetheequationas
However,whenF′(ux)<0,theequationiselliptic.Foragivensolution,the
changefromhyperbolictoellipticoccursonacurveF′(ux(x,t))=0,wherethe
equationisparabolic.
4.TheSemilinearWaveEquation:utt=Δu+f(u,∇u,ut).Foru=u(x,t),x∈Rn,
theprincipalpartislinearandhyperbolic,buttheequationisnonlineariff:
Rn+2→Risnonlinear,forexamplef=f(u)=u2.
2.2.3.DispersionRelations
Fortime-dependentlinearPDEwithconstantcoefficients,wecansometimesget
moreinformationaboutsolutionsfromadispersionrelation,whichisconnectedto
theFouriertransforminthesamewayastheprincipalsymbol(2.4).Itiseasiest
toseehowthisworksinonespacedimensionandtime,whereu=u(x,t).The
basicideaistoconsideraFouriermodeu0(x)=eiξxasaninitialcondition.The
parameter ξ ≥ 0 is the wave number; it is the spatial frequency of u0. It is
convenient to use the complex form, because then derivatives are also
exponentials. Solutions will be of the form u(x, t) = eiξx+σt for some complex
numberσ.Butσ=σ(ξ)dependsonthewavenumber.Thisdependenceiscalled
thedispersionrelation.Ingeneral,σ(ξ)isnotahomogeneousfunction,unlikeL(p)
[ξ],becauseσ(ξ)involvestheentirePDE,notjusttheprincipalpart.
For the linear transport equation ut + cux = 0, we find σ = −icξ.
Correspondingsolutionsu(x,t)=eiξ(x−ct)ofthePDEaretravelingwaves(whichis
no surprise, since all solutions of this equation are traveling waves). The linear
waveequationutt=c2uxxhasσ(ξ)=±icξ,correspondingtothetravelingwaves
u(x,t)=eiξ(x±ct).
Fortheheatequationut=kuxx,wehaveσ=−kξ2.Therefore,everyFourier
mode decays exponentially, provided k > 0, and the rate of decay increases
quadratically with frequency. However, if k < 0, then each Fourier mode has
exponential growth in time, and the growth σ(ξ) is unbounded as a function of
wave number ξ. This corresponds to ill-posedness, as it implies that a general
initial condition (which involves arbitrarily high wave numbers) will blow up
immediately. The same issue arises for initial value problems for elliptic
equations,suchasLaplace’sequation.(SeeSection2.3.3.)
The linearized KdV equation ut + cux + βuxxx = 0 is an example of a
dispersive equation. We find that σ=−iω is imaginary for all wave numbers,
and ω = cξ − βξ3. The corresponding solutions u(x, t) = eiξ(x−(c−βξ )t) are
traveling waves, but the speed c − βξ2 depends quadratically on the wave
number. From another point of view, ω is the temporal frequency, so that
different Fourier modes oscillate in time at different frequencies. This is
dispersion in the mathematical sense of different spatial wave numbers giving
rise to traveling waves with different speeds and to oscillations at different
frequencies.
2
ThelinearBenjamin-Bona-Mahoney(BBM)equationut+cux+βuxxt=0is
also dispersive, but the dispersion relation involves a bounded function ω.
Anotherexampleofadispersiveequationisthebeamequationutt+k2uxxxx=0.
For dispersive equations the traveling wave speed ω = ω(ξ) is called the
phase speed or phase velocity. Another speed of interest is the group velocity,
defined as
. The group velocity of dispersive equations is different
fromthephasevelocity.Fornondispersiveequations,suchasthelineartransport
and wave equations, both velocities are the same as the single traveling wave
speed or transport velocity. The roles of group velocity and phase velocity in
linear and nonlinear wave equations are discussed in detail by Whitham in his
classictext[46].
2.3.InitialValueProblemsandtheCauchy-Kovalevskaya
Theorem
Uptothispointwehaveonlyconstructedsolutionswithexplicitformulas.Inthis
sectionweoutlineanapproachthatconstructssolutionsaspowerseries,leading
toaversionofthecelebratedCauchy-Kovalevskaya2Theorem.Weconsiderinitial
valueproblemsinafairlygeneralcontext,thatofthesecond-orderequation(2.1):
AninitialvalueproblemconsistsofthePDE,togetherwithinitialconditions:
Weassumethatallfunctionsa,b,c,f,g,hareallC∞.
Inthisproblem,yistime-like,inthesensethaty=0isaninitialtime,and
wewanttosolvetheinitialvalueproblematleastforashorttimeinterval.The
analysisofthissectionappliestobothpositiveandnegativey.Inthissectionwe
discusstheexistenceofsolutionsthatcanberepresentedasaformalpowerseries
abouty=0.Suchaserieswouldtaketheform
Remark.If(2.7)isaconvergentseries,thenuhasyderivativesofallorders,and
Herethesuperscriptindicatesrepeatedderivatives:
.Let’smakethekey
assumptionin(2.5)thatc(x,y)isnonzeroforallxinsomeintervalI(andally
nearzero).3Then(2.7)canbewritten(bydividingbyc):
where
.
Claim 2.1. For any g, h ∈ C∞(I), (2.9) plus initial conditions (2.6) uniquely
determinetheC∞functionsuk(x),k=0,1,….
Remarks.Whiletheclaimseemstobeauniquenessresult,itisalsoanexistence
result,becauseitassertsthatthefunctionsuk(x)exist.
Wearenotgoingtoprovetheclaim,butitisinstructivetoconsiderwhyitis
true.Thetermsin(2.8)withk=0andk=1aregivenbytheinitialconditions
(2.6).Differentiatingthesem≥1timeswithrespecttoxgives
g(m)(x),
and
.Inparticular,thisgivesusGontheright-handsideof(2.9)
wheny=0.Hencewehavefound∂yyu(x,0),whichisu2(x).
Togetuk(x)fork≥3,wedifferentiatethePDE(2.9)withrespecttoxandy,
successively calculating derivatives of higher and higher order in terms of
derivatives of the functions a, b, c, f, g, h, and G. For example, to calculate
, we differentiate the PDE with respect to y and set y = 0. Then
(from the chain rule) the right-hand side has a term with ∂yuxx(x, 0). But we
alreadyknowthisfrom(2.6):
.
2.3.1.LimitationsofthePowerSeriesRepresentationofFunctions
Toexaminetheissueofconvergenceoftheseries(2.7)toasolution,wefocuson
some properties of power series. Taylor’s Theorem with remainder (in one
variable)istheformula
Astringentcondition(see(2.11))isneededtobeabletopasstothelimitN→∞
andensurethattheinfiniteseriesconverges.
Example 3. (A function ζ(x) that is C∞, but the Taylor series for ζ fails to
convergetoζ(x)exceptatx=0)Let
Notethat
sothepowerseries
convergestozeroforallx,butnottothefunctionζ(x),whichisnonzeroforx>
0.
The term real analytic is reserved for C∞ functions with convergent Taylor
series:Afunctionf∈C∞(I)iscalledrealanalyticontheintervalIif,foreveryx0
∈I,thepowerseries
convergestof(x)forallxinsomeneighborhoodofx0.
Proposition2.2.Letf∈C∞(I).IfthereexistpositiveconstantsCandϵsuchthat(for
allx∈I)
thenfisrealanalyticonI.
Thisresultmakessenseifyouarefamiliarwiththeroottestforconvergence
ofseriesofnumbers.Theconverseofthepropositionistruewitharestriction:iff
isrealanalyticonI,thentheestimate(2.11)isuniformforx(Cindependentof
x)incompactsubintervalsofI.
Wecanextendtheconceptofrealanalytictofunctionsoftwovariablesinan
opensetΩ⊂R2,whichwillberelevantforthetheorembelow.
Definition. If u ∈ C∞(Ω),u is real analytic if for every (x0,y0) ∈ Ω, there is a
neighborhoodN(x y )of(x0,y0)suchthatforall(x,y)∈N(x y ),thedoubleseries
0, 0
0, 0
convergestou(x,y).
2.3.2.TheCauchy-KovalevskayaandHolmgrenTheorems
These two theorems are part of the classical culture of the study of PDE. The
Cauchy-KovalevskayaTheoremestablishestheexistenceofrealanalyticsolutions
ofinitialvalueproblemsforPDE(orsystemsofPDE)withanalyticcoefficients.
The Holmgren Theorem states that, under the conditions of the CauchyKovalevskaya Theorem, the real analytic solution is the unique C2 solution
locally.
Theorem2.3.(Cauchy-Kovalevskaya)Supposethatthefunctionsa,b,cin(2.5)are
realanalyticinI×(−δ,δ),fisrealanalytic,andg,harerealanalyticinI.Assume
(as before) that c(x, 0) ≠ 0 for x ∈ I. Then the series (2.7) converges to a real
analyticsolutionoftheinitialvalueproblem,for(x,y)insomeneighborhoodΩofI×
{0}.
Remark.Therealanalyticsolutionofthetheoremisthesumoftheseries(2.7),
butitalsohasadoubleseriesexpansion(2.12),sinceitisrealanalyticinatwodimensionalopenset.
There is only one real analytic solution, since a real analytic function is
determined by its derivatives at one point, and by Claim 2.1, these derivatives
are uniquely determined. The next result shows that even if the analyticity
assumptiononsolutionsisrelaxed,thesolutionisstillunique.
Theorem2.4.(Holmgren)Undertheabovehypotheses,thereisaneighborhoodΩof
I×{0}inR2withthepropertythatifv∈C2(Ω)satisfies(2.5)and(2.6),thenv(x,
y)isthesolutionTheorem2.3.
ThesetheoremsareprovedelegantlyintheclassictextofGarabedian[16].
2.3.3.AnImportantCautionaryExample
Despite its generality, the Cauchy-Kovalevskaya Theorem is of limited utility in
the theory of PDE, because the assumption of real analyticity of the data is too
restrictive. For example, we cannot find a power series solution to solve initial
value problems with the function (2.10) as initial data, because the function is
notrealanalytic.
However,initialvalueproblemsraiseothersignificantissues,connectedwith
Hadamard’s notion of a well-posed problem, as discussed in Section 2.1. The
followingclassicexampleillustratesHadamardill-posednessfortheinitialvalue
problemforLaplace’sequation:
Let k > 0 be a parameter that is fixed for now. The parameter k is a spatial
frequency, and in this context is referred to as the wave number. The
correspondingwavelength(oftheperiodicfunctioncoskx)is2π/k.TheCauchyKovalevskaya Theorem implies there is a unique solution of this initial value
problem,andindeedwecanfinditusingtheimportanttechniqueofseparation
ofvariables.Foreachk>0,wehavetheexplicitsolution
Nowconsiderthesolutionsask→∞.Weobservethat:
1.Theinitialcondition:
(alongwithallx-derivatives).
2.Foranyy≠0,
These two observations mean that there is not continuous dependence of the
solution on the initial data. Moreover, these are the analytic solutions of the
Cauchy-Kovalevskaya Theorem, which guarantees a solution even when the
initial value problem is not well-posed. It is also interesting to note that the
solutions grow exponentially in y for each k > 0, and the rate of growth
increases exponentially with k. In this sense, the general solution is not just
unstable (growing exponentially), but is catastrophically unstable, a
manifestationofill-posedness.
2.4.PDEfromBalanceLaws
The theory describing the mechanics of continuous materials, such as solids,
fluids,andgases,iscalledcontinuummechanics.Itisbasedonconservationlaws
ofmass,momentum,andenergy.Theindependentvariablesarex,representinga
point in the material, and time t. Typical dependent variables are density,
velocity,stress,andinternalenergy.Theyaredefinedateachpointandateach
timeinaspecifiedregionofspace-time.
Abalance law is an equation expressing a conservation principle; it equates
therateofchangeofaquantityinaregionwiththesumoftwoeffects:therate
at which the quantity is entering or leaving through the boundary (the flux
through the boundary), and the rate at which the quantity is being created or
destroyed in the region. The derivation of a PDE from a balance law typically
involvesthefollowingsteps:
1.WritethebalancelawinanarbitraryboundedregionVwithsmoothboundary
∂V.
2.UsetheDivergenceTheoremtorelatethefluxthroughtheboundary∂Vtoan
integraloverV,anddeducethatthesumoftheintegrandsintheintegralsover
Vmustbalance.Thisgivesanequationorasystemofequations.However,
boththequantityandthefluxareunknowns;consequently,therearemore
variablesthanequations.
3.Closethesystembyrelatingthevariablesthroughadditionalequations(not
necessarilyPDE)calledconstitutivelaws,resultinginthesamenumberof
equationsasvariables.
Let’s consider a region U ⊂ Rn (n = 1, 2, or 3, generally), and a quantity
(suchasmass,momentum,orenergy)thatistobeconserved,representedbya
density function u. That is, the quantity is represented by u(x, t) measured at
eachpointxinUateachtimet.Forexample,thematerialdensityu=ρisthe
density function for mass (since it is the mass per unit volume), u = ρv is the
densityfunctionformomentum(wherevisavelocity),andthetemperatureu=
θisthedensityfunctionforheatenergy.
LetVbeanopensubsetofU,withsmoothboundary∂Vhavingunitoutward
normalν=ν(x).TheamountofuinVisaquantitythatdependsontime:
ThetimerateofchangeofAisthen
SupposethequantityucanflowinoroutofV,andcanbecreatedordestroyed
withinV.ThentherateofchangeofuinVisbalancedbythefluxofuacrossthe
boundary∂Vplusthecreation(duetoasource)orthedestruction(asink)ofuin
V.
Thenetfluxthroughtheboundaryisrepresentedbyanintegral∫∂VQ(x,t)·ν
dS, where the vector-valued function Q(x,t),x ∈ U, is called the flux function.
Note that ∫∂V Q(x, t). ν dS > 0 if Q points out of V; this has the effect of
decreasing the amount A(t). The creation or destruction of u in V is likewise
specified by a function, this time a scalar function f(x, t); the net rate of
creation/destructionisgivenby∫Vf(x,t)dx.
Step1.Nowwecanwriteabalancelaw:
Step2.NextweapplytheDivergenceTheoremtoconvertthesurfaceintegral
intoavolumeintegralandcombineterms(sincetheyareallintegralsoverV):
Ifweassumetheintegrandiscontinuous,then(2.13)impliesthatitiszero
everywhereinU.Thus,wehavethePDE
Inthisequation,weregardu=u(x,t)astheunknown,butthereare
additionalfunctionsQ(x,t)andf(x,t).Thesemustbedeterminedfrom
additionalequationsthatcouldspecifyQandfasfunctionsofxandt.
However,Qinparticularismoreoftenrelatedtouandderivativesofu,orto
additionaldependentvariables.Thisleadstothefinalstep.
Step3.Specifyconstitutivelawstoclosethesystem,asinthefollowing
examples.
Example 4. (Balance laws and the heat and wave equations) The heat and
waveequationsareexamplesofPDEderivedfrombalancelaws.Conservationof
heatenergye=ρcurelatesthetemperatureu(x,t)totheheatfluxQandsource
termsF(x,t).Here,thedensityρofthematerial,anditsspecificheatcaretaken
tobeconstant.Thebalancelawleadstotheequation
Fourier’sLawofheatconductionexpressesthethermodynamicpropertythatheat
energy flows from higher temperatures to lower. Specifically, this constitutive
lawlinkstheheatfluxlinearlytothetemperaturegradient:
Whenthethermalconductivityκ>0isconstant,weobtainfrom(2.14)theheat
equationwithsourcetermf:
Thesameequationalsomodelsthediffusionofasoluteinsolution,withsolute
concentrationu.Inthiscontext,theheatequationiscalledthediffusionequation.
TheproportionalitybetweenfluxandconcentrationgradientisthentermedFick’s
Law.
We will derive the one-dimensional wave equation carefully in Section 4.1,
but here is the idea of how the wave equation arises from conservation of
momentum.
Figure2.1.Trafficflow:carstravelingonasectionofhighway.
Thebalancelawequatestherateofchangeofmomentumwiththedivergenceof
themomentumflux:
Hereρ>0isthedensity,whichwetaketobeconstant,and
isavelocity,
thetimederivativeofdisplacementu.Insomeapplications,suchaselasticity,a
reasonable constitutive law specifies that the momentum flux is proportional to
the gradient of the displacement: Q=−k∇u,wherek > 0 is the constant of
proportionality.Thisleadsdirectlytothewaveequation:
withc2=k/ρ.
Example5. (Traffic flow) Traffic flow models help to illustrate how the
conservationlawandconstitutiveequationareformulatedseparately.Sincethese
modelsareonedimensional,theFundamentalTheoremofCalculusreplacesthe
DivergenceTheoreminstep2.Considerasinglelanehighwayandletu(x,t)be
thedensityofcarsatlocationx∈Ronthehighwayattimet:
Thatis,eachtimetandeverypointxonthehighwayisassociatedwithatraffic
densityu(x,t).Supposethecarsaremovingtotheright,asshowninFigure2.1.
Toformulatethebalancelaw,weconsiderthenumberofcarsinasectionof
highwaybetweenfixedlocationsx=aandx=battimet:
ThetimerateofchangeofNshouldbeequaltothenetrateatwhichcarsenter
at x = a and leave from x = b. (We assume no cars are manufactured or
scrappedinthemiddleofthehighway,sotherearenosourceorsinkterms:f≡
0.)LetQ(x,t)denotethefluxofcarspastaparticularpointxattimet:
Then, since cars enter the section [a,b] at a rate Q(a,t), and leave at the rate
Q(b,t),wehave
That is, the rate of change of the number of cars in the region (section of the
highway)isbalancedbythefluxofcarsthroughtheboundary.
NowusetheFundamentalTheoremofCalculustoobtain
Ifweassumecontinuityoftheintegrand,then
Equation(2.16)hastwounknowns,Qandu,andweneedanadditionalequation.
Intrafficflowmodels,typicallyweaddaconstitutivelawthatrelatesthefluxQ
tothedensityu.First,itmakessensethatQshouldbetheproductofspeedand
density,thenumberofcarsperunittimeacrossafixedlocation:
To complete the description of Q as a function of density alone, we need to
specifythespeedvasafunctionofdensity.Wecanattempttofitdatafromreal
observationsoftraffic,or,asisoftendonewhenfirstformulatingamathematical
model,weintroduceafunctionalformthatisconsistentwithnaturalqualitative
or physical properties. In the current context a simple model takes the traffic
speedvtobealinearanddecreasingfunctionoftrafficdensity:
(see Fig. 2.2). Here the parameters α, β have the interpretation of maximum
densityandmaximumspeed.Theequationut+Qx=0isnowasingleequation
fortheunknownu(x,t):
Equation(2.17)isknownastheLighthill-Whitham-Richardsmodel.Whitham
[46] identifies various scenarios, such as the timing of traffic lights, in which
solutions of the equation describe the behavior of traffic. Modeling traffic flow
with PDE is of considerable interest, as the equations are easy to work with
numerically. However, there are significant challenges in devising realistic
models that incorporate important behavior, such as multilane traffic and how
trafficdividesatintersectionsorentrancesandexitstofreeways.
Equation(2.17)isrelatedtotheinviscidBurgersequation
Figure2.2.Trafficflow:speedvvs.densityu.
inthatbothhaveaquadraticflux,buttheconvexityisdifferent.Thisdifference
in convexity is relevant when considering blow-up of ux (i.e., the steepening
observedinFig.1.2).Intrafficflow,inwhichthefluxisconvexdown,therewill
beajam(indicatedbyblow-upofuxtoinfinity—thecarsgetreallyscrunched!)if
the traffic density u increases ahead (i.e., is an increasing function of x).
However,intheinviscidBurgersequation,forwhichthefluxu2/2isconvexup,
we saw in Section 1.4.2 that there is blow-up of ux (to −∞) when u is a
decreasingfunctionofx(seeFig.1.2).
PROBLEMS
1.(a)Determinethetypeoftheequationuxx+uxy+ux=0.
(b) Determine the type of the equation uxx+uxy + αuyy + ux + u = 0 for
eachrealvalueoftheparameterα.
(c)Determinethetypeoftheequationutt+2uxt+uxx=0.Verifythatthere
are solutions u(x, t) = f(x − t) + tg(x − t) for any twice differentiable
functionsf,g.
(d) The equation (1 + y)uxx−x2uxy+xuyy = 0 is hyperbolic or elliptic or
parabolic,dependingonthelocationof(x,y)intheplane.Findaformulato
describe where in the x-y plane the equation is hyperbolic. Sketch the x-y
plane and label where the equation is hyperbolic, where it is elliptic, and
whereitisparabolic.
2.Showthatwiththechangeofvariablesy=Bx,theprincipalsymbolof(2.4)
corresponding to (2.3) has coefficients cij given by C = BABT, where C = (cij).
One approach to this is to write everything in coordinate form, such as
, BA = (bakj),
, and use the chain rule to
convertxjderivativestosumsofykderivatives.
3.Fortheseries(2.7),writeformulasforu3(x)andu4(x)intermsofderivatives
ofthefunctionsa,b,c,f,g,h,andG.
4.Showthatζ∈C∞(R),whereζisgivenby(2.10).
5.Findthedispersionrelationσ=σ(ξ)forthefollowingdispersiveequations.
(a) The beam equation utt=−uxxxx. Why is the equation dispersive and not
dissipative?Whatmakesthisequationdispersive,whereasthewaveequation
isnotdispersive?
(b)ThelinearBenjamin-Bona-Mahoney(BBM)equationut+cux+βuxxt=0.
Deduce that the equation is dispersive, and show that the corresponding
solutionsu=eiξx+σ(ξ)taretravelingwaves.Writeaformulafortheirspeedasa
function of wave number ξ. Identify a significant difference between this
formulaandthewavespeedsofKdVtravelingwaves.
6.SupposeinthetrafficflowmodeldiscussedinSection2.4thatthespeedvof
carsistakentobeapositivemonotonicdifferentiablefunctionofdensity:
v=v(u).
(a)Shouldvbeincreasingordecreasing?
(b)Howwouldyoucharacterizethemaximumvelocityvmaxandthemaximum
densityumax?
(c)LetQ(u)=uv(u).ProvethatQhasamaximumatsomedensityu∗inthe
interval(0,umax).
(d)Cantherebetwolocalmaximaoftheflux?(Hint:MakeQ(u)quartic.)
1. Jacques S. Hadamard (1865–1963) is well known for contributions to number theory, matrices,
differentialequations,geometry,elasticity,geometricaloptics,andhydrodynamics.Hispapersof1901and
1902discussill-posedandwell-posedproblems,respectively.
2.Augustin-LouisCauchy(1789–1857)mademanycontributionstomathematics,includingfundamental
developments in real and complex analysis, modern algebra, and the theory of elasticity. Sofia Vasilyevna
Kovalevskaya(1850–1891)madeimportantcontributionstoanalysis,differentialequations,andmechanics.
3.Sinceweshallbediscussingsolutionsonlylocallyin(x,y)wecouldsimplyassumec(0,0)≠0.
CHAPTERTHREE
First-OrderPDE
First-orderequationsenjoyaspecialplaceintheoryofPDE,astheycangenerally
be solved explicitly using the method of characteristics. Although this method
applies more generally to fully nonlinear equations, such as Hamilton-Jacobi
equations,wewillrestrictattentiontolinearandquasilinearequations,inwhich
the first-order derivatives of the dependent variable u occur linearly, with
coefficients that may depend on u. The method of characteristics reduces the
determination of explicit solutions to solving ODE. We develop the theory in
severalstages,withincreasingsophistication,butreallytheideaisthesameall
along: first-order PDE become ODE when the PDE are regarded as specifying
directionalderivativesinseveraldimensions.
3.1.TheMethodofCharacteristicsforInitialValueProblems
Initialvalueproblemsinonespacevariablexandtimettaketheform
Let’sassumethatcandraregivenC1(continuouslydifferentiable)functions,and
the initial condition f:R → R is a given C1 function. The coefficient c will be
establishedasawavespeed,andthenotationrsimplystandsfortheright-hand
sideofthePDE.
Wecansolve(3.1)atleastforashorttimeinterval(andperhapsonlylocally
in space) using the method of characteristics, which reduces the initial value
problem(3.1)toaninitialvalueproblemforasystemofODE.Inthismethod,we
dependontheobservationthatif{(x(t),t):t≥0}isasmoothcurve,thenalong
thecurve,u(x(t),t)hasrateofchange
givenbythechainrule.Comparing(3.2)withthePDEin(3.1),itlooksasthough
we can make progress by setting
and interpreting c as a speed. The lefthand side of the PDE can also be interpreted as the derivative of u(x,t), in the
direction(c,1)inx-tspace.1
NowthePDE(3.1)canbereplacedbytheODEsystem
These ODE are called the characteristic equations. Note that the characteristic
equationsareautonomousonlyifcandrareindependentoft.
Initial conditions for the ODE system are derived from the initial condition
u(x,0)=f(x)forthePDEproblem(3.1).ToseewhattheODEinitialconditions
should be, let’s write x(0) = x0, and u(t) in place of u(x(t),t). Then the initial
conditionsfor(3.3)are
FromthetheoryofODE,weknowthattheinitialvalueproblem(3.3),(3.4)hasa
uniquesolution(x(t),u(t)),atleastlocallyintimeforeachx0.Toemphasizethat
wehaveasolutionforeachx0,let’swritethesolutionas
.The
semicolon indicates that x0 is regarded as a parameter in the ODE initial value
problem,butnowwearegoingtotreatx0asasecondvariable,sothatxandu
arefunctionsofthetwovariablest,x0.
The parameter x0 specifies the curve in the x-t plane
, which
werefertoasthecharacteristicthroughx=x0,t = 0. As long as curves with
different values of x0 do not cross, the family of characteristics fills a region of
theupperhalf-plane{(x,t):t≥0},therebyparameterizingpointsintheregion
withx0,t.AteachpointP:(x,t)ofthisregion,weknowthesolutionu,since
on each characteristic. Figure 3.1 illustrates the characteristic
originatingatx0thatpassesthroughpointP.Oncewehaveidentifiedthevalue
ofx0,then
.
Thereisanicephysicalinterpretationofthisconstruction.Theparameterx0,
called the Lagrangian variable, labels a material point. Then
is the
Eulerianvariabledescribingthelocationattimetofthatmaterialpoint.Thevalue
u of the variable can be thought of either in Lagrangian variables, for which
,orinEulerianvariables,forwhichuisobservedatafixedlocation:u
=u(x,t),thesolutionweseek.
Mathematically,togetthesolutionuexplicitlyateachpoint(x,t),weneedto
invert the change of variables
. To do so, we eliminate x0 and
write
asthesolutionoftheequation
.Then
isthesolutionof(3.1).
Figure3.1.Characteristic
initialvalueproblem(3.1).
:t≥0},alongwhich
,forthe
For this kind of initial value problem, the method of characteristics is
summarizedas:
1.Rewritetheinitialvalueproblem(3.1)asasystemofODEconsistingofthe
characteristicequations(3.3)withinitialconditions(3.4).
2.SolvetheODEandinitialconditionsforx(t),u(t),withparameterx0=x(0)to
getthesolutionalongeachcharacteristic.
3.Solveforx0asafunctionofx,t.Thiseffectivelychangesvariablesfromt,x0to
x,t.
4.Writethesolutionu=u(x,t).
Example1.(Initialvalueproblem)Solve
Inthisexample,yistime-likeinthesensethattheinitialconditionisposedaty
= 0. The PDE written as ∇u. (1, 1) = u shows that the left-hand side is the
directionalderivativeofuinthedirection(1,1).Considerthelinesx=y+k
parallelto(1,1),wheretheparameterkplaysthesameroleasx0above.Therate
ofchangeofualongeachlineis
Therefore,
whereA(k)isanarbitraryfunction.Thus,sincek=x−y,
isthegeneralsolutionofthePDE,dependingonthearbitraryfunctionA(k)ofa
singlevariable.
To complete the solution, we use the initial condition to determine A(k).
Settingy=0in(3.6)gives
Thus,thesolutionoftheproblemis
Since the left-hand side of (3.5) is a directional derivative, it is an ordinary
derivative in that direction. Thus, u′ = u in this direction, explaining the
exponentialgrowthofthesolutionalongeachcharacteristicx=y+k.Likewise,
thesolutionwouldbeu(x,y)=cos(x−y)iftheright-handsideofthePDEwere
zero.
Inthisexample,wefoundthecharacteristicsbeforedeterminingthebehavior
ofualongthem.Generally,thecharacteristicsfor(3.1)willalsodependonthe
solution,ifcdependsonu.
3.2.TheMethodofCharacteristicsforCauchyProblemsinTwo
Variables
Inthissectionwepresentamoregeneralversionofthemethodofcharacteristics
for first-order quasilinear PDE in two independent variables. First-order
quasilinearPDEintwoindependentvariablestaketheform
wherea,b,caregivenC1functionsfromR2×RtoR.Inthisequation,neither
of the variables necessarily has a special role, such as time. Consequently, the
notationissomewhatdifferentfromtheprevioussection.
Ratherthanposinganinitialcondition,weposeamoregeneralsidecondition
for(3.7)intheform
wherex0,y0,z0aregivenC1functionsonanintervalI.Thisissometimesreferred
toastheinitialcurveΓ.Problem(3.7),(3.8)isreferredtoastheCauchyproblem.
WeshallshowthatforC1solutions,thePDEisreallyanODEindisguise(as
wesawinExample1).Supposeu(x,y)isasolutionoftheCauchyproblem.Then
thegraphz=u(x,y)isatwo-dimensionalsurfaceinx-y-zspacethatincludesthe
curveΓ.Equation(3.7)statesthatthevectorfield(a(x,y,z),b(x,y,z),c(x,y,z))
is tangent to the solution surface z = u(x, y), since the solution surface has
normal
Figure3.2.InitialcurveΓ,thecharacteristiccurvetangentto(a,b,c),andthe
solutionsurface.
The solution surface can therefore be generated by integrating along the vector
field,startingateachpointofthecurveΓ={(x,y,z):x=x0(s)…,s∈I}.(See
Fig.3.2.)Ifτisthevariableofintegrationalongtheseintegralcurves,thenthe
surfacegeneratedisparameterizedby(s,τ):x=x(s,τ),y=y(s,τ),z=z(s,τ).
Torecoveru(x,y),wetransformfrom(s,τ)backto(x,y)inzandsetu(x,y)=
z(s, τ), establishing the existence of the inverse using the Inverse Function
Theorem(seeAppendixA).
ThisproceduretosolvetheCauchyproblem(3.7),(3.8)isdividedintothree
steps:
1.Generatethesolutionsurfacefromintegralcurves.Inthisstepwesolvetheoneparameterfamilyofinitialvalueproblems
foreachs∈I.Denotethesolution(x,y,z)(s,τ).FromODEtheory,the
solutionexists,isC1,andisunique,atleastinaneighborhoodofΓ.The
solutioncurvesinR3areknownascharacteristiccurves.Wereservetheterm
characteristicstomeantheprojectionofthecharacteristiccurvesontothex-y
plane.
2.ApplytheInverseFunctionTheorem.Inthisstepwesolvetheequations
for(s,τ)asafunctionof(x,y):(s,τ)=(s,τ)(x,y).Thesolutionis
guaranteedbytheInverseFunctionTheorem.
3.Writethesolutionsurfaceasagraphz=u(x,y).
Nowweareabletowritethesolutionasafunctionofx,y:
This procedure will work as long as the transformation (3.10) is invertible. We
can guarantee this locally by appealing to the Inverse Function Theorem.
Specifically, let P = (x0(s0), y0(s0), z0(s0)) be a point on Γ. For (3.10) to be
invertiblenear(x,y)=(x0(s0),y0(s0)),werequiretheJacobianmatrix∂(x,y)/∂(s,
τ)tobeinvertibleatthispoint.Thatis,atPwerequire
wherewehaveused(3.8),(3.9).Thisconditionmeansthatthetangent(a,b)to
thecharacteristicat(x0(s0),y0(s0))isnotparalleltothetangent
ofthe
projectionγ of Γ at P onto the x-y plane. Consequently, when (3.11) holds, we
saythatthecurveΓisnoncharacteristicatP. Thus, provided the initial data are
noncharacteristicinthesenseof(3.11),wehaveauniqueC1solutionu(x,y)of
(3.7),(3.8)for(x,y)near(x0(s0),y0(s0)).
Example2.(ACauchyproblem)SolvetheCauchyproblem
Herea=z,b=1,c=1,andtheinitialconditionisu=0ontheliney=x.
Weparameterizetheinitialconditionasfollows:
Characteristicequationsare
withcorrespondinginitialconditionsx(0)=s,y(0)=s,z(0)=0.Thus,z=τ,
sothatx′=z=τ.Nowwecansolveforxandy:
Eliminatings,wegetaquadraticequationfor
.Thus,
But τ = z = u(x, y), and to satisfy the initial condition, we have to take the
negativesquareroot:
Thesolutionisvalidonlyfor
.Infact,thesolutionsurfacez
=u(x,y)isthelowerhalfofthesmoothparabolicsurface(z−1)21+2x−2y,
whichhasafoldalongtheline
.Sincethesurfacebecomesvertical
atthefold,thesolutionu(x,y)hasasingularityontheline
,wherethe
derivativeux−uyblowsup.
3.3.TheMethodofCharacteristicsinRn
In this section we repeat the method of characteristics for a single quasilinear
first-order equation to show how the method works for any number of
independent variables. Characteristic curves are of course one dimensional, and
thus contribute one dimension to the solution surface, which is n − 1
dimensional.Theremainingdimensionsinthesurfaceareprovidedbytheinitial
conditions.
Considerx∈Rn;u=u(x)∈R.Thefirst-orderequationweconsiderhasthe
generalform
wherea:Rn×R→Rn,avectorofcoefficients,andc:Rn×R→R,ascalar,
aregivenC1functions.
The Cauchy problem involves an (n − 1)-dimensional hypersurface γ ⊂ Rn
thatprovidesinitialconditionsforcharacteristiccurves:
Characteristiccurvesin(x,z)space(Rn+1)aresolutioncurvesofthesystem
Notethatforeachs,wehaveexistenceanduniquenessofsolutionsof(3.14)for|
τ|small,sinceaandcareC1.Moreover,sincethedataareC1,thesolutionsare
C1insalso.
Asbefore,wewritethesolutionsintheform
Thesolutionu(x)=z(s,τ)isexpressedinphysicalvariablesxifwecaninvert
(3.15a) to get s = s(x),τ = τ(x). This is guaranteed by the Inverse Function
Theorem,atleastlocally,ifweassumethehypersurfaceisnoncharacteristic,i.e.,
∂x/∂(s,τ)isinvertibleonγ(whereτ=0),withz=u0(s),andrecallthat∂x/∂τ
=a:
Incomponents:
Thenwehavethesolution
Moreprecisely,themethodofcharacteristicsandtheInverseFunctionTheorem
havebeenusedtoprovethefollowingresult.
Theorem3.1.Supposethedatax0,u0areC1in a neighborhood ofs = 0 andare
noncharacteristicinthesenseof(3.16)ats=0.ThenthereexistsaneighborhoodN
ofx0(0)andaC1functionu:N→RthatsolvestheCauchyproblem(3.12),(3.13)
inN.
Example 3. (Particle size segregation in an avalanche) Avalanches and rock
slides are examples of granular flow, typically involving particles of different
sizes.Inthisexample,wewriteaPDEforthetransportoftwosizesofparticles(a
bidispersemixture)thathavethesamedensity.Weassumethatastheavalanche
flows down the hillside, it establishes a constant depth, and that the velocity
varies linearly with depth. We ignore all but the component of velocity that is
parallel to the hillside. In these circumstances, Gray and Thornton [20]
formulatedamodelthatdescribesthedistributionofparticlesintheavalanche.
Let x, y denote the spatial variables, and let v(y) = y denote the parallel
velocity.TheseareshowninFigure3.3.Thedependentvariableu=u(x,y,t)is
thevolumefractionofsmallparticles.Intheflow,largeparticlestendtorise,and
smallparticlestendtofall.GrayandThorntonarguedthatsmallparticlesfallata
speed proportional to the volume fraction 1 − u of large particles, essentially
becausetheydependonspaceopenedupbythemotionoflargeparticles.Then
largeparticleshavetomoveupwardtobalancethemotionofthesmallparticles.
Withtheseassumptions,thePDEis
Figure3.3.Coordinatesandvelocityprofilev(y)foravalancheflowmodel.
where S > 0 is a constant of proportionality. Let’s suppose there is an initial
distributionofsmallparticlesgivenby
Characteristicequationsforthisequationcanbewritten
Thus,u=u0(x0,y0)isconstantonthecharacteristiccurvethrough(x0,y0)att=
0.Consequently,
Att=0,wehavex=x0,y=y0,sothat
Thus, characteristics are parabolas in the x-t plane. Now we solve for x0, y0 in
termsofu,x,y,t:
Finally,wehaveaformulaforthesolutionu=u(x,y,t),definedimplicitlyby
theequation
Thissolutiontechniquecanbeusedtostudythedynamicsofavalancheflowwith
variousinitialandboundaryconditions.
3.4.ScalarConservationLawsandtheFormationofShocks
In this section we consider the initial value problem for the inviscid Burgers
equation. We show that solutions generated by the method of characteristics
typicallybreakdowninfinitetime.Thisnonlinearwavebehavioroccursinsuch
applications as gas dynamics, combustion and detonation, and nonlinear
elasticity. The breakdown of solutions signals the formation of a shock wave,
acrosswhichthesolutionisdiscontinuous.
Considertheinitialvalueproblem
withinitialcondition
ThemethodofcharacteristicsofSection3.1isapplicablehere:
Thus, u is constant on each characteristic, and characteristics are therefore
straightlineswithspeedu:
Thesolutionu=u(x,t)isthengivenimplicitlybytheequation
LetF(u,x,t)=u−u0(x−ut).Generally,wecannotsolveforuexplicitly,but
wecanusetheequationtoprovelocalexistencenearanyinitialpointx=x0,by
applying the Implicit Function Theorem to the equation F(u, x, t) = 0. (See
problem11.)
3.4.1.BreakdownofSmoothSolutions
AswesawinSection1.4.2,thegraphofthesolutionu(x,t)steepenswhereithas
negative slope, because larger positive values of u travel faster than smaller
values.Fornegativevaluesofu,thecharacteristicstraveltotheleft,butthesame
istrue:thegraphsteepenswheretheslopeisnegative.Mathematically,wefind
ux→−∞atsomexastincreasestoatimet∗.Thenotionthatsomevaluesofu
travelfasterthanothers,leadingtosteepening,maybeexpressedinthefollowing
statement:
Characteristicsx=ut+x0thatoriginateatpointsx0inaninterval
where
crossoneanotherinfinitetime.
Figure3.4.InviscidBurgers’equation:crossingcharacteristicsassociatedwith
thebreakdownofasmoothsolution.
InFigure3.4weshowthecharacteristicsforthesolutionshowninFigure1.2
and see that characteristics ahead of the crest of the wave eventually cross one
another. If this first occurs at a time t=t∗, then the method of characteristics
gives a multivalued function of (x, t), for t > t∗ in the region where the
characteristicscross.Wesaythesolutionbreaksdownatt=t∗.
Our goal is to make this argument rigorous and to find a formula for the
breakdown time t∗. To do so, we derive an equation for ux by taking of the
PDE,thusderivinganODEfortheevolutionofuxalongcharacteristics.Firstwe
differentiate(3.21):
Letv=ux.Thenwehave
Alongcharacteristicsx=ut+x0wegettheODE
Thisequation(knownasaRiccatiequation due to the quadratic nonlinearity) is
solvedeasily.Noticethatitstatesthatvdecreasesint,andthemoreitdecreases
throughnegativevalues,themorerapidlyitcontinuestodecrease.
Now we differentiate the initial condition u(x, 0) = u0(x) to obtain a
correspondinginitialconditionforv:
Wesolve(3.24),(3.25)tofindvalongthecharacteristicx=ut+x0:
Wedistinguishtwocases:
1.If
,thenvstaysfiniteforallt>0.Consequently,ifu0ismonotonically
increasing,thenu(x,t)isdefinedforallx,t.Notefrom(3.23)thatu(x,t)only
takesonvaluesofu0(x0),x0∈R.Therefore,ifu0isboundedbym,M,m≤
u0(x)≤M,x∈R,thenm≤u(x,t)≤Mforallx∈R,t>0.
2.Ifu0isnotmonotonicallyincreasing,sothat
then
forsomevaluesofx0,
in(3.26).Thus,thesolutionbreaksdown(ux→−∞)atdifferenttimeston
eachcharacteristic(dependingonx0).Consequently,thesolutionu(x,t)ofthe
initialvalueproblembreaksdownattheearliestsuchtimet=t∗:
Notethattheminimumisachievedwhereu0hasminimumslope,whichwill
beataninflectionpointifu0isC2.
Tocontinuethesolutionbeyondt=t∗,wedefineweaksolutions,inwhichthe
functionu(x,t) is allowed to be discontinuous. We pursue this topic in Chapter
13, after first considering solution and analysis techniques for second-order
equations.
PROBLEMS
1.Usethesubstitutionv=uytosolveforu=u(x,y):
2.Solveforu=u(x,t):
3.Solveforu=u(x,t):
4.Solve(3.5)usingthegeneralmethodofcharacteristics.Youwillneedtosetup
the initial condition with a parameter s. Show that the initial curve Γ is
noncharacteristic.
5.Verifythatu(x,t)constructedingeneralinSection3.1isindeedasolutionof
(3.1). Start by working out what calculation you have to do to carry out this
check.Youwillhavetousethechainrulerepeatedlytocheckcarefully.
6.TakeanalternativedirectapproachtoExample1,reversingtherolesofxand
y,bysettingy=x+k. The PDE then becomes the ODE
u along
characteristics. Solve and incorporate the initial condition, finally obtaining the
solutionu(x,y).
7. For the avalanche flow equation (3.17), suppose an initial distribution of
particlesisgivenby
andaninletboundaryconditionisspecifiedby
Find the solution u(x, y, t), 0 < x, 0 < y < 1, t > 0 by the method of
characteristics. (The side conditions and hence the solution do not obey the
physical constraint 0 ≤ u ≤ 1, and hence are not intended to be physically
significant.Thisproblemisanexerciseinusingthemethodofcharacteristics.)
8.(a)Usethemethodofcharacteristicstosolvetheinitialvalueproblem
(b)Showthatthesolutionblowsupast→1:
9.SketchthegraphofthetrafficflowfluxQ(see(2.16),(2.17)inChapter2)asa
functionofdensityu.ExplaineachzeroofQintermsofthephysicalmodel.
10.FormulateconstitutivelawsforthetrafficfluxQ(seeExample2inChapter
2) as a function of density assuming that traffic speed v(ρ) is a quadratic
decreasingfunctionofdensityρ.Howmanyparametersarethereinthemodel?
Isitpossibletomakethefluxnonconcaveasafunctionofdensity?
11. Write the details of how to use the Implicit Function Theorem on (3.23)to
prove:Ifu0issmoothandboundedon(−∞,∞)thenforeachx0∈R,thereisan
intervalI⊂anintervalI⊂Rcontainingx0suchthatthesolutionu(x,t)exists,is
C1,andisuniqueforallx∈Iandallsmallenought.
12.Letu0(x)=H(x)x2,whereH(x)=0forx<0andH(x)=1forx≥0isthe
Heaviside function. Write the solution u(x, t) of (3.21), (3.22) as an explicit
formulafort>0.
13.Gettheanswer(3.26)bydifferentiatingtheimplicitsolution(3.23):
(Thissimplerapproachdependsonhavingtheimplicitequationforuavailable,
whichisnotthecaseforsystemsofequations.)
14.Usethemethodofcharacteristicstoproveglobal(forallt>0)existenceofa
smooth solution of (3.21), (3.22) when the initial data are given by a strictly
increasingbutboundedC1functionu0.
15. Carry through the analysis presented in Section 3.4 for a general scalar
conservationlaw
where f : R → R is a given C2 function. Derive an implicit equation for the
solutionu(x,t)oftheCauchyproblem,andformulateaconditionforthesolution
to remain smooth for all time. Likewise, if the condition is violated, find an
expressionforthetimeatwhichthesolutionfirstbreaksdown.
1.Strictlyspeaking,thedirectionis
bytratherthanbyarclength.
;themagnitude
setstheparameterizationtobe
CHAPTERFOUR
TheWaveEquation
Thewaveequation
is the prototype for second-order hyperbolic PDE, modeling the propagation of
sound waves; electromagnetic waves, such as light; and waves in elastic solids.
We show in detail how the wave equation describes the deformation of onedimensionalelasticsolids,specifically,thinrodsandelasticstrings.
Centraltothestudyoftheone-dimensionalequationisd’Alembert’ssolution,
an explicit formula for solutions of initial value problems. The method of
spherical means provides a corresponding explicit formula in two and three
dimensions. In three dimensions, this formula embodies Huygens’ principle of
light propagation. From the wave equation we derive an energy principle in
whichthetotalenergy(thesumofkineticandpotentialenergy)isconserved.
4.1.TheWaveEquationinElasticity
Weintroducethewaveequationwithasimplederivationfromone-dimensional
elasticitytheory.Thederivationillustratesthebasicnotionsofconservationlaws
and constitutive equations introduced in Chapter 2. Then we discuss a second
application,toanelasticstringvibratinginaplane.Conservationofmomentum
leadstoasystemofnonlinearPDE.Consideringsmall-amplitudevibrationsnear
a stationary string, we linearize the equations, thereby deriving two wave
equations with different wave speeds. One equation represents longitudinal
motion along the string, and the other represents transverse motion—the
vibrationsseeninaguitarorviolinstring.
4.1.1.LongitudinalMotionofaThinElasticRod
Considerathinelasticrodundergoingonlylongitudinaldeformation(extension
orcompression),withnobending.
We label locations of cross sections in the rod by using points in a reference
configuration,aninterval,say0≤x≤1.(SeeFig.4.1.)Thephysicalconfiguration
isalsoaninterval0≤u≤L,dependingonthedeformation.Thecrosssection
labeled x in the reference configuration has coordinate u(x, t) in the physical
configuration at time t. It is convenient, but not essential, to think of the
referenceconfigurationasbeingtherodinequilibrium,withnoforcesactingon
it. The function u is called the displacement; it is the unknown, or dependent
variable. We assume the density ρ (mass per unit volume in the reference
configuration) is constant, and that the cross-sectional area A of the rod is
constantalongitslength.
Figure4.1.Deformationuinaone-dimensionalrod.(a)Referenceconfiguration
(Lagrangianvariables);(b)physicalconfiguration(Eulerianvariables).
Figure4.2.Forcesonasmallsectionoftherod.(a)Referenceconfiguration
(Lagrangianvariables);(b)physicalconfiguration(Eulerianvariables).
Consider forces on a cross section labeled x0 in the rod, at a specific time t.
Thepartoftherodwithx>x0exertsaforceF(x0,t)onthepartwithx<x0,
andthepartwithx<x0exertsanequalandoppositeforce−F(x0,t)onthepart
withx>x0,sothatacrosseachcrosssection,forcesarebalanced(seeFig.4.2).
However, the variation of these forces along the rod means that the net force
actingonasegmentofrodmaybenonzeroandinducesachangeinmomentum.
Inourformulationoftheequationofmotion,itisconvenienttoexpressthe
force distribution as a function of Lagrangian variable x rather than Eulerian
variable u, even though we think of the force acting in the physical domain
ratherthanthereferenceconfiguration.Infact,ifweweretolabelforcesinthe
physicaldomainasf(u,t),thenF(x,t)=f(u(x,t),t).Moreover,itisconvenientto
consider the stress σ, which is force per unit area, rather than force F. In the
presentcontextF=Aσ.
Since ut is the velocity of a point (i.e., cross section) in the rod, the
momentum density (meaning momentum per unit volume) is the quantity ρut.
Now consider a short segment of the rod a < x < b. The momentum of this
section is
. The balance law states that the rate of change of
momentumisequaltothenetforce:
Noticethatifut(x,t)isconstantinx,thenthisispreciselyNewton’slaw:
wheremassmeansthemassofthelittlesectionofrod.
AsinChapter2, we can now derive a PDE from the balance law by writing
bothsidesoftheequationasintegralsovera<x<b:
Thus,providedutt,σxarecontinuous,wehavethePDE
whichexpressesconservationofmomentum.
Tothisequationweaddaconstitutivelaw,anequationthatrelatesσtouina
different way. In elasticity, this constitutive law states that the stress σ is a
function of the strain. The strain is the deformation gradient; in the onedimensionalcontextoftherod,wehave
In engineering, it is common to define strain to be ux − 1, so that zero strain
correspondstonodeformation:u(x,t)=x.Inbothcases,elasticityisexpressed
byafunctionalrelationshipbetweenσandux:
SubstitutingintothePDE(4.1),weobtain
Asweobservedearlier,thisequationishyperbolicifσ′(ux)>0,inwhichcase,
stress increases with strain, but it is elliptic if σ′(ux) < 0. The hyperbolic case is
more significant, especially for small deformations (more precisely, for small
strains), but the elliptic case is also important for large deformations; it is
associatedwithaneffectcalledstrainsoftening.
PerhapsthemostimportantformoftheconstitutivelawisHooke’slaw,which
states that increases in stress are proportional to increases in strain. This is
expressedintheformula
Notethatthiscanalsobestatedasstressisproportionaltostrainifwedefinethe
straintobeux−1.
Substituting(4.2)into(4.1),weobtaintheone-dimensionalwaveequation
inwhich
.
RemarksonHooke’slaw.Theconstantk>0isaconstitutiveparametercalled
theelasticmodulusthatdependsontheelasticpropertiesofthematerial;itcanbe
measured in experiments. The same experiments assess the range of strains in
whichHooke’slawisreasonable.
Theparameterchasdimensionsofaspeed,thatis,L/T,whereLandTarea
typicalreferencelength(perhapsthelengthoftherod)andatypicaltimescale,
respectively. Correspondingly, density (mass per unit volume) has dimensions
M/L3, where M is the mass of the rod. It follows that k has dimensions LM/T2,
that is, the dimensions of mass × acceleration, the same dimensions as force.
Note that this is consistent with (4.2), since both u and x have dimensions of
length,sothatuxisdimensionless.
Hooke’s law is familiar from elementary mechanics or the study of ODE. It
arisesinrelatingtheextensionofaspringtothetensioninthespring.Toseethe
connection with the rod, consider a uniform deformation given by u(x) = Lx.
Thenσ(ux) = k(L − 1). But L − 1 is the extension (if L > 1); the stress σ is
constant and corresponds to the tension in the spring. Thus, the tension is
proportional to the extension. Indeed, just as for springs, the constant k in
Hooke’slawcanbefoundbyperformingsimpleextensionexperiments.
4.1.2.TheVibratingString
Consider a thin elastic string, such as a guitar string or bungee cord, which we
treatasaone-dimensionalcurve.Forsimplicity,weassumethatthestringmoves
onlyintwodimensions,andthatthetensioninthestringishighenoughthatwe
can ignore gravity. Another scenario with no effect of gravity would be an
experiment with a string constrained to a horizontal frictionless table. The
effectivelyone-dimensionalelasticbodyiscalledastringwhenweassumethatit
canbebentwithnoresultingforce.Thenwesaythereisnoresistancetobending.
Let’sconsiderthemotionofapointonthestringlabeledbyx∈[0,1].Ateach
timet,thispointwillbelocatedintheplaneat(r1,r2)=r(x,t)∈R2(seeFig.
4.3). Then the tangent to the string is rx(x, t). Since there is no resistance to
bending and no gravity, the only force on the string is due to the tension ,
which acts tangentially and is the only nonzero component of the stress. If the
stringhasauniformcross-sectionalareaAandconstantdensityρ(gm/cm3),then
the equations of motion (Newton’s second law, or conservation of momentum)
are
Figure4.3.Theelasticstring;one-dimensionalstringdeformingintwo
dimensions.
Now we make the constitutive assumption that the tension depends only on
thestrain|rx|andwrite
.Thus,thestringequationsare
Suitableboundaryconditions,inwhichthestringisfixedattwolocations,are
Withtheseboundaryconditions,thereisanequilibriumsolutionr0=(xL,0)in
which the string is stretched between the two fixed ends, as in a guitar string
before it is plucked or strummed. We assume that the tension at equilibrium is
positive:T(L)>0,andalsothatitisincreasingwithstrain:T′(L)>0.
Nowconsidersmalldeviations(u,v)fromtheequilibriumsolution,andwrite
r=(xL+u,v).Weaimtofindequationsforthenewvariablesu,vasfunctions
of x, t. Of course, we can simply substitute this expressi on into the string
equationsandgetexactequationsforu,v.However,wewanttotakeadvantage
ofthesmallnessofu,v.Todothis,wesubstituteintothePDEsystem(4.4)and
thenuseaTaylorexpansionabouttheequilibriumsolution,whichisnowu=v
=0.
Whenwesubstituteinto(4.4)andexpandeachtermasaTaylorseriesinu,v
retainingonlyconstantandfirst-orderterms(linearinu,v),wegetalotofterms.
Forexample,weneed
where h.o.t. represents the remaining higher-order terms in the Taylor series;
specifically,
.Similarly,
Thus,
Finally,theequation(4.4)becomesapairofwaveequationsifweretainonly
termslinearinuandv;thatis,droptheh.o.t.termsin(4.6):
In these linear wave equations,
is the longitudinal wave speed, and
is the transverse wave speed. It can be argued that the longitudinal
motion represented by u is smaller than the transverse motion if the string is
displacedlaterally.Thus,agoodapproximationistotaketheequationforvalone
andrepresentthestringsimplybythetransversedisplacementv(x,t),0<x<
1.
Thisisausefulwaytothinkofsolutionsofthewaveequation;foreachfixed
timetthegraphofv(x,t)representsthestring.Astimevaries,thegraphevolves
asastringinmotion.
4.2.D’Alembert’sSolution
In 1747, d’Alembert1 published a paper on vibrating strings that included his
famoussolutionofthewaveequationinonespacevariablexandtimet:
Thefirst,andfundamental,stepinderivingd’Alembert’ssolutionistoshowthat
thegeneralsolutionof(4.8)is
whereFandGarearbitraryC2functions.Thelinesx−ct=const.,x+ct=
const., where F(x − ct), G(x + ct) (respectively) are constant, are called
characteristics.
Sincec>0isconstant,wecanfactorthepartialdifferentialoperator
andwritethePDEas
Thensince(∂t+c∂x)F(x−ct)=0,and(∂t−c∂x)G(x+ct)=0,weseethat
(4.9)isasolution.
Itwillbeusefulwhendiscussingsolutionsofthewaveequationtointerpret
(4.9)asthesuperpositionoftwowaves:thegraphofF(x−ct)asafunctionofx
forvarioustimestisawavetravelingwithspeedctotheright,andG(x + ct)
representsawavemovingtotheleftwithspeedc.Thus,thewaveequation(4.8)
modelswavesofspeedcmovinginbothdirections,totheleftandright,justas
thelineartransportequationmodelswavesofspeedc>0movingtotheright
only. We can add the two waves in (4.9), because the PDE (4.8) is linear and
homogeneous.
To see that every solution can be represented in the form (4.9) for some
choice of functions F,G, we introduce characteristic variables suggested by the
factorizedequation(4.10):
andwritez(ξ,η)=u(x,t).Thenwehave
Addingandsubtractingasin(4.10),theequationbecomes
Integratingoverη,wefind
forsomefunctiong(whichisconstantwithrespecttoη).
Thus, z(ξ, η) = G(ξ) + F(η), where G(ξ) = ∫ g(ξ) dξ, and F is another
arbitraryfunction.Backintheoriginalvariables,wearriveat
4.2.1.InitialValueProblem(CauchyProblem)
We use the general solution (4.9) of the wave equation to solve the Cauchy
problem
JustasforODE,sincethePDEissecondorderint,tohaveawell-posedproblem
(cf.Section2.1)wehavetospecifyboththeinitialdisplacementuandtheinitial
velocityut.
Theorem4.1.IfϕisC2andψisC1,thentheuniqueC2solutionof(4.11)isgivenby
Proof.ThegeneralsolutionofthePDEis
Thentheinitialconditionsgive
Integratingthesecondequationyields
.
NowwecansolveforFandG,leadingto(4.12).Weleaveitasanexerciseto
verifythattheinitialconditionsaresatisfied.
It is clear from (4.12) that u(x, t) is a C2 function. Uniqueness is a
consequenceofthefactthatFandG in the general solution are determined by
theinitialcondition.
Formula(4.12)isknownasd’Alembert’ssolution.Notethatsinceψspecifiesut
at t = 0, it is consistent that it should be integrated in a formula for u. In
integratingψ,wegainaderivative.Thus,wehavethefollowingregularityofthe
solution:
Inotherwords,thesolutioninheritsregularityfromtheinitialdata.Thereisno
gain or loss of regularity, a property typical of hyperbolic PDE. In fact, (4.12)
makes sense even if ϕ or ψ are less regular than in the theorem; the
correspondingfunctionu(x,t)isthenknownasaweaksolution,eventhoughthe
derivativesofthesolutionseeminglyrequiredbythePDEmaynotexist.
D’Alembert’s solution allows us to establish another part of well-posedness,
namely,continuousdependenceonthedata.
Proofofcontinuousdependence.Letu=u1,u=u2 be solutions of problem
(4.11)withinitialdataϕk,ψk,k=1,2,thatareboundedanduniformlyclosein
thesenseofcontinuousfunctions:
whereϵ>0issmall.From(4.12),wehave
Inthiscalculation,wehaveusedthetriangleinequalityandtheintegralestimate
|∫f(x)dx|≤∫|f(x)|dx.
Itfollowsthatif|ϕ1−ϕ2|and|ψ1−ψ2|areuniformlysmall,then|u1(x,t)−
u2(x, t)| is small at each x, t < ∞. Note that the estimate gets worse with
increasingtime,sothattomakeu1−u2uniformlysmallinxandt,wehaveto
takeafinitetimeinterval,unlessweinclude
inthesmallness
condition.
Figure4.4.Initialdisplacementϕ(x).
Example 1. (Representing d’Alembert’s solution graphically) For simplicity,
let’s take the initial velocity to be zero, ψ(x) ≡ 0, choose c = 2, and take the
supportofϕtobetheinterval[1,3].Thesupportofafunctionϕisdefinedtobe
theclosure(includingboundarypoints)ofthesetwhereϕisnonzero.Thus,supp
.
Thesolutionof(4.11)inthisexampleis
Theinitialdataarepiecewiselinear,withchangesinslopeatx=1,2,3(Fig.
4.4). Correspondingly, the solution (at a fixed t) will be piecewise linear, with
changesinslopeexpectedatvaluesofxforwhich
Thegraphofϕ(x+2t)asafunctionofxhastheshapeofatrianglemovingto
theleftwithspeed2,andϕ(x−2t)hasthesameshapeandspeed,butmoving
totheright.Thesolution(4.14)istheaverageofthesetwographs.
We can represent the solution in the x-t plane, as shown in Figure4.5. The
leadingedgeoftheleft-movingtriangularwaveliesonthecharacteristicx+2t
=1;itgetstox=0when
.Startingatx=1,andmovingleftwithspeed2,
thewavetakesuntil
toreach0.
Domain of dependence and region of influence. The structure of the
characteristics in the x-t plane in Figure 4.5 suggests how the initial data
propagateandinfluencethesolution.Likewise,wecanconsiderthedependence
ontheinitialdataofthesolutionatapoint(x,t).
Thebackwardcharacteristicsthroughapointapoint(x0,t0)witht0>0arethe
lines
Thebackwardcharacteristicsintersectthex-axisatx=x0±ct0;thesolutionof
the Cauchy problem at (x0, t0) depends only on the initial data in the interval
between these points. The interval is referred to as the intervalofdependence of
thepoint(x0,t0).Morecommonterminologyistorefertothetrianglewithbase
given by the interval of dependence and sides given by the backward
characteristicsasthedomainofdependenceofthepoint(x0,t0).
Figure4.5.Thex-tplaneforExample1.Notethattheslopeofthecharacteristics
isthereciprocalofthewavespeed.
Figure4.6.(a)Domainofdependence(x0,t0)and(b)regionofinfluenceof[a,
b]forthewaveequation.
The region of influence of a point (x0, t0) is the set bounded by the forward
characteristics:
Wecanalsospeakoftheregionofinfluenceofasubsetofthex-tplane,butmore
commonlywerefertotheregionofinfluenceofaninitialintervala≤x0≤b,
witht=0.Theregionofinfluenceoftheinterval[a,b]onthex-axisistheset
boundedbycharacteristicsx+ct=a,x−ct=b.
These notions give us a graphical means of understanding how initial data
propagateforwardintime,asshowninFigure4.6.Inparticular,ifthedatahave
compact (i.e., bounded) support in an interval [a, b], then the solution is
necessarily zero outside the region of influence of the initial interval [a, b], as
can be seen by drawing backward characteristics from any point outside this
regionofinfluence.Wesaythatinitialdisturbances(meaningwhereϕorψare
nonzero)propagatewithfinitespeedc.
4.2.2.TheWaveEquationonaSemi-InfiniteDomain
TheCauchyproblemshowshowinitialdisturbancespropagateaswavesinfree
space. To describe how these waves are reflected at a boundary, we formulate
andsolveaninitialboundaryvalueproblemonthequarter-plane{(x,t):x>0,
t>0}withasingleboundary.
Considertheinitialboundaryvalueproblem
Theboundaryconditionspecifyingu(0,t)meansthattheendofthestringisheld
inplace.Wecouldinsteadspecifytheslopeux(0,t),whichwouldmeanthestress
isspecified.Inparticular,theboundaryconditionux(0,t)=0isreferredtoasa
stress-freeboundarycondition.
Forx>ct,wehaved’Alembert’ssolution(4.12)
withϕ,ψevaluatedonlyforpositivevaluesoftheirarguments.
Toobtainanexpressionforthesolutionintheregion0<x<ct,wehaveto
usetheboundarycondition,since(4.16)doesnotapplyforx−ct<0.Notice
that characteristics with positive speed c propagate into the domain from the
boundary x = 0 as t increases (see Fig. 4.7). These characteristics carry
information from the boundary condition. Moreover, each point in the quarterplane also has a characteristic moving left with speed c that originated on the
initiallinex>0,t=0.Therefore,thesolutionforx<ctwillinvolveboththe
initialconditionandtheboundarycondition.
The solution can be found from the general solution (4.13), obtaining
expressionsforthefunctionsF,Gfromtheinitialandboundaryconditions,much
aswasdoneintheproofofTheorem4.1.Theresultofthiscalculationis
Observethatthisformulasatisfiestheboundaryconditionu(0,t)=0.Formula
(4.17)canbeinterpretedasthesolutionoftheCauchyproblemwithinitialdata
ontheentirereallineobtainedbyextendingbothϕandψtobeoddfunctions,so
thatϕ(−x)=−ϕ(x),ψ(−x)=−ψ(x),x>0.Then(4.17)usestheoddnessof
the extension to express the solution entirely in terms of the given data on the
positivex-axis.
Moreover,thereisanotherinterpretationofthisconstruction.Thesolutionof
theCauchyproblemwithoddinitialdatainvolveswavesmovingleftandrightin
theupperhalf-plane.Thosewithx<0andmovingrighthavethepropertythat
along the line x = 0 (the t-axis), they exactly cancel waves moving left. This
cancellationexplainshowtheboundaryconditionu(0,t)=0issatisfied.
Figure4.7.Characteristicsforthequarter-planeproblem.
Example2.(Representingaquarter-planesolutiongraphically)Let’ssuppose
suppϕ⊂[a,b]andsuppψ⊂[a,b].Thesolutionisrepresentedinthex-tplanein
Figure4.7, where we have drawn forward characteristics from the boundary of
thesupportoftheinitialdata,includingtheirreflectionsfromtheboundaryx=
0. The reflected characteristics record the switch from x − ct to ct − x in the
solution;equally,wecanthinkofthereflectedcharacteristicsasoriginatingfrom
thex-axis and carrying information from the extended initial data. Finally, the
reflected characteristics carry information from the boundary (specifically, the
boundary condition) into the interior of the domain. This last point of view is
helpful when considering nonzero boundary data. In Figure 4.7 the domain is
divided into sectors in which we can write the solution in more detail. For
example,thesolutioniszerointhreeoftheregions.Ineachoftheotherregions,
we use backward characteristics to see which part of the support of the initial
dataisusedincalculatingthesolution.
For (x, t) in the triangular region labeled Δ in Figure 4.7, the backward
characteristics hit the x-axis at x − ct and x + ct, both of which lie in the
interval [a, b] Thus, there is no simplification, and u is given by d’Alembert’s
formula(4.16).
InregionI,x−ct<a,andct−x<a,whilea<x+ct<b,sothatboth
(4.16)and(4.17)reduceto
whichisawavetravelingtoleft,afunctionofx+ct.
In region II, u(x, t) is likewise a function of x − ct; the graph is a wave
travelingtotheright.
InregionIII,partofthewaveisreflectedbytheboundaryandinteractswith
thewavetravelingtowardtheboundary.Infactthereisjustenoughcancellation
sothattheboundaryconditionissatisfiedatx=0.Thereflectedwaveemerges
as a function of (x−t) in region IV. Thus in region III, where both waves are
present,anda<ct−x<x+ct<b,u(x,t)isgivenbythefullformula(4.17).
In region IV, after the left-moving wave has been fully reflected and is now
movingtotheright,wehavex+ct>b,andthus
InregionV,x−ctandct−xarelessthana,whilex+ct>b.(Thiscaseis
also shown in Fig. 4.7.) Therefore, there is no contribution from the initial
displacementϕ,andthecontributionfromtheinitialvelocityisconstant:
Similarly,intheregionbetweenIVandthet-axis,wefindu≡0.Weseethisby
notingthatbothx+ctandct−xarelargerthanb.Equivalently,sincex−ct
<−b<b<x+ct,wefindthatuistheintegraloftheoddextensionofψfrom
−btobandhenceiszero.
Itisinstructivetographthesolutioncarefullyforvariousvaluesoft,say,for
Example2,withϕtriangular(seeFig.4.4)andψzero.
Example3.(Nonzeroboundarycondition)Considertheinitialboundaryvalue
problemwithnonzeroboundarycondition
The solution is similar to (4.16), except there is an additional term that
propagates the boundary data into the domain x > 0. To derive the additional
term,firstobservethatifϕandψarezero,thenthestringisinitiallyhorizontal
andatrest.Thedisplacementh(t),specifiedattheboundary,propagatesintothe
interior, and induces motion of the string. Consequently, this disturbance
propagates as a wave u(x, t) = F(x − ct) with speed c. Setting x = 0 in this
solution,wematchtheboundarycondition:F(−ct)=h(t).Consequently,F(ξ)=
h(−ξ/c),forξ<0,andthesolution
follows immediately. The full solution is obtained by simply superimposing the
solution with h ≡ 0. Note that right-moving characteristics carry only half the
information of the solution for the Cauchy problem, but they carry all the
boundary information, since the boundary data travel only to the right in the
physicaldomainx>0.
4.3.TheEnergyE(t)andUniquenessofSolutions
In this section we define an energy function for the wave equation, show that
energy is conserved for the Cauchy problem (4.11), and use this property to
establishuniquenessofsolutionsoftheCauchyproblem.
Let’sassumethatu=u(x,t)isasmoothsolutionoftheCauchyproblemand
thederivativesut(x,t),ux(x,t)aresquareintegrable(i.e.,inL2(R))foreacht≥0.
Thenthetotalenergydefinedby
isfinite.2
NotethatE(t)isthesumofthekineticandpotentialenergies.Thepotential
energy
istheenergystoredinthestringduetotension,andthe
kinetic energy
is akin to the quantity
in classical
mechanicsofarigidbodywithmassmandvelocityv.
ToseehowE(t)isconnectedtotheone-dimensionalwaveequation(4.8),we
multiplythePDEbyutandintegratebyparts:
Thus,wehave
Thatis,
Therefore,wehaveconservationoftotalenergy:E(t)=constant,fromwhichwe
deduce
This identity is an important tool for existence and regularity of solutions, but
alsoforuniquenessofsolutions,aswenowdiscuss.
Uniquenessofsolutions.ConsidertheCauchyproblem
in which the inhomogeneity f(x, t) is a specified function representing a timedependent distribution of force along the one-dimensional elastic body. For
example, if the PDE represents small transverse vertical vibrations of an elastic
string,thenf(x,t)=−gcouldbetheforcedistributionduetogravity.(Notethat
thedensityρhasbeenabsorbedintoc2.)
We can use the energy calculation to prove uniqueness of C2 solutions of
(4.19). Consider two C2 solutions u1, u2 with the same data ϕ, ψ, f. To prove
uniqueness,weshowu1=u2.Defineu(x,t)=u1(x,t)−u2(x,t).Thenusatisfies
thehomogeneousversionof(4.19),withzeroinitialdata:
SinceE(t)=E(0)=0forthisproblem,wehave
Therefore,
Thus,uisconstantinxandt.Butu(x,0)=0,sotheconstantiszero.Henceu=
u1−u2≡0.
4.4.Duhamel’sPrinciplefortheInhomogeneousWaveEquation
Duhamel’s principle is used to solve inhomogeneous initial boundary value
problems when we have the solution of the homogeneous problem in hand.
Considertheinitialvalueproblem
Tosolve(4.20)usingDuhamel’sprinciple,let
>0)of
bethesolution(foreachs
Tounderstandwhy mightbehelpful,considerthespecialcaseinwhichf(x,t)
=F(t)isindependentofx,andweseekasolutionv(t)independentofx.Thenwe
have
Integratingtwiceandreversingtheorderofintegration,weseethat
But for each s > 0, w(s,t) = (t − s) F(s) solves the x-independent version of
(4.21),namely,
and
.
This calculation suggests that
solves (4.20). To complete
the solution, it remains to find . But satisfies a Cauchy problem for the
homogenous wave equation with initial condition at time t = s. We can adapt
d’Alembert’ssolutionbytranslatingtbysind’Alembert’sformula,withϕ(x)=
0;ψ(x)=f(x,s).Thisgivesaformulafor :
Nowlet
.Thatis,
The double integral is an integration of f over the triangular domain of
dependenceof(x,t)showninFigure4.6.
Claim4.2.Thefunctionu(x,t)satisfies(4.20).
Proof.Let(x0, t0) be fixed with t0 > 0. The proof involves integrating the PDE
(4.20)overthedomainofdependenceΔ={(x,t):x0−c(t−t0)<x<x0+
c(t−t0),0<t<t0}andusingGreen’stheoremintheplane(seeAppendixA).
With the exact differential du = uxdx + utdt, the integral on the boundary is
reducedtou(x0,t0).
It is straightforward to use this procedure to solve the more general initial
valueprobleminwhichtheinitialdatacanbenonzero:
Wesolvethisproblembyconsideringeachoff,ϕ,ψseparately,settingtheothers
tozero;thesolutionisthenthesumofthecorrespondingsolutions:
4.5.TheWaveEquationonR2andR3
Themethodofsphericalmeansusestherotationalandtranslationalinvarianceof
the wave equation to find solutions that are the analog in Rn, n ≥ 2, of
d’Alembert’s solution in one dimension. The resulting formulas allow us to
understand the Huygens principle of wave propagation. Huygens originally
expressed his principle geometrically, using spheres centered at points of a
wavefront with radius given by an incremental time and arguing that the
intensities would cancel except along the expanding surface formed as the
envelopeoftheoverlappingspheres.
Here we show briefly how to derive an explicit formula for the solution of
initialvalueproblemsinR2andR3.ItisinfacteasiertostartwithR3.Suppose
u(x,t)isasolutionofthewaveequation
withCauchydatau(x,0)=ϕ(x),ut(x,0)=ψ(x).Forr>0,definethespherical
means
,whereS(x,r)denotesthespherewithcenter
xandradiusr.(See AppendixA for the integral average notation .) If we can
findv(r,t),thenwerecoveru(0,t)=limr →0v(r,t)Butthiswillgiveaformula
foru(x,t)foranyx,t,bycenteringthespheresatageneralpointx∈R3instead
ofatx=0.Herearethemainstepsinconstructingtheformula.
1.Observethatv(r,t)isarotationallyinvariantsolutionofthewaveequation
withinitialdatagivenbytheaveragesofthedatafor
,
.
Noticethatv(r,t)issymmetricabouttheorigin,butwecouldequallywellhave
centered the spheres at any point x in space, taking integral averages over the
resultingspheresS(x,r),leadingtothesamestatementoftheCauchyproblemfor
v,butwithdifferent
.
2.Letw(r,t)=rv(r,t).Thenwsatisfiestheone-dimensionalwaveequation
withinitialdata
,andboundaryconditionw(0,t)=0.
Consequently, d’Alembert’s solution can be used to solve this quarter-plane
problem, effectively using the even extensions of
or equivalently, the
oddextensionsof
.
3.Sinceweareinterestedonlyinu(0,t)=limr→0v(r,t)=limr→0w(r,t)/r,we
writethesolutionwithr<ct:
Noticethatwewritethefinalϕtermasthederivativeofanintegral,which
turnsouttobemoreconvenientthantheequivalentformusedinSection
4.2.1.
4.Nowwecomputethelimitasr→0:
sincew(0,t)=0.Computingthisderivativeusing(4.22)andincludingthe
formulasfortheintegralaverages(seeAppendixA),wefind
5.Finally,weobservethatbytranslationinvariance,thecorrespondingformula
appliesbycenteringthespheresatanypointx∈R3:
Solutionintheplane(n=2).Theformula(4.23)canbeadaptedtofindthe
solution of the wave equation in two space dimensions. The idea is to consider
solutionsv(x1,x2,t)=u(x1,x2,x3,t)thatareindependentofx3Then vsatisfies
thewaveequationintwodimensionsandinthreedimensions.Sinceinitialdata
ϕ, ψ are in two dimensions, we consider them as functions of x1, x2, x3 but
independentofx3.Thenwecalculatetheintegralsin(4.23)byrepresentingthe
spheresovertheprojectionsontothex1−x2plane,whicharedisks.Thisprocess
givestheformula
Everythinghereistobeinterpretedintwodimensions.Thus,x=(x1,x2),y=
(y1,y2),B(x,ct)isthetwo-dimensionaldiskcenteredatx,andwithradiusct,and
dy=dy1dy2.
Because of the way (4.24) in two dimensions is related to (4.23) in three
dimensions, it is not surprising that the integrals are over the disks B(x, ct).
However,thishasaprofoundconsequence,becausenowthesolutiondependson
values of ϕ and ψ inside the disk B(x,ct), in contrast to the three-dimensional
case, in which the dependence is only on values of the data on the expanding
sphereS(x,ct).Thus,theHuygensprincipledoesnotholdintwodimensions.For
example,wavesgeneratedbydroppingastoneintotheflatsurfaceofabodyof
water generates not just a circular expanding wave, but also lots of concentric
ripplesbehindtheleadingwave.
PROBLEMS
1.Considertheinitialvalueproblem
Letϕ(x)bethefunctionwithgraphshowninFigure4.4,andψ(x)≡0.Inthex-t
planerepresentationofthesolutioninFigure4.5,wefind u≡0inthemiddle
section,with
. Show that if we keep the same ϕbutmake ψ nonzero, with
supp ψ = [1, 3], then u will still be constant in this middle section. Find a
conditiononψthatisnecessaryandsufficienttomakethisconstantzero.
2.ConsiderC3solutionsofthewaveequation
For c = 1, define the energy density
momentumdensity).
, and let p = utux (the
(a)Showthatet=px,pt=ex.
(b)Concludethatbotheandpsatisfythewaveequation.
3.Supposeu(x,t)satisfiesthewaveequation(4.25).Showthat:
(a)Foreachy∈R,thefunctionu(x−y,t)alsosatisfies(4.25).
(b)Bothuxandutsatisfy(4.25).
(c) For any a > 0, the function u(ax, at) satisfies (4.25). Note that the
restrictiona>0isnotnecessary.
4.(a)Letu(x,t)beasolutionofthewaveequation(4.25)withc=1,validfor
allx,t.Provethatforallx,t,h,k,
(b)Writeacorrespondingidentityifusatisfies(4.25)withc=2.
5.Considerthequarter-planeproblem
Let ϕ(x) be the function with graph shown in Figure 4.4, and let ψ(x) ≡ 0.
Sketchthesolutionu(x,t)asafunctionofxfor
.
6.Considerthequarter-planeproblemwithahomogeneousNeumannboundary
condition
Supposesuppϕ=[1,2]=suppψ.
(a)Solveforu(x,t),x≥0,t>0.
(b)Wherecanyouguaranteeu=0inthefirstquadrantofthex-tplane?
(c)Considerϕ≡0;writeaformulaforu.
(d)If0isinthesupportofϕorψ(e.g.,iflimx→0+ϕ(x)≠0),writeconditions
thatguaranteeuis(a)continuousand(b)C1.Explainyouranswersintermsof
the behavior of the data around the boundary of the domain. (Any
compatibilityconditionwillbeeffectivelyattheorigin,butyouwillneedto
matchu,ux,andutacrossx=t.)
7.Considerproblem6,butinthemoregeneralcaseinwhichux(0,t)= h(t)is
not identically zero. Here, the general solution can be employed with the
boundaryconditiontofindF(ξ)for ξ<0intermsof G(−ξ)andh.Usingthis
approach,derivethesolution
forx<t.Deriveasuitablecompatibilityconditionattheoriginthatensuresthe
solution is continuous when the data are continuous. What about the first
derivativesacrossx=t?
8.Considerthewaveequationthatincludesfrictionaldamping:
inwhichμ>0isadampingconstant.Showthatifu(x,t)isaC2solutionwithux
→ 0 as x → ∞, then the total energy
is a decreasing
function.
Incidentally, can you devise a C2 function f(x) with the property f(x)
approachesaconstantasx→±∞,butf′(x)doesnotapproachzero?
9.Considerthequarter-planeproblem(4.15).
(a) Formulate the mechanical energy E(t) for solutions, and show that it is
conserved.Specifyanyassumptionsyouneedontheinitialdata.
(b)Forthenonzeroboundaryconditions(4.18)ofExample3evaluateE′(t)in
termsofthedataϕ,ψ,h.
10. Let f(x, t) be a continuous function, and let Δ(x, t) denote the domain of
dependence of the point (x, t) for (4.25). Use the Fundamental Theorem of
Calculustoshowdirectlythat
satisfies
11. Consider the wave equation in three dimensions, with initial conditions in
whichϕ(x)=f(|x|)isrotationallysymmetric,thefunctionfsatisfiesf(r)=0,r
≥ϵ,andψ≡0.Showthatthesolutionu(x,t)is(a)rotationallysymmetric,and
(b)zerooutsideacircularstripcenteredattheoriginandhavingwidthϵ.
1.JeanleRondd’Alembert(1717–1783).
2.Sincetheconstantdensityρhasbeenabsorbedintoc2,thephysicalenergyisactuallyρE(t).
CHAPTERFIVE
TheHeatEquation
Theheatequation
istheprototypeofparabolicPDEandmodelsdiffusionprocesses,includingheat
flow and the spread of a solute in a fluid. The heat equation also plays a
significantroleinmodelsofcombustion,fluidflowwithtemperaturedependence
(forexample,whendensitydependsontemperature),andpopulationdynamics.
Inchemicalandbiologicalsystems,diffusiveprocessesarecommonlymodeledby
random walks and Brownian motion, which are closely related to the heat
equation.
The heat equation has the remarkable property that even for rough initial
data, solutions are immediately smoothed. This property is in contrast to
hyperbolic equations, such as the wave equation, for which rough initial data
remains just as rough through its evolution. Solutions of the heat equation also
exhibitinfinitepropagationspeed,meaningthatachangeintemperatureinone
locationisimmediatelydetectedeverywhere.(Buttheeffectdecaysexponentially
with distance from the source of the change.) Since characteristics are defined
onlyforhyperbolicequations,themethodofcharacteristicsdoesnotapplytothe
heat equation. Instead, we introduce several new PDE techniques that are
applicableingeneraltolinearPDE.
The fundamental solution lies at the heart of the theory of infinite domain
problems. On bounded domains, the fundamental solution is adapted to take
account of boundary conditions. The adapted functions are called Green’s
functions.
The maximum principle applies to the heat equation on domains bounded in
spaceandtime.Thisimportantpropertyofparabolicequationsisusedtoprovea
varietyofresults,suchasuniquenessofsolutionsandcomparisonprinciples.
The energy method for the heat equation has a few key differences from the
methodforthewaveequation.Forexample,thephysicalheatenergyisnotvery
useful,andinsteadweintroduceamathematicalenergyfunction.Typically,this
energy is not conserved and decays in time. The decay of the energy leads to
straightforward uniqueness results, just as for the wave equation. The energy
decay is also useful for obtaining estimates that are part of the existence and
regularitytheoryforsolutionsofparabolicequations.
Separationofvariablesisaprocedureforsolvingcertaininitialboundaryvalue
problems.Theprocedureisstraightforwardfortheheatequation,andwithsmall
modifications it also applies to the wave equation and Laplace’s equation. The
methodyieldssolutionsrepresentedasinfiniteseriesofeigenfunctionsassociated
with the PDE and boundary conditions. We first demonstrate the technique on
specific examples. More generally, in the next chapter we analyze eigenvalue
problemsforODEandPDE,andtheconvergenceofFourierseries.
5.1.TheFundamentalSolution
Thefundamentalsolutionisimportantforproblemsoninfinitedomains.Inthis
section we define the fundamental solution Φ(x,t) and show how it is used to
solvetheCauchyproblem:
Toderivethefundamentalsolution,weusethescaleinvariancepropertyofthe
heat equation. Let a > 0 be a constant and introduce the change of variables
. Then the heat equation is unchanged but is expressed in the new
variables:
Thisscaleinvariancesuggeststhatweseeksolutionsoftheself-similarform
for some α ∈ R. Substituting into the heat equation, we see that whatever the
valueofα,thefunctionvsatisfiesanODEwithnonconstantcoefficients:
Toselectα,weintroducethepropertyofconservationofheatenergy,which
we wish to have satisfied by our solution. Suppose u is a solution of the heat
equationwiththepropertythat
,andux(x,t)→0asx→±∞.
Then,integratingthePDE,wefind
sothatthetotalheatenergyisconserved:
However,
whichsuggestsweshouldscalethefunctionvby
thatis,choose
:
.Withthisscaling,heatisconservedinthesenseof(5.2).
Now we solve (5.1). Since it is a second-order equation, there will be two
independentsolutions.FirstrewritetheODEas
Thus,
Sincewearereallyonlyseekingonesolution,itisconvenienttosettheconstant
tozero,resultinginahomogeneousfirst-orderequationthathasgeneralsolution
Convertingbackto(x,t),with
,weobtainthesimilaritysolution
Usually,wechooseaparticularvalueofAsothattheconstantin(5.2)isone;
forthischoiceofconstant,wehavethefundamentalsolutionoftheheatequation:
(see Fig. 5.1). In higher dimensions, x ∈ Rn, the fundamental solution takes a
similarform:
Thenu(x,t)=Φ(x,t)satisfiesut=kΔu.
PropertiesofthefundamentalsolutionΦ(x,t).
1.Φ(x,t)>0forallx∈Rn,t>0.
2.ΦisC∞in(x,t),t>0.
3.∫R Φ(x,t)dx=1forallt>0.
n
Figure5.1.GraphofthefundamentalsolutionΦ(x,t)fortheheatequation,with
t>0.
Properties1and2followdirectlyfromtheformulaforthefundamentalsolution.
Forn=1,property3isverifiedbydirectcalculation:
Forn>1,property3followsfrom
.
Fromproperties1and3,weseethatΦ(x,t)isaprobabilitydistribution.In
factΦisanormaldistributionforeacht>0withinterestingdependenceontin
thelimitst→∞andt→0.Theareaunderthegraphis1forallt>0,yetast
→∞,maxxΦ(x,t)→0;thetailspreadsouttomaintain∫Φ=1.Ast→0the
maximum(atx=0)blowsuplike ,buttheintegralremainsconstant.Wealso
observeΦ(x,t)→0forx≠0,ast→0+.
5.2.TheCauchyProblemfortheHeatEquation
WearenowreadytosolvetheCauchyproblem
usingthefundamentalsolution
whichsatisfies(5.5a)fort>0.
Bytranslationinvariance,Φ(x−y,t)isasolutionof(5.5a)forally.Thus,
isalsoasolutionof(5.5a).
BylinearityandhomogeneityofthePDE,wecantakelinearcombinationsof
solutions,whichsuggeststhat
shouldbeasolution.Moreover,propertiesofΦsuggestthatast→0+,u(x,t)→
g(x),sinceΦ(x−y,t)collapsestozeroawayfromy=xandblowsupaty=x
whilepreserving∫Φ=1.
BecauseoftheexponentialinΦ(x,t),theintegralsforu,ut,uxx all converge
fort>0providedg∈C(R)isbounded.Then
sothatusatisfiesthePDEfort>0.
Itismorecomplicatedtoproverigorouslythattheinitialcondition(5.5b)is
satisfied,sincet=0isasingularpointforΦ(inthatΦ(x,t)isnotdefinedatt=
0).Togetaroughideaofhow(5.5b)holdsasalimitast→0+,let’sfixx.Then,
forδ>0,
The second line has two integrals. In the first integral, g(y) ≈ g(x) for small δ
sincegiscontinuous.Thesecondintegralapproacheszeroast→0+,becauseΦ
→0uniformlyandexponentiallyawayfromy=xast→0+.Inthefollowing
theoremwestatethislimitcarefullyandproveitbyestimatingtheintegrals.The
finalintegralgivesg(x),sinceitisindependentofy,andtheintegralofΦovera
smallintervalaroundy=xbecomesunityast→0+,sinceΦapproacheszero
sufficientlyfastelsewhere.
Theorem5.1.Letg∈C(R)bebounded,andletu(x,t)begivenby(5.6).Then
1.uisC∞in(x,t)fort>0;and
2.usatisfiestheheatequationut=kuxx,x∈R,t>0;and
3
forallx0∈R.
Proof.Property1followsbecauseΦisC∞fort>0,andsincederivativesofΦ
all decay exponentially as |x| → ∞, the integrals converge. Property 2 follows
fromΦt=kΦxx,t>0.
Toproveproperty3,weconsiderthedifference|u(x,t)−g(x0)|,andestimate
the integrals, guided by the discussion above. This is the first time we have
encounteredthesekindsofestimates,soweprovidethedetails.
Letϵ > 0. We wish to show |u(x,t)−g(x0)|<ϵfor(x,t) close to (x0, 0).
Since∫Φdx=1,wecanexpressthenumberg(x0)asanintegral,sothat
Let δ > 0 (we choose δ below), break up the integrals in (5.7), and use the
triangleinequality:
Nowweuseδintwowaystoshowthetwointegralsaresmall.
Inthefirstintegral,bycontinuityofgwecanchooseδ>0smallenoughthat
|g(y)−g(x0)|<ϵfor|x0−y|<δ,whichisthedomainoftheintegral.Weare
left with an integral of Φ(x − y, t), which is bounded uniformly by 1, even
thoughΦ(x−y,t)blowsupast→0.
Inthesecondintegralweobservethat
,provided
, while g(y) is bounded. To make this observation more precise, we
writetheright-handsideof(5.8)asIδ+Jδ.Chooseδ>0sothat|g(y)−g(x0)|
<ϵfor|y−x0|<δ.Then
TheintegralJδissomewhattrickier.Sincegisbounded,thereisK>0suchthat
|g(y)|≤K,forally.Thus,
Considerxsatisfying
.(Recallthatweareconsideringthelimitasx→
x0.)Then
intherangeofintegration|x0−y|>δ.However,thisisnot
agoodenoughestimateoftheexponential,becausewewouldstillbeleftwithan
integraloveraninfiniteintervalofasmallbutpositivequantity.Togetamore
useful estimate, we observe (see Problem 1) that in the region of integration,
.Then
for t > 0 sufficiently small. Here we have used the change of variables z =
andtheconstant
.Nowitfollowsthat|u(x,t)−g(x0)|<
2ϵfor
sufficientlysmall.Thisprovesproperty3andcompletesthe
proofofthetheorem.
The solution (5.6) is a convolution (Φ(·, t) ∗ g)(x) for each t > 0. For
integrablefunctionsϕ,ψonRtheconvolutionproductofϕandψisafunctionϕ
∗ψdefinedby
The convolution product provides a useful construction of the product of two
integrable functions. It is used widely in signal processing and Fourier analysis.
Property3ofthetheoremshowsthatΦ(·,t)∗g→gast→0.Inthissense,the
functionsΦ(·,t)convergeast→0toageneralizedfunctionδdefinedinformally
by
WemakethispreciseinChapter9,whereδisdefinedastheDiracdeltafunction,a
distribution.
Recall that for first-order equations and for the wave equation, solutions
propagatewithfinitespeed,meaningthattheregionofinfluenceofanypointis
boundedoverfinitetime.However,theheatequationhastheproperty,typicalof
parabolic equations, that solutions propagate with infinite speed: the region of
influenceofapointisimmediatelyallofspace.
To make this concept a bit more precise, consider an initial temperature
distribution u(x, 0) = g(x) that is nonnegative and has compact support (see
Appendix A). An example is the function g(x) = ϕ(x) in Figure 4.4 that we
considered for the wave equation. For the wave equation, such an initial
disturbance propagates with finite speed, the wave speed. But for the heat
equation, the solution u(x,t) is immediately positive everywhere, meaning that
forallt > 0, no matter how small, u(x,t) > 0 for all x∈R. You can see this
directlyfromthesolution(5.6).Thus,theinitialdata,confinedtoaboundedset,
has induced a positive temperature everywhere immediately. Of course, the
temperature drops off exponentially with distance from the support of g and is
tiny outside the support for small t, but nonetheless, the initial temperature
distributionhaspropagatedwithinfinitespeed.
Thisisprettyclearlynotphysical(wecannothavesignalspropagatingfaster
thanthespeedoflight!)andisalimitationoftheheatequation,andinparticular
of Fourier’s law of heat transfer. From a different point of view, it is a
consequence of taking a finite speed effect, namely, the exchange of thermal
energy between molecules, and describing the process in a continuum in a
seemingly natural way that introduces this anomaly. Nonetheless, the heat
equation is an extremely useful PDE that describes heat transfer accurately in
manysituations.
5.2.1.UsingtheFundamentalSolutiontoSolveQuarter-PlaneProblems
Tointroduceaboundarycondition,considerthequarter-planeproblem
As in the strategy for the wave equation, we reflect the initial data, so the
solutionsatisfiestheboundarycondition.Let betheoddextensionofg(x):
anddefine
Then
sinceΦ(y,t)isanevenfunctionofy,and isanoddfunction.Notethatu(x,t)is
an odd function of x ∈ R. That is, the symmetry in the initial data is carried
throughtothesamesymmetryinthesolution.
Nowreplace
withg(y)using
fory<0.Then
For a quarter-plane problem with a homogeneous Neumann boundary
condition, corresponding to an insulated end at x = 0, the calculation involves
theevenextensionoftheinitialdata:
The solution is obtained by extending g using the even extension , so that the
firstderivativeiszeroatx=0.Then
satisfies the heat equation. To check the boundary condition, we calculate
. Now Φ(y,t) is even in y, so Φx(y,t) is odd.
Thus,ux(0,t)=0,asrequired.
5.3.TheEnergyMethod
In this section we show the sense in which heat energy is conserved on both
bounded and unbounded domains. The function H(t) = ∫ u(x, t) dx is
proportional to the physical heat energy when u(x, t) is the temperature
distribution.WealsoconsideramathematicalenergyintegralE(t)=∫u2(x,t)dx,
whichisusefulfortheheatequation,justasthemechanicalenergywasforthe
waveequation.Thetreatmentofenergyintegralsisverysignificantforthestudy
ofPDE,especiallynonlinearPDE,whereestimatesofmathematicalenergieshelp
establishexistenceanduniqueness.
ConsideraC2solutionu(x,t)oftheheatequationonaboundedinterval:
Theheatenergy
satisfies
Thus, the evolution of heat energy is controlled by the heat flux −kuxthrough
the ends x = a, b. If the ends are insulated, then ux = 0 there, and the heat
energyisconstant.
Onanunboundeddomain,thecalculationworkssimilarly,exceptitisnatural
to assume there is no heat loss at infinity and that the temperature u(x, t) is
integrable as well as smooth. Suppose ut = kuxx, −∞ < x < ∞, t > 0. Let
.Then
Toanalyzethemathematicalenergy
uandintegrateover[a,b]:
,wemultiply(5.11)by
Therefore,
Thus,theenergyintegral
isdecreasingintimetif
.For
example,ifeitheruoruxiszeroateachendpointx=a,x=b,thentheenergy
integralE(t)decreasesint.Inthiscase,wehavetheimportantcomparisontothe
initialdata:
Thatis,
Onaninfinitedomain,weassumethesolutionu(x,t)issquareintegrable,so
that
isdefined.Then,sinceitisreasonabletoassumeu(x,t)→
0 as x → ±∞, and ux is bounded in x, we obtain E′(t) ≤ 0. Once again the
energyisdecreasing,andwehavethecomparison(5.12)totheinitialenergy.
5.3.1.UsingtheEnergyMethodtoProveUniqueness
Decayofthemathematicalenergyallowsustoproveuniquenessofsolutionsof
theinitialboundaryvalueproblem
wheref,g,h,ϕ are given functions. Because the following result and proof are
verysimilartotheuniquenessargumentforthewaveequation,wedemonstrate
the method for the initial boundary value problem rather than for the Cauchy
problem.
Theorem5.2.Ifu1,u2solve(5.14)andareC2functions,thenu1=u2everywhere.
Proof.Letu=u1−u2.Thenusatisfies(5.14)withzerodata:f≡g≡h≡ϕ≡
0.Thus,from(5.13),
Therefore,u≡0,sothatu1≡u2.
5.3.2.TheEnergyPrincipleinHigherDimensions
Thesameargumentworksinthemorerealisticcontextofhigherdimensionsbut
uses multivariable calculus. Consider a bounded open subset U of Rn and the
initialboundaryvalueproblem
Webeginbydefiningtheenergyintegralasinonedimension:
Then
In conclusion, the mathematical energy E(t) decreases in time unless the
boundaryconditionsinjectenergyintoU.Inparticular,witheitherhomogeneous
DirichletorNeumannboundaryconditions,E(t)isdecreasingintime.
5.4.TheMaximumPrinciple
Maximum principles provide a powerful alternative to the energy method for
analyzing parabolic equations. We prove the maximum principle for the heat
equation, noting that similar ideas apply to more general linear and nonlinear
second-orderparabolicequations,butnotingeneraltosystemsofequations,nor
tohigher-orderequations.
Themaximumprinciplestatesthatthemaximumofany(smooth)solutionof
the heat equation occurs either initially (at t = 0) or on the boundary of the
domain.Foratimeinterval[0,T]withT>0andaspatialdomainU⊂Rn,we
usethenotationUT=U×(0,T],shownschematicallyinFigure5.2.Notethat
UTincludesboththeinteriorandthetopportionoftheboundaryU×{t=T}.
ThepartoftheboundaryofUTdefinedby
Figure5.2.DomainUTforthemaximumprinciple.
is known as the parabolic boundary. Here, ∂U denotes the boundary of U.
Solutionsoftheheatequationshouldhavetwospatialderivativesandonetime
derivative,sowedefinetheappropriatespaceoffunctionsonUT:
To compare values of u in UT with values on the boundary, in the following
theoremwerequirethatushouldbecontinuouson .
Theorem5.3.(TheMaximumPrinciple)Let
Then
satisfy
.
Remarks. Suppose u has a local maximum at (x, t) ∈ U × (0, T). Then (by
calculus)ut=0,∇xu=0andΔu≤0.IfweknewΔu<0at(x,t),thenthePDE
wouldimmediatelygiveusacontradiction:
Although this is not a proof, since we have to handle the degenerate case in
which Δu = 0, it has the main idea. The actual proof merely perturbs u to
removeapossiblydegeneratemaximum.
isclosedandbounded,sothecontinuousfunctionuachievesitsmaximum
somewherein .Thatis,thereisan(x0,t0)suchthatu(x0,t0)=
.
ProofofTheorem5.3.Let
.Ourgoalistoprovethatu(x,t)≤M
forall(x,t)∈UT.TodealwiththepossibilityΔu=0atamaximum,weperturb
uabit.Letv(x,t)=u(x,t)+ϵ|x|2,ϵ>0.Then(recallingthatx∈Rn),
sinceut−kΔu=0.
SupposevhasalocalmaximuminUT,atP0=(x0,t0)witht0<T.Thenvt=
0, ∇v = 0, and Δv ≤ 0 at P0. But this contradicts (5.15), so v cannot have a
maximumintheinteriorofUT.
Nowsupposevhasamaximumonthelinet=T,atP1=(x1,t=T).Then
∇v = 0, Δv ≤ 0, and vt ≥ 0 at P1.1 Now we have vt − kΔv ≥ 0, again
contradicting(5.15).
Therefore the maximum of v on
occurs on
for all
.Wehaveproved
where
.Thus,
Sinceϵ>0isarbitrary,wehaveu(x,t)≤Mforall
.
Remarks. The weak maximum principle is easy to prove. The related strong
maximumprinciple is somewhat harder to prove. The strong maximum principle
statesthat,providedUisconnected,themaximumofuisachieved onlyonthe
parabolicboundary,unlessuisconstantthroughout [12].
Byapplyingthemaximumprincipleto−u,whichalsosatisfiestheconditions
ofTheorem5.3,weseethatthereisacorrespondingminimumprinciple:
5.5.Duhamel’sPrinciplefortheInhomogeneousHeatEquation
Duhamel’s principle gives a formula for the solution of an inhomogeneous PDE
usingsolutionsofthehomogeneousequation.Fortheheatequationonthewhole
realline,Duhamel’sprincipleutilizesthefundamentalsolution.
Letf(x,t)beagivenfunctionrepresentingaheatsourceorsink,andconsider
theinitialvalueproblem
ThefundamentalsolutionΦ(x,t)ofthehomogeneousheatequationsatisfiesut=
kuxxfort>0.Butthenforanyy∈Rands>0,theshiftedfunctionΦ(x−y,t
−s)satisfiestheequationfort>s.Wecanmultiplybyanamplitudef(y,s),so
thatwehaveacollectionofsolutionsΦ(x−y,t−s)f(y,s)oftheheatequation,
withy∈Randt>s.Summing(i.e.,integrating)thesesolutionsovery∈Rwith
sfixed,wehavethat
isasolutionforallt>s,satisfying
Now the idea is to integrate
.Then
.
with respect to s from 0 to t. Define
In this calculation we differentiated under the integral sign, ignoring the
singularityofΦ(x−y,t−s)atx=y,t=sontheboundaryofthedomainof
integration.However,thissingularitycanbehandledwiththeappropriatelimit,
andtheresultisthesame.(SeeEvans[12],p.50forthedetails.)
Finally,weobserve
Insummary,theformula
solves the initial value problem (5.16). General initial conditions can be
incorporatedeasily,justasforthewaveequation.
In this chapter, we have constructed solutions of initial value problems for the
heatequation;derivedsomebasicprinciples,suchasthemaximumprinciple;and
examined certain properties, such as uniqueness of solutions. The solution of
initial value problems is described using the fundamental solution of the heat
equation,whereasforthewaveequation,thesolutionsaregivenbypropagating
disturbanceswithwaves,expressedthroughd’Alembert’ssolution.Usingthetwo
formulas, we can make some comparisons, which are a good guide to the
differences between solutions of hyperbolic equations and those of parabolic
equations.
Solutions of hyperbolic equations preserve the regularity of the initial data,
propagating disturbances and singularities along characteristics, which have
finite speed. By contrast, solutions of the heat equation and other parabolic
equations immediately smooth initial data, and information is propagated with
infinitespeed.
Nonlinear equations may have different properties. For example, the porous
mediumequationut=Δ(um)isdegenerateat u=0form>1,aconsequence
being that initial data with compact support spread but with finite speed; the
solutionu(x,t)continuestohavecompactsupportforeachtimet>0.
PROBLEMS
1.FillinthefollowingdetailsusedintheproofofTheorem5.1,property3.
(a)Usethetriangleinequalitytoprovethatifδ>0,andx,x0,y∈Rnsatisfy
, then
. (Hint: Draw a picture in the onedimensionalcase,n=1,tovisualizehowtheargumentworks.)
(b)Provethat
.
2.Showthat
.Consequently,theheatequationforrotationally
symmetricfunctionsu(x,t)=ϕ(r,t),r=|x|,is
3.(a)Letg:[0,∞)→Rbeaboundedintegrablefunction.Provedirectlythat
isanoddfunctionofx∈Rforeacht>0.
(b)Leth:R→Rbeanoddboundedintegrablefunction.Provethat
isanoddfunctionofx∈Rforeacht>0.Thatis,thesymmetryintheinitial
dataiscarriedthroughtothesamesymmetryinthesolution.
4.WritethesolutionoftheCauchyproblemfortheheatequation
withinitialcondition
intermsoftheerrorfunction
5.Solvetheheatequation(5.17)withinitialconditionu(x,0)=epx,wherep>
0isaconstant.Youcanusetheidentity
.
6.Solvetheheatequation(5.17)withinitialconditionu(x,0)=H(x)e−x.
7.Provetheenergyinequalityforut=∇·(k(x,u)∇u),wherek=k(x,u)∈Ris
agivenpositivefunction.
8.ConsidertheCauchyproblemfor(5.17),withinitialconditionu(x,0)=x2.
(a)Showthatifu(x,t)isthesolution,thenv(x,t)=uxxx(x,t)satisfiestheheat
equationwithv(x,0)=0.
(b)Findu(x,t)asanexplicitformula.
9.Considertheinitialvalueproblem
withconstantd,andgivenintegrablefunctiong.
(a) Use the change of variable u(x, t) = e−dtv(x, t) to find u using the
fundamentalsolution.
(b)Whatistheeffectoftheconstantd?
(c)Supposed=d(t)isagivencontinuousfunction.Whatwouldbeasuitable
changeofvariabletosolvetheproblem?
10.Deviseachangeofvariablecorrespondingtoamovingframeofreferenceto
solve the initial value problem for the convection-diffusion equation with
constantspeedc
11. Formulate and prove a statement regarding conservation of energy for the
waveequationonaboundeddomaininRn:
underhomogeneousDirichletorNeumannboundaryconditions.
1.Thefinalinequalityiseasilyprovedbycontradiction:ifvt<0atP1,thenv(x1,t)>v(x1,T)fort<T
closetot=T,contradictingtheassumptionthatvhasamaximumatP1.
CHAPTERSIX
SeparationofVariablesandFourier
Series
Inthischapterwefindexplicitsolutionsoftheheatequationandwaveequation
onboundeddomainsusingthepowerfulmethodofseparationofvariables.The
methodallowsustoexpresssolutionsofconstant-coefficientequationsasinfinite
series of functions. Recall that infinite series of functions were used for the
Cauchy-KovalevskayaTheoremofSection2.3,wheresolutionswereexpressedas
powerseries.Inthischapterthesolutionsareseriesoftrigonometricfunctionsof
the spatial variable x with coefficients that depend on time t. Such series are
called Fourier series. We introduce them here, show how they relate to the
methodofseparationofvariables,andstudythemmoreextensivelyinthenext
chapter.
Fourier series solutions have several advantages over power series. Fourier
series can be used to represent functions that are not analytic, for example,
continuous functions. Moreover, solutions expressed as Fourier series are
typically designed to have each term in the series satisfy both the PDE and the
boundary conditions, provided the boundary conditions are homogeneous.
Consequently, the only issues with such an approach are: (1) how to construct
theseriesand(2)convergence.Wedealwiththeconstructioninthischapter,and
giverigorousconvergenceresultsinthenextchapter.
6.1.FourierSeries
In this section we introduce the notation and basic formulas for Fourier series.
Consideracontinuousfunctionf:R→Rthatisperiodicwithperiod2L.Thenf
canberepresentedbyaFourierseries,whichtakestheform:
An important property of the trigonometric functions in this series is that they
areorthogonalontheintervalx∈[−L,L].Thus,forallj,k=0,1,2,…,
and
where
istheKroneckerdelta.Bymultiplying(6.1)byoneofthesineorcosinefunctions
and integrating, the coefficients an, bn are readily obtained from f, in the Euler
formulas:
withn=0,1,2,….ThecoefficientsanarecalledFouriercosinecoefficients,and
thebnareFouriersinecoefficients.
For an odd function f, all the cosine coefficients are zero, and the sine
coefficientsare
TheresultingseriesofsinefunctionsiscalledtheFouriersineseriesoff.Notethat
thecoefficients(6.5)aredefinedeveniffisspecifiedonlyontheinterval[0,L].
ThentheFouriersineseriesistheFourierseriesoftheoddperiodicextensionoff,
whereinfisextendedtobeanoddfunction thatis2Lperiodicandsuchthat
(x)=f(x),0<x<L.
Similarly,iffisanevenfunction,thenithasaFouriercosineseries,inwhich
allthesinecoefficientsarezero,andthecosinecoefficientsaregivenbyintegrals
overtheinterval[0,L]:
Iffisanintegrablefunctionon[−L,L],thentheFouriercoefficientsarewell
defined,andwesaythatfhasaFourierseries.Theseriesmayconvergetofonly
in a weak sense that we make precise in Chapter7, where the convergence of
Fourierseriesisdiscussedindetail.
6.2.SeparationofVariablesfortheHeatEquation
Themethodofseparationofvariablesworksmostsimplyfortheheatequationin
one space dimension with homogeneous Dirichlet boundary conditions. If you
follow the method in this case, then using separation of variables in other
circumstancesbecomesamatterofchangingsomedetails.
Consider the initial boundary value problem with homogeneous Dirichlet
boundaryconditionsandinitialdatagivenbyacontinuousfunctionf:[0,L]→
R,
Wesummarizethemethodintwosteps.Inthefirststep,weseeksolutionsin
thespecialseparatedform
afunctionofxtimesafunctionoft.SubstitutingintothePDEandtheboundary
conditions, we obtain ODE for the functions v,w with corresponding boundary
conditions.Thisstepleadstoafamilyofsolutions{un(x,t),n=1,2,…}.
In the second step, we write a linear combination of the un (actually, an
infiniteseries):
This series is a representation of the general solution of the PDE and boundary
conditions.Itremainstochoosethecoefficientsbn∈R,n=1,2,…tosatisfythe
initialcondition.
Claim6.1.Thisprocessleadstotheseriessolutionof(6.7):
Proof. As indicated above, the proof is organized around constructing the
solutionintwosteps.
Step1.Substituteu(x,t)=v(x)w(t)intothePDE:
Dividingbykv(x)w(t)andsimplifying,weget
aconstant.Notethatphysicalconstantssuchaskarealwaysincludedwiththe
timeportion,sothatthespatialpiecebecomesastandardequation.Also,the
minussignontheright-handsideof(6.9)isincludedsothatvaluesofλall
turnouttoberealandpositive.
From(6.9) we have two ODE for v(x),w(t), with an additional unknown,
theparameterλ:
Tosummarize,u(x,t)=v(x)w(t)satisfiesthePDEifandonlyifv(x),w(t)
satisfy(6.10)forsomeλ∈R.
Nowwesubstituteintotheboundaryconditionsforu,andnotethattoget
solutionsuthatarenotidenticallyzero,wemusthavecorrespondingboundary
conditionsonv.Forexample,u(0,t)=0=v(0)w(t),sothateitherw(t)=0
for all t (implying that u(x, t) ≡ 0) or v(0) = 0. Incorporating boundary
conditions at both x = 0 and x = L completes the eigenvalue problem to be
solvedforv(x):
Theeigenvalue/eigenfunctionpairsfor(6.11)are
Withtheseeigenvalues,weturnto(6.10a)forw(t),settingλ=λn.Thegeneral
solutionofthissimplefirst-orderODEisanarbitrarymultipleof
Notetheimportantpropertythatallthewndecayintime,becausethe
eigenvaluesarerealandpositive.
Nowwecompletestep1bysettingu=un(x,t)=vn(x)wn(t),n=1,2,….
Step2.Tosatisfytheinitialcondition,weformtheseries
Substitutingt=0intotheseriesandusingtheinitialconditionin(6.7),we
get
NowwecanextracttheformulaforbnfromtheEulerformulas(6.4).
Having completed a procedure to generate a formula, it is natural to ask
whether the formula in the claim really solves the initial boundary value
problem.Inparticular,wehavenotprovedthattheseriesconverges,orindeed
whetherallinitialfunctionsfcanberepresentedbyaFouriersineseries.Evenif
theseriesforu(x,t)converges,weneedtoknowwhetherthelimitissufficiently
differentiabletobeasolutionoftheheatequation.Wepostponetheseimportant
considerationsfornow,withtheremarkthattherapiddecayofthecoefficients
bne−kλ tof
fort>0impliesthattheseriesconvergesuniformlyto
n
aC∞function,anddifferentiationterm-by-termcanbeusedtoverifythatu(x,t)
doesindeedsolvetheproblem.
Itisworthnotingthattheboundaryconditionsaresatisfiedfort>0,even
thoughtheymaynotbesatisfiedbytheinitialdata.Forthewaveequationsucha
discontinuity is propagated along characteristics, but for the heat equation the
discontinuity is immediately smoothed and does not propagate into the interior
ofthedomain.
We next illustrate the method of separation of variables with two worked
problems.Inthefirstproblemwesettheconstantsk,Landchoosetheinitialfto
showthattheinfiniteseriesmayhaveonlyafinitenumberofnonzeroterms.
Example 1. (Homogeneous Dirichlet boundary conditions) Solve the initial
boundaryvalueproblem
Separation of variables with these homogeneous Dirichlet boundary conditions
andL=1giveseigenfunctionssinnπx.Therefore,sincek=7,
Nowwechoosethecoefficientsbnsotheseriesagreeswiththeinitialcondition
when t = 0. Since the initial condition is a Fourier sine series with just two
terms,theseriesforu(x,t)alsohasonlytwoterms.Let
Then
isthesolution.
Example2. (Homogeneous Neumann boundary conditions) In this example
wechangetheboundaryconditionandconsequentlygetdifferenteigenfunctions.
Considertheinitialboundaryvalueproblem,inwhichwehaveL=π,k=4:
Recall that in the first step of separation of variables, we consider the PDE
and boundary conditions only; then in step 2 we form a series of separated
solutionstosatisfytheinitialcondition.Neumannboundaryconditionssupporta
constantsolution,whichdiffersfromtheDirichletcase:anonzeroconstantdoes
notsatisfythehomogeneousDirichletboundaryconditions.
Letu(x,t)=v(x)w(t).TheODEforv,wwithboundaryconditionsarenow
Wesolvetheeigenvalueproblemforλ=λn,v=vn.Firstnotethatλ0=0gives
v″(x)=0,sothatvislinear.Butv′(0)=0=v′(π)impliesthatvisflatatthe
ends,soitmustbeflateverywhere.Thatis,visconstant:
Forλ>0,thegeneralsolutionoftheODEisacombinationofsinesandcosines.
Usingtheboundaryconditions,wearriveat
The time-dependent part of the solution is the same as for Dirichlet boundary
conditions,sincetheeigenvaluesarethesame,exceptthattheconstantsolution
isincludedhereaswell:
Tocompletetheseparationofvariablesstep,weformthesolutions
NowwecanformtheseriessolutionofthePDEandboundaryconditions:
Finally, we satisfy the initial condition by determining the coefficients an using
orthogonality of the eigenfunctions. Setting t = 0 in the series, the initial
conditionbecomes
Multiplying by cos mx and integrating both sides from 0 to π, we see that all
termsbecomezero,exceptwherem=n.Consequently,integratingbypartsand
usingcosnπ=(−1)n,wehave
Thus,thesolutionis
inwhichwehavewrittenn=2k−1,k≥1.WediscussconvergenceofFourier
seriescomprehensivelyinChapter7,butitiseasytoshowthatthespecificseries
(6.14)converges.
Claim6.2.Theseries(6.14)convergesuniformlyfort≥0.
Proof.Toprovetheclaim,weestimatethekthtermoftheseries:
Since
is a convergent series of constants, the Weierstrass M-test(see
AppendixB)impliestheseries(6.14)convergesuniformlyinxforeacht≥0.
To examine whether the series satisfies the PDE, we need to be able to
differentiatetermbyterm.Fort>0,thecoefficientofthekthtermintheseries
decaysexponentiallyink,sincetheexponentiale−4(2k−1)2tismultipliedonlybya
rational function (a ratio of polynomials). The same is true of the coefficients
after differentiating any number of times in x and t. Consequently, from the
standard result on uniform convergence and differentiating series of
differentiable functions term by term, the series converges uniformly to a C∞
function.
Thesituationatt=0isratherdifferent.WhiletheFourierseriesconverges
uniformlyforeachx∈Rtothecontinuousevenperiodicextensionoff(x)=x,
thisfunctionhasjumpdiscontinuitiesinthederivative.Consequently,theseries
differentiated with respect to x does not converge uniformly at t = 0, and the
twicedifferentiatedseriesconvergesonlyinamuchweakersense,specifically,in
thesenseofdistributions(seeSection9.2).
Because Neumann boundary conditions correspond to insulated ends, heat
energyisconserved,whichwecanseeexplicitlyintheseriessolution.Theheat
energyattimetisproportionalto
Thenwehave
fromtheboundaryconditions.Therefore,
Now,ast→∞,theseriesconvergestoaconstant:
.Fromconservationof
energy,wecalculate
Therefore,
Thisconclusionagreeswiththecalculationofa0fromorthogonality.
6.2.1.RobinBoundaryConditions
Robin-type boundary conditions relate the normal derivative of the function on
the boundary to the function itself. We relate the analysis of these boundary
conditionstophysicalconsiderations,suchasthedirectioninwhichheatisbeing
transportedacrosstheboundary.Considerthefollowingproblem:
The physical interpretation of the boundary conditions depends on the signs of
thecoefficients:
Since heat energy flows from warm to cool, at x = L if u > 0, a radiating
boundaryconditionwillgiveux>0,sothatheatwillflowoutoftheboundary
(theheatfluxisnegative),whereasforanabsorbingboundarycondition,ux<0,
andheatwillflowintothedomainthroughtheboundaryatx=L.Consequently,
for radiating boundary conditions, all eigenvalues are positive, because heat is
transportedoutofthedomain.Forabsorbingboundaryconditions,theremaybe
oneormorenegativeeigenvalues.
Separationofvariablesleadstotheeigenvalueproblem
We consider the implications of radiating and absorbing boundary conditions
separately.
Radiatingboundaryconditionsa0>0,aL>0.Forpositiveeigenvaluesλ=
β2>0,thegeneralsolutionoftheODEis
NowweusetheboundaryconditionstofindequationsforA,B,andβ:
Figure6.1.PlotsoftanβL(solidcurves)andβ(a0+aL)/(β2−a0aL)(dashed
curves)withL=π,a0=1,aL=2.
DividingbycosβLandsubstitutinginforA,wefindeitherB=0or
whichisanequationforβ,sincealltheotherconstantsaregiven.WerejectB=
0,becauseitleadsonlytothetrivialsolutionu≡0.
FromFigure6.1,weobservethattherearesolutionsβ1<β2<…:
Asn→∞theβnsareapproachingthelowerlimittoleadingorder:
Thus,asn → ∞ the eigenvalues
approach the eigenvalues
that
wefoundforeitherNeumannorDirichletboundaryconditions.
Thecorrespondingeigenfunctionsaregivenby
Absorbingboundaryconditions.Inthecaseofabsorbingboundaryconditions,
a0<0,aL<0,heatenergyisabsorbedthroughtheboundaries.Consequently,
we might expect the temperature to rise when both boundaries are absorbing.
Correspondingly,atleastoneeigenvalueshouldbenegative.Positiveeigenvalues
λ=β2satisfy(6.16)andcanberepresentedasintersectionsofthegraphsoftan
βLandtheright-handsideoftheequation,similartoFigure6.1.
Theissueislessclearwhenoneboundaryisabsorbingandoneisradiating,a
situationexploredintheexercises.
6.2.2.InhomogeneousBoundaryConditions
Up to now the method of separation of variables has relied on homogeneity in
boundaryconditionsandinthePDE.Hereweconsiderasampleinitialboundary
value problem showing how to handle inhomogeneous problems. Consider the
initialboundaryvalueproblem
Thestrategyistofirstignoretheinhomogeneitiesf,u0,uπsothatwecanidentify
suitable eigenfunctions vn(x). Then we form a series
with timedependentcoefficientswn(t) that are determined in a new way to accommodate
theinhomogeneities.
From the homogeneous version of problem (6.17), we know that the
eigenfunctionsaresines,duetotheboundaryconditions.Thus,weconsiderthe
Fouriersineseriesforu:
TosatisfythePDE,wealsoexpandf(x,t)asaFouriersineseries:
wherethecoefficientsaredefinedby
The series (6.18) apparently does not satisfy the inhomogeneous boundary
conditions,becausewhenevaluatedatx=0orx=πtheseriesgivesthevalue
zero.However,weshallseeinthenextchapterthatFourierseriescanconverge
toafunctionthatisdiscontinuous.Wheretheboundaryconditionisnonzero,itis
satisfied by the series (6.18) in the sense that u(x, t) approaches the specified
value(u0(t)oruπ(t))asxapproachestheboundary:
Tofindthecoefficientswn(t),wecouldsubstitutetheseries(6.18)intothePDE
andequateterms.However,thisprocedurewouldomittheeffectoftheboundary
conditions,soinstead,weusetheEulerformulasforthecoefficientswn(t):
GuidedbytheexpectationofanODEforwn(t),wedifferentiateandthenusethe
PDE:
integratingbypartstwice.WecanrewritethisasaninhomogeneousODE,
where
. Setting t = 0 in (6.19), we obtain the
initialcondition
Thesolutionoftheinitialboundaryvalueproblem(6.17)isgivenbytheseries
(6.18),wherethecoefficientswn(t)areobtainedbysolvingtheODEinitialvalue
problem(6.20),(6.21).
6.3.SeparationofVariablesfortheWaveEquation
Separation of variables works for the wave equation on bounded domains in
muchthesamewayitworksfortheheatequation.Differencesarisebecausethe
waveequationhastwotimederivatives,ratherthantheonetimederivativethat
appears in the heat equation. Previously we emphasized the traveling wave
structure of solutions of the wave equation through d’Alembert’s solution,
whereasseparationofvariableshighlightstheroleofvibrations,whicharetimeperiodicsolutions.Thetypicalphysicalcontextforthesesolutionsisavibrating
stringoffinitelength,wheretheeigenfunctionsforthewaveequationaremodes
ofvibrationfordifferentfrequencies.
Considertheinitialvalueproblemforthewaveequation
We solve this problem by following the separation-of-variables procedure used
forthecorrespondinginitialboundaryvalueproblem(6.7)fortheheatequation.
Because there are two t derivatives in the PDE, we need two initial conditions
insteadofone.Also,theconstantk>0intheheatequationisreplacedbythe
constantc2>0inthewaveequation.
Claim6.3.Thesolutionof(6.22)istheseries
with
Proof. We substitute u(x,t) = v(x)w(t) into the PDE and boundary conditions,
anddividebyc2w(t).Thisleadsto
which is a constant. Now we have two ODE for v(x), w(t), with an additional
unknown,theparameterλ:
Just as for the heat equation, the PDE boundary conditions become boundary
conditionsforv(x):
Wealreadyknowtheeigenvaluesλandeigenfunctionsvforthisproblem:
Thedifferencebetweentheheatequationandthewaveequationsolutionsisin
the time-dependent part w(t). The ODE (6.25a), with λ = λn, has the general
solution
Nowweletun(x,t)=vn(x)wn(t),n=1,2,…,whichisafamilyofsolutionsof
thePDEandboundaryconditions.Nextweformtheinfiniteseries
anditremainstoshowhowthecoefficientsan,bnaredeterminedfromtheinitial
conditions.
Theinitialconditionu(x,0)=ϕ(x)is
leadingtotheexpressionforanintheclaim.
Similarly, the initial condition ut(x, 0) = ψ(x) leads to the formula for bn,
since(differentiatingtheserieswithrespecttot),
Thus,
fromwhichtheformulaintheclaimfollows.
The form of these solutions of the wave equation leads us to note several
differences between solutions of the wave equation and those of the heat
equation:
1.Theindividualtermsintheseriesforthewaveequationoscillateintime.
Indeed,thisisthebehaviorofthesolutionasawhole;itisperiodicintime,
withperiod2πL/c.
2.Usingtrigonometricidentities,youcanwrite(6.23)intheformu(x,t)=F(x
+ct)+G(x−ct).Afterall,weknowthateverysolutionofthewave
equationinonedimensionisofthisform.ItisagoodexercisetofindFandG
intermsofϕandψ,andtheirperiodicextensions.
6.4.SeparationofVariablesforaNonlinearHeatEquation
Separationofvariablesisveryusefulforlinearequations.However,fornonlinear
equations, it is not nearly as useful, partly because the technique depends on
linear combinations of solutions also being solutions, which is not the case for
nonlinearequations.Althoughtheregenerallywillnotbesolutionsinwhichthe
independentvariablesareseparated,theporousmediumequationhasastructure
that admits separated solutions (See Evans [12], p. 170.). This equation is a
simplified model of fluid flow in a porous medium, such as compacted sand or
soil.
Letm>1beagivenconstant,andconsidertheporousmediumequationfor
theunknownfunctionu=u(x,t)>0,x∈Rn,t>0:
Letu(x,t)=v(x)w(t).Then
andtherefore,
Note that the function F(u)=umis homogeneous of degree m, meaning F(αu) =
αmF (u), for α > 0. The homogeneity allows us to separate variables. We thus
obtainthepairofequations
Tosolve(6.27),weintegrateoncetoget
.Hence,
inwhichλ,μareconstants;weneedλ>0toensurethatw(0)isdefined.
Next, we find spherically symmetric solutions of the v equation (6.28).
Considerv(x)=rα,r=|x|,withαaconstanttobedetermined.Then
(whereϕ(r)=rmα).Equatingpowersofryields
Equatingcoefficientsresultsin
Finally,combiningvandwgives
where α, μ are given by (6.29), (6.30), but λ > 0 is a free parameter. The
solutionu(x,t)iswelldefinedast→0+.However,astincreases,thenegative
fractionalpowermeansthatublowsupastincreasesto
,sincem>1.
6.5.TheBeamEquation
Thebeamequation
isafourth-orderPDEforthedeflectionofahomogeneousbeamunderanapplied
external load distribution q(x, t). The beam equation is written here in
nondimensional form, in which the dimensionless parameter G = EI/ρ > 0 is
related to the elastic modulus E, the bending modulus I, and the density (mass
perunitlength)ρ.Thisequationmodelssmalldeflectionsfromastraightbeam
and is based on assuming that cross sections bend around the center line and
exertforcesononeanotherduetobending,leadingtocompressionononeside
ofthecenterlineandstretchingontheoppositeside.
ThedispersionrelationisobtainedbyseekingsolutionsrelatedtotheFourier
transformof(6.31).Letq≡0andu=exp(iωt+iξx).Then
gives the temporal frequency ω as a function of the spatial frequency ξ.
However, it is somewhat more informative to consider the full separated
equations,aswedidinSections6.2and6.3forsecond-orderequations.
Letu(x,t)=v(x)w(t).Substitutingintothebeamequation(6.31)anddividing
byvw,wegettwoODE:
whereω2>0isconstant,and
thegeneralsolutionforv(x)is
.Thusw(t)=Acosωt+Bsinωt,and
witharbitraryconstantsa,b,c,d.
Sincetheequationisfourthorderinspace,weneedfourboundaryconditions,
andtwoinitialconditionsarealsoneeded,sincetheequationissecondorderin
time. For example, for a beam fixed at both ends x = 0, x = L, the boundary
conditionsare
Theseclamped-beamboundaryconditionstranslatetothefourboundaryconditions
on v(x) that require v and v′ to be zero at each end. Then by processing the
conditionsonvgivenby(6.32),wefindthespatialwavenumbersμ=μn,n=
1,2,…,aredeterminedfromtheequation
Itiseasytoseethatμn∈((n−1)π/L,nπ/L),and(sincesechξ→0asξ→∞),
μn∼nπasn→∞.
Acantileveredbeamistakentobeclampedatoneendx=0,andfreeatthe
other.Theboundaryconditionsare
Inthiscase,wealsofind(seeproblem8)μn∈((n−1)π/L,nπ/L).
Inthischapterwehaveseenhowtoimplementthemethodofseparationof
variablesinseveralexampleproblems.Inthenextchapterweplacethemethod
in a rigorous theoretical framework. In particular, we explore eigenvalue
problems and prove results on the convergence of infinite series of
eigenfunctions,includingFourierseries.
PROBLEMS
1.Solvetheinitialboundaryvalueproblem
2.Solvetheinitialboundaryvalueproblem
3. Consider the eigenvalue problem (6.15) with Robin boundary conditions.
Supposeλ=0isaneigenvalue.
(a)Findtheeigenfunction.
(b)Findanecessaryconditiononthecoefficientsa0,aL.
(c) Prove that this condition is also sufficient to guarantee that λ = 0 is an
eigenvalue.
4.Considertheeigenvalueproblem(6.15)withRobinboundaryconditions.
(a)Provethatthereisatleastonenegativeeigenvalueintheabsorbingcasea0
<0,aL<0.
(b) Find a condition on a0, aL that is necessary and sufficient to have two
negativeeigenvalues.
5. Consider the eigenvalue problem (6.15) with Robin boundary conditions. In
the radiating case a0 > 0, aL > 0, prove the properties shown graphically in
Figure6.1.(Hint:Asinthetext,itiseasiertoletλ=β2.Youcanalsousetan(nπ
+θ)=tanθ.)
(a)Thereareaninfinitenumberofeigenvaluesλn,n≥1.
(b)
.
(c)
.
(d)Ifβn=(n−1)π/L+θn,withθn→0asn→∞,findtheleading-order
behaviorofθn:θn=A/n+O(1/n2).
6. This problem is a graphical summary of problems 3–5. Sketch the hyperbola
a0aLL+a0+aL=0inthea0,aLplane,showingtheasymptotes.Inyoursketch,
label the (three) regions corresponding to values of (a0,aL) for which there are
two, one, or zero negative eigenvalues. Label the point corresponding to
Neumann boundary conditions. Where in the plane are Dirichlet boundary
conditionsrepresented?
7.Forconstants0<m<L,let
where pj, rj, j = 1, 2, are positive constants. Assuming that eigenfunctions are
continuously differentiable, find an equation for the eigenvalues λ for the
eigenvalueproblem
Youmayassumethattheeigenvaluesarerealandpositive.
8.Thebeamequationisassociatedwitheigenvalueproblemsforthefourth-order
ODE
Usingthegeneralsolution(6.32),findanequationforthepositiveeigenvaluesλ
=μ4forthecantileveredbeam,withboundaryconditions
Fromthisequation,orusingagraph,showthatthereareaninfinitenumberof
positiveeigenvaluesλn,correspondingto
Statetheleading-orderdependenceofλnonnasn→∞.
9.(a)CalculatetheFourierseriesofthefunctionf(x)thatisoddand2πperiodic,
withf(x)=1,0<x<π.
(b)Assumingtheseriesconvergestof(x)=1atx=π/4,calculatethesumof
theseries
CHAPTERSEVEN
EigenfunctionsandConvergenceof
FourierSeries
Inthepreviouschapterweusedthetechniqueofseparationofvariablestosolve
avarietyofinitialboundaryvalueproblemsfortheheatandwaveequations.We
showedhowthetechniqueleadsnaturallytoeigenvalueproblemsforthespatial
dependence of the solutions. The resulting eigenfunctions (sines and cosines in
thepreviouschapter)arethencombinedintoinfiniteserieswithtime-dependent
coefficientstorepresentthesolutionofthePDEanditsinitialconditions.
Inthischapterweshowthatthetechniquecanalsobeusedfornonconstantcoefficientequations,suchas
in which the coefficients p,q are specified functions. However, the eigenvalues
and eigenfunctions are generally not available explicitly for such equations.
Nonetheless, with the aid of the Sturm-Liouville Theory of eigenvalue problems
inSection7.1,weplaceonafirmfootingthemethodofseparationofvariables
toconstructinfiniteseriesoffunctionsrepresentingsolutionsofthePDEandits
boundary conditions. There remains the issue of convergence of the infinite
series,evenfortheexplicitseriesofthepreviouschapter.Forexample,sofarwe
haveexaminedconvergenceofsuchaseriesinjustoneexample.InSection7.2,
weintroducenotionsofpointwise,uniform,andL2convergence,provingtypical
convergence results in subsequent sections. Finally in Section7.6, we introduce
the Fourier transform, and show how it is related to Fourier series before
demonstrating the use of the transform in PDE, and discussing its utility in a
rangeofareas.
7.1.EigenfunctionsforODE
Consider the following ODE eigenvalue problem, known as a Sturm-Liouville
problem.Letp,q,rbegivenrealC2functionsontheboundedinterval[a,b].We
assumep(x)>0,r(x)>0,a≤x≤b.Theeigenvalueproblemconsistsofthe
ODE
inwhichλ∈Cisaparameter,togetherwithlinearandhomogeneousboundary
conditions. The treatment of eigenvalue problems in the previous chapter
corresponds to the special case p≡1,q ≡ 0, r ≡ 1. In this section we derive
propertiesoftheeigenvaluesλandeigenfunctionsvthathelpmakethemethod
of separation of variables work for linear PDE with variable coefficients. In
particular,weshowthattheorthogonalityofthesineandcosineeigenfunctions
inthepreviouschapterwasnotanaccidentandcanbeestablishedeasilyinthe
moregeneralcontextof(7.1).
WedefineadifferentialoperatorLby
.Thenforany
v∈C2[a,b],Lv is a new function (in C[a,b]).Towrite(7.1) as an eigenvalue
problem
werestrictthedomainofLtofunctionsthatsatisfytheboundaryconditions.
We consider complex-valued functions, so as to include the possibility of
complexeigenvaluesλ.Inthiscontext,weneedtheweightedL2spaceofsquareintegrablecomplex-valuedfunctions,withweightr(x)andinnerproductdefined
by
Thecorrespondingnorm(seeAppendixB)is
Forsimplicity,weassumer≡1andleavetoproblem2theconsiderationofmore
generalfunctionsr(x)>0.
Notice that L has a special form, in which the first- and second-order
derivativesarecombinedintoasingletermthatcanbeintegrated.Thisallowsus
touseintegrationbypartsasfollows.Foru,v∈C2[a,b],
Iftheboundaryconditionsaresuchthattheboundarytermvanishes:
thenwehavetheidentity
Inthiscase,wesayLissymmetric. Notice that this depends on two properties,
namely,thespecialformofthesecond-orderdifferentialoperatorandcondition
(7.4)ontheboundaryconditions.
Theorem7.1.LetLbesymmetric.Then
1.alleigenvaluesofLarereal,and
2.eigenfunctionsofdifferenteigenvaluesareorthogonal.
Proof. The proof is the same as the proof of the corresponding properties of a
symmetric matrix. It depends on the calculation (7.3) formalized in (7.5) and
elementarypropertiesofthecomplexinnerproduct.
1.SupposeLu=λu,u≠0,andsupposeusatisfiestheboundaryconditions.
Then
But‖u‖≠0,sinceu≠0,sowemusthave
.Therefore.λisreal.
2.Letu,vbeeigenvectorsfordifferenteigenvaluesλ≠μ:Lu=λu,Lv=μv.
Then
Therefore,(λ−μ)(u,v)=0,sothat(u,v)=0.
Remarks. For the eigenvalue problem to make sense, the boundary conditions
have to be linear and homogeneous. Clearly, both homogeneous Dirichlet or
Neumann boundary conditions are symmetric. We leave as an exercise the
propertythatRobinboundaryconditionsoftheform
arealsosymmetric.
As we noted earlier, an important property of eigenvalues for the heat
equationisthattheyarepositive,exceptpossiblyforafinitenumberofthem.For
symmetricSturm-Liouvilleproblemswecanfindasufficientconditionforallthe
eigenvaluestobenonnegative.
SupposeLu=λu.Then
if
For Dirichlet or Neumann boundary conditions, the first of these conditions is
satisfied, but it need not be for Robin boundary conditions (7.6) without
restrictions on the coefficients a0,a1. However, if both coefficients are positive,
then the boundary term in (7.7) is nonnegative, so the eigenvalues are
nonnegative(providingthesecondconditionq(x)≥0issatisfied.)
InChapter6,Claim6.2,weproveduniformconvergenceoftheFourierseries
solvinganinitialboundaryvalueproblemfortheheatequation.Inthenextthree
sections we prove more general convergence results for series: pointwise
convergence,uniformconvergence,andmean-squareconvergence.
7.2.ConvergenceandCompleteness
Before proving results on the convergence of Fourier series and other series of
functions,weintroducesomegeneralconceptsofconvergenceandcompleteness,
anddiscusstheirsignificanceforPDE.Theconvergenceofasequence
inRn
toalimitvmeansthat||vk−v||→0ask→∞.Here,||x||denotestheusual
Euclideannormofthedistanceofx∈Rntotheorigin.Theideaofconvergence
of a sequence of numbers carries over naturally to functions. If
is a
sequenceoffunctionsdefinedonasetS,thenthesequenceconvergespointwiseon
Stoafunctionfifthesequenceofnumbers
convergestof(x)foreachx∈
S.Inthisdefinition,thefunctionscanbevectorvalued,sothatfk:S→Rn,and
theycanbedefinedonamultidimensionalsetS⊂Rm.
Wealsoneedtointroduceconvergenceinnorminthesensethat||fk−f||→0
ask→∞,where||f||denotesanappropriatenormonaspaceoffunctions.For
example,ifC[a,b]denotesthespaceofcontinuousfunctionsf:[a,b]→Rona
boundedinterval[a,b]⊂R,thenonenormonC[a,b]isthesupnorm||f||max=
maxa≤x≤b|f(x)|. This is also called the uniformnorm,because||fk − f||max → 0
means fk → f uniformly, in which case f ∈ C[a, b], a fundamental result in
analysis. More generally, a sequence
in a normed space X is a Cauchy
sequenceif||fj−fk||→0asj>k→∞(uniformlyinj),sothatforeveryϵ>0,
thereisN>0suchthatif||fj−fk||<ϵforallj,k>N.IfXhastheproperty
that Cauchy sequences converge to a limit in X, then X is complete. Complete
normed vector spaces (such as Rn, and C[a, b]with the sup norm) are called
Banach spaces. In a Banach space, a sequence converges if and only if it is a
Cauchysequence.
AnothernormonC[a,b]istheL2norm
.Thisnormismore
closely analogous to the Euclidean norm, if you think of f(x),a ≤ x ≤ b as a
vectorinaninfinite-dimensionalspace,andthenormasthesquarerootofasum
of values f(x)2, thus measuring the distance of f from the origin f ≡ 0.
Incidentally,theanalogofthesupnorminRnisthenorm||x||max=max{|xj|,j
=1,2,…,n}.Interestingly,allnormsinRnareequivalent,sothatconvergence
intheEuclideannormandin||·||maxareequivalent.Incontrast,thesupnorm
andtheL2normonC[a,b]arenotequivalent.Thefollowingexampleshowshow
convergenceinthesetwonormsisdifferentandmoreoverthatthespaceC[a,b]
withtheL2normisnotcomplete.
Example1.(PointwiseandL2convergencebutnotuniform)Letfk:[0,1]→
Rbethesequenceofcontinuousfunctionsdefinedby
Then,ask→∞,
pointwise,and||fk−f||2→0.However,fkdoesnotconvergetofintheuniform
(sup)norm,sincesup0≤x≤1|fk(x)−f(x)|=1forallk.
Example 2. (Pointwise but not uniform convergence) The sequence of
continuous functions gn(x) = xn on [0, 1] converges pointwise to the
discontinuousfunction
Consequently,gncannotconvergeuniformlyon[0,1].Infact,
S(x)|
= 1 does not go to zero as n → ∞. For the same reason, the convergence also
fails to be uniform on the open interval 0 < x < 1, even though the limit is
continuousthere.However,theconvergenceisuniformontheinterval[0,a],for
anya∈(0,1),since|xn|≤an→0asn→∞.
When dealing with PDE, we often work in spaces of functions that are not
necessarily continuous, such as L2 spaces. Since spaces of continuous or
continuouslydifferentiablefunctionsaresurelybettersuitedtofindingsolutions
ofdifferentialequations,itisnaturaltoask“Whybother?”Moreover,forelliptic
andparabolicequations,thesolutionstendtobeverysmooth,havingevenmore
derivativesthantheequationwouldseemtorequire.
Theshortanswertothisquestionisthatexistenceofsolutionsismucheasier
to prove in the weaker L2 spaces. A longer explanation is that, at least for the
beginning theory, existence results and proofs rely on having a space that is
completeinanormwithacorrespondinginnerproduct.Suchaspaceiscalleda
Hilbertspace.Spacesofcontinuousfunctionsonboundedclosedsetsarecomplete
inthesupnorm,butthisnormdoesnothaveacorrespondinginnerproduct.In
contrast,L2(U)isaHilbertspacewithinnerproduct(7.2).Havingestablishedthe
existence and uniqueness of weak solutions in such a space, then regularity
resultsfortheweaksolutionareestablishedseparately.
The precise definition of L2(U) as a Hilbert space requires some measure
theory.Inparticular,theRiemannintegralisnotsufficientlygeneral,andamore
sophisticatedtheoryofintegration,basedonLebesguemeasure,ismostsuitable.
In this theory, the Riemann integral is generalized to enable some irregular
functions to be integrable in the Lebesgue sense. Functions that are Lebesgue
integrable are called measurable. More precisely, all measurable functions are
Lebesgueintegrable(seeAppendixB).
FortheL2(a,b)normtomakesense,weneedtoclarifyhowtheproperty||f||
=0⇒f=0istobeunderstood.Thisreliesonthenotationofsetsofmeasure
zero. A set S ⊂ R has measure zero if for every ϵ > 0, S can be covered by a
countablefamily{Ij:j=1,2,…}ofopenintervalswithlengths(i.e.,measures)
mj,j=1,2,…,suchthat
and
.Forameasurablefunctionf:
[a, b] → C, we say f(x) = 0 almost everywhere if there is a set S ⊂ [a, b] of
measurezerosuchthatf(x)=0forallx∈[a,b]withx∉S.
AkeypropertyoftheLebesgueintegralappliestoL2functions:Iff∈L2(a,b),
then||f||=0impliesthatf(x)=0almosteverywherein[a,b].ThespaceofL2
functions is then defined by identifying functions that are equal almost
everywhere (meaning that their difference is zero almost everywhere). This can
be formalized by making this identification into an equivalence relation and
definingL2tobethesetofequivalenceclasses.Thenthenormisdefinedonan
equivalenceclassbydefiningitonanyfunctionintheequivalenceclass.Similar
considerations apply to the Lp spaces and Sobolev spaces introduced in Chapter
10.
The properties of L2(a,b) that we need can be summarized in the following
theorem, which we state without proof. The theorem is proved in Rudin’s text
[38], where a more extensive treatment of Lebesgue integration can also be
found.
Theorem7.2.ThefollowingpropertiesholdfortheL2(a,b)norm:
1.L2(a,b)iscompleteintheL2norm,and
2.everyL2(a,b)functionfcanbeapproximatedinnormbycontinuousfunctions.
Thatis,thereisasequence{fk}ofcontinuousfunctionson[a,b]suchthat||f−
fk||→0ask→∞.
Aseries
sums
offunctionsconvergespointwiseinanintervalIifthepartial
convergeasasequenceofnumbers,
foreachx∈I.TheconvergenceisuniformonIif
thatis,||SN−S||max→0asN→∞.NotethatifSN→Suniformly,thenSN→S
pointwise.
The series converges in L2if||SN−S||2 → 0 as N → ∞. This is also called
mean-squareconvergence.
7.3.PointwiseConvergenceofFourierSeries
AclassicresultinFourieranalysisistheproofofpointwiseconvergenceforthe
fullFourierseries(Chapter6)ofa2π-periodicfunctionf.Tostart,let’sassumef
isC1onR.Thenwewanttoshow
inwhichthecoefficientsaregivenbytheEulerformulas(Chapter6):
To prove pointwise convergence of the series, we fix x ∈ [−π, π], and
considerthepartialsums:
WeinserttheEulerformulasforthecoefficientsandmanipulatethefinitesum:
Tomakeprogresswiththeintegral,considerthesum
Itisconvenienttousecomplexexponentials(recallthate±iξ=cosξ±isinξ):
Thisisapartialsumofageometricserieswithratioeiθ,sothat
TheDirichletkernelisdefinedtobe
forPN(θ),wededucethefollowingproperties:
.Consideringtheform(7.8)
1.DN(θ)isevenand2πperiodic,
2.
3.
,and
.
Fromtheseproperties,asN→∞,DN(0)→∞whilethearea
remains
constant.WefoundsimilarbehaviorinSection5.1fortheheatkernelΦ(x,t)ast
→0+.Consequently,asN→∞,wemightexpectthatDN(θ)approachestheDirac
deltafunction.Ifso,thenforacontinuousfunctionf,
However,incontrasttotheheatkernel,DN(θ)doesnotapproachzeroforθ≠0,
asN→∞.Instead,totakethelimit,weappealtotheRiemann-LebesgueLemma.
Lemma7.3.(TheRiemann-LebesgueLemma)Iff∈L2(a,b),then
WhenfisC1on[a,b],theresultfollowseasilybyintegrationbyparts,since
both|f|and|f′| are bounded, and integrating sin βx introduces a factor of 1/β.
Theproofiscompletedbyapproximatingfbysmoothfunctions,asinTheorem
7.2.Detailscanbefoundinmanyanalysistexts[38].
ProofofconvergenceofSN(x)tof(x).ReturningtothepartialsumsSN(x),we
useproperty(2)ofDN(θ)towrite
where
fromL’Hôpital’sruleatθ=0.Sincegiscontinuous,wecanapplytheRiemannLebesgueLemma7.3,whichestablishesthat|SN(x)−f(x)|→0asN→∞.
With a bit more care, we can prove pointwise convergence for piecewise
smoothperiodicfunctions.Theseareperiodicfunctionsfthat,ineverybounded
interval, are continuously differentiable apart from a finite number of points
whereeitherthefunctionoritsderivativehasajumpdiscontinuity(meaningthe
leftandrightlimitsexistbutarenotequal).
Theorem7.4.Letfbepiecewisesmoothand2πperiodic.ThentheFourierseriesforf
convergesateachxtotheaverageoftheleftandrightlimitsoff:
where
Note that at points x of continuity, the theorem guarantees that the series
convergestof(x).
Proof.Usingthenotationdefinedinthetheorem,weestimatetheerrormadeby
theNthpartialsum.Letxbeaspecificpoint,−π<x≤π.Then
Inthiscalculation,wehavesplittheintegral,andusedthefactthatPN(θ)iseven
inθ,followedbythetriangleinequality.Notethattheintegrandineachintegral
iscontinuousatθ=0.
To show that each integral approaches zero, we use the Riemann-Lebesgue
Lemma 7.3. Mimicking the development of (7.9) and (7.10), we have to check
thatthecorrespondingg(θ)heresatisfiestheconditionsofthelemma.Consider
thefirstintegral.Parallelingthef∈C1case,welet
Thengispiecewisecontinuous,andhenceintegrable,on−π≤θ≤0.Asimilar
constructionappliestothesecondintegral,I+.
7.4.UniformConvergenceofFourierSeries
Theorem 7.4 shows that a Fourier series can converge pointwise even if the
function it represents is not smooth everywhere. This can be important for the
study of PDE. For example, solutions of the wave equation describing the
vibrationsofaviolinstringthathasbeenpluckedatonepointarecontinuousbut
not differentiable. Discontinuities in the derivative propagate along
characteristicsandarereflectedattheendsofthestring.Suchasolutioncanalso
berepresentedbyaFourierseriesthatconvergespointwise.
InthissectionweproveinTheorem7.5thatthepointwiseconvergenceofthe
previous section is in fact uniform if f is sufficiently smooth. This result uses
Bessel’s inequality, which will also arise in the context of L2 convergence. Let
beasetoforthonormalfunctions,meaningthat(vj,vk)=δjk,j,k
=1,2,….
For
asequenceofrealnumbers,themean-squareerrorEN
obtainedfromapproximatingf by the linear combination
isdefinedto
be
Then
Completingthesquare,wefind
Consequently,EN is minimized by choosing the coefficients cn to be the Fourier
coefficients,definedinthisgeneralcontextby
Withthischoice,andsinceEN≥0,wealsoconclude:
whichisknownasBessel’sinequality.Intermsofthecoefficientsof(7.11),
Thus,theinfiniteseries
isconvergentandthelimitisnolargerthan||f||2.
Inparticular,wehavetheimportantresultthat
aresultcloselyrelatedtotheRiemann-LebesgueLemma7.3.
We are ready to prove the following theorem on uniform convergence of
Fourierseriesforsmoothfunctionsf.TheessenceoftheproofistoapplyBessel’s
inequalitytogetaboundontheFouriercoefficientsforthederivativef′.Thisis
enoughtogiveastronguniformestimateontheFourierseriesforf.
Theorem7.5.Letf:R→RbeaC12π-periodicfunction.ThentheFourierseriesoff
convergesuniformly.
Proof. Since f′ is continuous, it is integrable on [−π, π]. Therefore, f′ has a
Fourierseries.Moreover,sincefisperiodic,theconstanttermintheseriesforf′
iszero.Thus,
andwehaveBessel’sinequality(7.12):
Therefore,
.
Not surprisingly, the Fourier coefficients an, bn of f are related to the
coefficientsAn,Bnoff′byintegration:
sothat
andsimilarly,
Toshowuniformconvergence,weverifytheCauchycriterion,forwhichwe
estimatehowclosethepartialsumsbecome:
UsingtheSchwarzinequality,wecanmakethissumsmallasp,q→∞:
Remark. The integration by parts in (7.14) to relate Fourier coefficients of the
function to those of derivatives generalizes to higher derivatives when they are
available. Since each integration pulls down another factor , we see that the
regularity of the function is closely linked to the rate of convergence of the
Fouriercoefficients.
7.5.ConvergenceinL2
ForPDE,itisoftennaturaltoworkinL2spaces,usingboththeinnerproductand
completeness.
Lemma7.6.Letf∈L2(a,b),andsuppose
is an orthonormal set in L2(a,b).
Thentheseries
isconvergentinL2and
Proof.FromBessel’sinequality,weseethattheseriesofcoefficients
isconvergent.Thus,theseriessatisfiestheCauchyconditionforconvergence,so
thatusingorthogonality,wehave
That is,
is a Cauchy sequence in L2(a, b). Completeness of the
spacemeansthattheseriesconvergestoanL2(a,b)function.Moreover,applying
orthogonalityandBessel’sinequalityagain,wehave
Nowthatweknowtheinfiniteseries
isconvergent,wewouldlike
toknowwhetheritconvergestofinL2(a,b).Anecessaryconditionisthatiffis
orthogonaltoallthevn,thenf=0.Thatis,(f,vn)=0forallnimpliesf=0.In
fact,wecannowprovideasimpleproofthatnotonlyisthisconditionnecessary
andsufficient,butitisequivalenttohavingequalityinBessel’sinequality.
Theorem7.7. If
is an orthonormal set in L2(a, b), then the following three
conditionsareequivalent:
1.If(f,vn)=0foralln,thenf=0.
2.Foreachf∈L2(a,b),
.
3.Parseval’sidentityholdsforeveryf∈L2(a,b):
Proof.Wewritetheproofinthreeparts,showingcondition1⇒2⇒3⇒1.
Condition1⇒2:Letf∈L2(a,b).ThenLemma7.6showsthat
converges.Toseethatitconvergestof,considerthefunction
Thengisorthogonaltoeveryvk:
Thus,bycondition1,wehaveg=0.
Condition2⇒3:Suppose
.Thenorthogonalityof{vn}implies
Condition3⇒1:SupposeParseval’sidentityholdsforf∈L2(a,b).If(f,vN)=
0foralln,then||f||=0,whichimpliesf=0.
A given orthonormal set of functions
is complete if any of the three
conditionsofthetheoremhold.Inthiscase,condition2meansthatthesetisan
orthonormalbasisforL2(a,b).
Example 3. (An orthonormal basis) The set
is an
orthonormal basis for L2(−π, π). This can be proved by approximating f ∈
L2(−π, π) by a continuous function h on [−π, π]. Then h is approximated
uniformly by the partial sums of its Fourier series, hence also in L2. Using
orthogonalityandthetriangleinequality,wecanshowthatfisapproximatedby
thepartialsumsofitsFourierseries.
7.5.1.MoreGeneralResultforSturm-LiouvilleProblems
Itisconvenienttostate(withoutproof)twofurtherpropertiesoftheeigenvalue
problem, in addition to those of Theorem 7.1. Recall that L is the differential
operatorassociatedwiththeSturm-LiouvilleproblemofSection7.1.
Theorem7.8.LetLbesymmetric.Then
1.Theeigenvaluesformacountablesetthatisboundedbelowbutnotabove:
Eacheigenvaluehasfinitemultiplicity,λ1isasimpleeigenvalue,andλn→∞asn
→∞.
2.TheeigenfunctionsvncanbechosentoformacompleteorthonormalsetonL2(a,b).
For a proof of this result, see Strauss [45]. Note that part 2 of the theorem
tellsusthatFourierseriesofL2(a,b)functionsconvergeinL2.
Parseval’s identity establishes that the correspondence between the L2
functionfanditssequenceofFouriercoefficients{cn},isanisometry,inthesense
thatthemappingf→{cn}preservesthenorm.
7.5.2.GibbsPhenomenon
TheFourierseriesofadiscontinuousfunctiondoesnotconvergeuniformly,even
thoughitconvergespointwise.TheGibbsphenomenon1istheobservationthatat
eachdiscontinuity,theFourierseriesovershootsthefunctionbyroughly9%.The
overshootappearsasapersistentlyexcessivemaximumintheoscillatingpartial
sumsoftheFourierseries.
Withoutlossofgenerality,wecanlookatthisphenomenonbyconsideringa
simplediscontinuousfunction,namely,astepfunction.Althoughthisseemslike
aspecialexample,itinfactembodiestheessenceofhowtheGibbsphenomenon
ismanifested.Thisisbecauseapiecewisecontinuousfunctioncanbewrittenas
thesumofasmoothfunctionandaweightedsumofafinitenumberofHeaviside
functions.
Letfbethe2π-periodicextensionofthefunction
Sincefisodd,ithasaFouriersineseries,whichweeasilycalculate,andbythe
pointwiseconvergencetheorem,wehave
wherebn=(2/nπ)(1−(−1)n),n=1,2,….Thus,
WegetagoodideaoftheGibbsphenomenonbygraphingthepartialsumsSN(x)
formoderatelylargeN.(SeeFigure7.1.)Notethatthegraphy=SN(x)(Nodd)
crosses the line y = 1 exactly N + 1 times in the interval (0, π), and the
maximumappearstobeatapproximately
(andatπ−xN.)Infact
Figure7.1.TheGibbsphenomenon.ThepartialsumS21(x)ofFourierseries
(7.16)isshown.
(seeStrauss[45]).Thus,SN(x)overshootsf(x)=1by0.18,orapproximately9%
ofthejumpinf(x)acrossx=0.Incidentally,thisexampleshowsthedangerin
reversingtheorderoftwolimits:
7.6.FourierTransform
Fourier transforms and other transforms, such as the Laplace transform, are
usefultoolstotransformequationsordataintoformsthatareeasiertoanalyze
or solve. In this section we explore some of the properties of the Fourier
transform, drawing comparisons to Fourier series and showing how the Fourier
transformisusedtosolvecertainPDE.
The Fourier series of a periodic function, say of period 2π, decomposes the
function into pieces each with a different integer frequency k, with the Fourier
coefficientsbeingtheamplitudeatthatfrequency.Thispointofviewisclearest
inthecomplexform
withthe(complex)amplitudesckgivenby
Of course, for a real-valued function f,
. We can regard ck as a function
fromthesetZofintegerstoC,thecomplexnumbers.Theperiodicfunctionfhas
been transformed into the sequence
. Then the series (7.17) is the inverse
transform;itgetsusfromthesequence
backtof.Wehaveseenhowthis
transform to Fourier coefficients and back again is useful when solving PDE on
boundeddomains.TheFouriertransformisasimilartechniqueforfunctionson
unboundeddomains.
TheFouriertransformofafunctionf:R→Rthatisabsolutelyintegrable,
isafunction
thatlooksabitlike(7.18):
Thus,
isthecomplexamplitudecontributiontothefunctionfatfrequencyω.
Wereconstructthefunctionffromtheseamplitudeswiththeinversetransform:
Notice that the infinite sum in (7.17) has been replaced by an integral. In the
Fouriertransform,thefrequencyisacontinuousvariableω,andthereisasign
changeintheexponentialbetweenthetransformandtheinversetransform.
Wehavealsoplacedthefactor1/(2π)intheinversetransformratherthanin
thetransform.Amoresymmetricplacementcommonlyusedistohaveafactorof
in both the transform and its inverse. It is also consistent to have the
1/(2π) in only the transform. Of course, just as for the convergence of Fourier
series to the periodic function f(x), the identity (7.20) requires proof. You can
findtheproofinstandardtextsonFourieranalysis(forexample,seeStade[44]);
our purpose in this section is to illustrate how the transform is used. Along the
way,wefindsomeinterestingproperties.
One more bit of intuition about the Fourier transform. Consider an
experimentthatisperformedoveralongtime,andtheresultisameasurement
f(t),t0<t<t1(knownasatimeseries)thatisverynoisy.Themeasurementwill
be at discrete values of time t, but for the purpose of discussion (and you can
imaginehowtomakesenseofthis),weregardfasafunctionoft,andextendfto
thewholerealline.NowtaketheFouriertransformtogetanewfunction
,
andplotthegraphof .Recallthatthetransformrecordstheamplitudeoffat
each frequency, so if the graph of shows clear maxima at a finite number of
frequencies ωn,n = 1, …, N, then the function is dominated a combination of
periodic functions at those frequencies, with the amplitudes given by
.Thisisacentralideainsignalprocessing,wheresomesensecan
be made out of a noisy signal—the time series—by investigating the Fourier
transform.
AnothertransformyoumaybefamiliarwithistheLaplacetransform,which
ismostusefulinthecontextofinitialvalueproblemsforODEandPDE,because
itisdefinedforintegrablefunctionsonthehalf-line[0,∞).Othertransformsare
useful for different contexts. For example, the Radon transform of functions in
twoorthreedimensionsisusedintomographytoexpressthefunction(typically
adensity)intermsofscatteringdatathatmightcomefromascan,suchasanxray,ultrasound,ormagneticresonanceimaging.
Example4.(Computingatransform)Toshowhowthetransformiscomputed
inasimpleexample,letf:R→Rbedefinedby
Then
To apply the transform to differential equations, we transform derivatives
usingintegrationbypartsandobservingthatf(±∞)=0(since|f|isintegrable
onR),toeliminatetheboundaryterms:
Example5.(TheCauchyproblemfortheheatequation)ConsidertheCauchy
problemfortheheatequation
Thesolutionu=u(x,t)wasobtainedfromthefundamentalsolutioninSection
5.2. Here we show how the same formula can be obtained using Fourier
transforminxforeacht.
The solution of (7.21) by Fourier transform depends on some further
propertiesofthetransform.Iff,gareinL2(R),then
Thecompactnessofthisformulaisthereasonforputtingthe1/(2π)normalizing
factorintheinverseratherthaninthetransform.Theproofof(7.22)involvesa
simplechangeofvariablesz=x−y:
Wealsohave
We transform everything in (7.21) with respect to x, using
transformut:
Thus,
that
to
.Leta=1/(4kt)in(7.23).Then(7.22)and(7.23)imply
ThisisexactlytheformulathatwefoundinSection5.2.
PROBLEMS
1. Let Lu(x) = a(x)u″(x) + b(x)u′(x) + c(x)u(x), where a, b, c are given C2
functions on the interval 0 ≤ x ≤ 1. Consider C2 functions u, v on [0, 1]
satisfying Dirichlet conditions u = v = 0 at x = 0, 1. Find an ordinary
differentialoperatorL∗suchthat
2.Letr:[a,b]→(0,∞)beC2,andconsidertheeigenvalueproblem(7.1).Using
theweightedL2 space with inner product (7.2), prove that eigenvalues are real
andthecorrespondingeigenfunctionsareorthogonal.
3.Letthepiecewisecontinuousfunctionf:R→Rbedefinedby
Write f(x) as a sum of a continuous function and a linear combination of
Heavisidefunctions.
4. Fill in the details of Example 3 in Section 7.5: prove L2 convergence of the
FourierseriesofanL2function,usingapproximationanduniformconvergence.
5.Letf,g:R→R be differentiable and have the property that they and their
firstderivativesareabsolutelyintegrableoverR.Provethat
6. Consider the Fourier transform of a band-limited function f(x), in which
.
(a)Showthat
for|ω|≤π.
(b) Hence deduce that f(x) only depends on the sequence {f(n)} of sampled
valuesoff:
7.ThepointwiseconvergenceofFourierseriescanbeusedtodeducethesumsof
certainkindsofinfinitesequences.
(a) Start with calculating the Fourier series of the even 2π-periodicfunction
f(x)=|sinx|.UsetheFourierseriestocalculatethesums
(b)Chooseasuitableeven2-periodicfunctionf(x)whoseFourierseriesallows
youtodeterminethesums
1.JosiahWillardGibbs(1839–1903)mademanycontributionstothermodynamics,statisticalmechanics,
and other areas of science and mathematics, including vector analysis. The Gibbs phenomenon was
discoveredandexplainedbytheEnglishmathematicianHenryWilbrahamin1848;nearly50yearslaterit
wasfoundindependentlybyGibbs.
CHAPTEREIGHT
Laplace’sEquationandPoisson’s
Equation
In this chapter we consider Laplace’s equation and its inhomogeneous
counterpart,Poisson’sequation,whichareprototypicalellipticequations.1They
may be thought of as time-independent versions of the heat equation, with and
withoutsourceterms:
WeconsidertheseequationsinadomainU⊂Rn,n≥2,butalsoonallofRn.
Applicationsoftheseequationsincludetheclassicalfieldofpotentialtheory,of
importance in electrostatics and steady incompressible fluid flow. In
electrostatics,f(x)inPoisson’sequationrepresentsachargedensitydistribution,
inducingtheelectricpotentialu(x).Intwo-dimensionalsteadyfluidflow,uisthe
velocitypotentialorstreamfunction,bothofwhichsatisfyLaplace’sequation.
SeveralpropertiesofsolutionsofLaplace’sequationparallelthoseoftheheat
equation: maximum principles, solutions obtained from separation of variables,
andthefundamentalsolutiontosolvePoisson’sequationinRn.
8.1.TheFundamentalSolution
To solve Poisson’s equation, we begin by deriving the fundamentalsolutionΦ(x)
fortheLaplacian.Thisfundamentalsolutionisratherdifferentfromtheonefor
the heat equation, which is designed to solve initial value problems, and
consequentlyhasasingularityattheinitialtimet=0.Thefundamentalsolution
fortheLaplacian,beingtimeindependent,isusedtorepresentsolutionsinspace
alone.Todothis,Φ(x)ischosentohaveasingularityatapointx0inthedomain;
since the Laplacian is translation invariant, we can take x0=0. Moreover, the
Laplacian is invariant under rotations, so we can seek a rotationally invariant
fundamentalsolution.
Motivatedbythisdiscussion,weseekrotationallysymmetricsolutionsu(x)=
v(r),r=|x|,ofLaplace’sequation.Thenwehave
Therefore,
Integrating,weobtainlogv′=−(n−1)logr+C.Thatis,
,whereA=
logC is the constant of integration. Integrating again, we get a two-parameter
familyofsolutions
ThefundamentalsolutionΦ(x)isdefinedbysettingb=0andchoosingthe
constantatonormalizeΦ(x)dependingonthevolumeα(n)oftheunitball:
The purpose of the normalization is to make the formula for the solution of
Poisson’sequationonRnassimpleaspossible.(See(8.1).)NotethatalthoughΦ
hasanintegrablesingularityattheorigin(Φisintegrableonboundedsets,even
thoughitisnotdefinedatx=0),wewillseethatthesingularityof−ΔΦisnot
integrable,andisinfactthesingulardistributionδ,whichwedefinecarefullyin
Section9.2.
WesayafunctionuisharmonicinanopensetU⊂Rnifu∈C2(U)andΔu(x)
= 0 for each x ∈ U. By construction, Φ(x) is harmonic in every open set not
containingtheorigin.
8.2.SolvingPoisson’sEquationinRn
InthissectionweestablishaformulaforthesolutionofPoisson’sequationinall
space Rn using the fundamental solution, much as we used the heat kernel to
solvetheCauchyproblemfortheheatequation.
Letf∈C2(Rn)havecompactsupportanddefine
theconvolutionproductofΦwithf.
Remark. Note that ΔΦ(x) has a nonintegrable singularity at the origin.
Therefore,wecannotdifferentiateundertheintegralsigninthisformula.Ifwe
could, we would have Δu(x) = 0, since Φ is harmonic away from the origin.
However,aswenowshow,thenonintegrablesingularitymakestheconvolution
productworktosolvePoisson’sequation.
Theorem8.1.Iff∈C2(Rn)hascompactsupportandu(x)isgivenby(8.1),then
Proof.Changingvariablesin(8.1),wehave
Therefore,
Inthisintegral,andinsubsequentcalculations,weusesubscriptstoindicatethe
variableofdifferentiationorintegration.Thus,ΔyindicatesthattheLaplacianis
takenwithrespecttotheyvariables.WewouldliketointegratebypartstoputΔ
backonΦ(y).However,Φ(y)hasasingularityaty=0,sowehavetotreatthe
integral as an improper integral. For ϵ > 0, let
, where B(x, r)
n
denotestheopenballofradiusrcenteredatx∈R .Then
Thefirstintegralapproacheszeroasϵ→0,sinceΦisintegrableattheorigin.We
use Green’s second identity on the second integral, observing that (since f has
compact support) the integrand is identically zero outside a large enough ball
centeredatx.Thus,
Incidentally,thereisnocontributionfromtheboundaryofthesupportoff,since
theintegrandiszerothere.ThefinalintegraliszerosinceΦisharmonicinUϵ.
Theintegralontheboundaryhastwoterms.Thefirsttermconvergestozeroasϵ
→ 0, since Φ is integrable at the origin. We need to prove that the second
integralconvergestof(x).
SincetheunitnormalνisoutwardwithrespecttoUϵ,onthesphere∂B(0,ϵ)
wehaveν=−y/ϵ.Therefore,sinceΦ(y)isafunctionofr=|y|,
Figure8.1.AboundedregionU.
Notethatthisformulaholdsforn≥2eventhoughtheformulaforΦisdifferent
forn=2.Thus,
where |∂B(0, ϵ)| = nα(n)ϵn−1 denotes the measure of the sphere in Rn. This
integralconvergestof(x),becausefiscontinuous.
8.3.PropertiesofHarmonicFunctions
In this section we state and prove the mean-value property of harmonic
functions, and use it to prove the maximum principle, leading to a uniqueness
result for boundary value problems for Poisson’s equation. We state the meanvaluepropertyintermsofintegralaverages.
Theorem8.2.(Mean-ValueProperty)Supposeu∈C2(U).ThenuisharmonicinUif
andonlyifithasthemean-valueproperty:
foreveryballB(x,r)⊂U.
Proof.Supposeuisharmonic.Then,forB(x,r)⊂U(seeFig.8.1),
Nowlet
Sincelimr→0ϕ(r)=u(x),wecompletetheproofbyusing(8.2)toshowthatϕ(r)
isconstant.Todoso,wecalculatedirectlythatϕ′(r)=0.Lety=x+rz,z∈
B(0,1),tofacilitatedifferentiatingtheintegral.Then
by(8.2).Thus,ϕ(r)isconstant,sothatϕ(r)=lims→0ϕ(s)=u(x).
WeusethisresulttoobtaintheintegralaverageovertheballB(x,r):
Conversely,supposeuhasthemean-valueproperty.Then,asabove,weget
sinceϕ(r)=u(x)isconstant.Thus,
Δu(x)=0,sinceΔuiscontinuousinU.
,andlettingr→0,weobtain
The mean-value property of harmonic functions is peculiar to solutions of
Laplace’s equation and has no counterpart for more general elliptic equations.
However, it simplifies the proofs of key results that do generalize, in particular
themaximumprinciple,whichwenowstateinbothweakandstrongforms.
Theorem 8.3. (The maximum principle) Let U ⊂ Rn be open and bounded, and
suppose
isharmonicinU.Then
1.Weakform:
.
2.Strongform:IfUisconnected,theneitheru=constantin ,or
Proof. We prove the strong form first. The weak form then follows easily.
SupposeUisconnectedandthereisapointx0∈Usuchthat
ChoosersothatB(x,r)⊂U.Thenbythemean-valueproperty,
Butu(y)≤Meverywhere,soequalityimpliesthatu(y)=MthroughoutB(x0,r).
Thus,thesetS={x∈U:u(x)=M}isnonemptyandopen.However,Sisalso
relatively closed in U: Let xn ∈ S converge to x ∈ U as n → ∞. Then, by
continuity of u, we have u(x) = limn → ∞ u(xn) = M, so x ∈ S. But the only
nonemptyopenandclosedsetinUisUitself,sowehaveS=U,implyingthatu
isconstantin .Theweakform1followsfrom2.
Remarks. The corresponding minimum principle follows by applying the
maximumprincipleto−u.
IfUisnotconnected(i.e.,U=U1∪U2withU1,U2disjointopensetsinRn),
then the weak maximum principle holds, but the strong maximum principle
breaks down. To see this, define u(x) = k, for x ∈ Uk, k = 1, 2. Then u(x) is
harmonicbutfailstosatisfyeitherconclusionofpart2ofTheorem8.3.
We can apply the maximum principle to the Dirichlet problem, which is the
followingboundaryvalueproblemonaboundedopensetU⊂Rn:
Theorem 8.4. Suppose g is continuous and
is a solution of the
Dirichletproblem.IfUisconnectedandgsatisfiesg(x)≥0forallx∈∂U,andg(x)
>0,forsomex∈∂U,then
Proof.Fromtheweakminimumprinciple,wehave
.Butthestrong
versiongiveseitheru(x)>min∂Ug,forallx∈U,oru(x)=constant.Ineither
case,theconclusionfollows.
8.3.1.UniquenessofSolutionsofBoundaryValueProblems
As with the heat equation, we can prove uniqueness of solutions of boundary
valueproblemsfromthemaximumprincipleorfromenergyconsiderations.LetU
⊂Rnbeopenandbounded.ConsidertheboundaryvalueproblemwithDirichlet
boundaryconditions:
wheref∈C(U),g∈C(∂U).
Theorem8.5.There is at most one solution
problem(8.3).
of the boundary value
Proof.Letu1,u2bothbesolutions.Applyingtheweakmaximumprincipletou1
−u2andtou2−u1,bothofwhichsatisfy(8.3)withf=0,g=0,wededuce
thatu1=u2.
Alternatively,theenergyapproachsetsu=u1−u2andappliesaversionof
Green’sidentity:
Thus,∇u=0inU,sothatuisconstant.Sinceu=0on∂U,weconcludethatu
=0in ,sothatu1=u2.
8.4.SeparationofVariablesforLaplace’sEquation
IfthedomainU⊂Rnhasspecialgeometry,thenseparationofvariablescanwork
on Laplace’s equation. Examples include rectangular, spherical, and cylindrical
domains.Herewetreattwoexamplestoillustratethedifferencesfromtheheat
andwaveequations.Wethenmakesomeremarksaboutotherdomains.
Of course, if the boundary conditions are homogeneous, then u = 0 is a
solution, generally the only solution. So it is more natural to consider linear
boundaryconditions
on∂Uthatareinhomogeneousoveratleastpart
oftheboundary.Ifα≠0,thentheenergymethoddiscussedinSection8.3above
canbeusedtoprovethatthisproblemhasatmostonesolution.
8.4.1.Laplace’sEquationinaRectangle
WeconsiderLaplace’sequationΔu=0inarectangulardomainU=(0,a)×(0,
b) ⊂ R2 with a mixture of Dirichlet and Neumann boundary conditions,
representingpartsoftheboundarywherewespecifyeitherthetemperatureuor
theheatflux,whichisproportionaltothenormalderivative.
In this problem we can formulate an eigenvalue problem if boundary
conditions on opposite sides of the rectangular boundary are homogeneous. We
use this observation to implement a solution strategy. We split the boundary
value problem into four problems, setting the boundary condition to zero on
three sides in each problem. To illustrate the process, consider the example in
Figure8.2.Forj=1,2,3,4,let(Pj)betheproblemwithfk=0,k≠j,andletuj
bethesolutionof(Pj)Thenbylinearity,thesolutionofthefullproblemis
Wesolve(P4)indetailtoillustratethisapproach.
Figure8.2.ExampleofaboundaryvalueproblemforLaplace’sequation.
Letu=u4=v(x)w(y). From the boundary conditions, we guess v(x)=sin
nπx/a,sothatu4(x,y)=sin(nπx/a)w(y).ThenΔu(x,y)=0leadstoanODEfor
w(y):
withgeneralsolution
Weneedtosatisfyahomogeneousboundaryconditionaty=b,
Thus,
Nowwecanformaseriestosatisfythenonzeroboundaryconditionaty=0:
Ony=0,
fromwhichwegetformulasforthecoefficientsAn:
Thesolutionu4isthengivenbytheseries(8.4).Similarly,wecanobtainu1,u2,
u3, and finally put these series together to get the solution u of the original
problem.Notethattheseriesforu2isasineserieslike(8.4),buttheseriesforu1
andu3havetheform
becauseofthecombinationofhomogeneousboundaryconditionsaty=0,b.
8.4.2.Laplace’sEquationonSphericalandCylindricalDomains
Insphericalandcylindricaldomains,itisnaturaltousecurvilinearcoordinates
(i.e., polar coordinates and cylindrical coordinates, respectively). Since the
Laplacian in these coordinates has nonconstant coefficients, the ODE that result
will also have nonconstant coefficients. Moreover, the dimension of the space
makesadifferencetothetypeofequationthatresults.Thisleadstothestudyof
special functions, specifically, Legendre functions (solutions of Legendre’s
equation) and Bessel functions (solutions of Bessel’s equation). These special
functionsaretypicallyexpressedasseriessolutionsofODE,usingthemethodof
Frobenius. Some details may be found in the PDE book of Strauss [45], in
engineering mathematics books, such as Jeffery [26] and Kreyszig [30], and in
textsthattypicallyhave“PDE”and“BoundaryValueProblems”intheirtitles.
HerewegivethedetailedsolutionofLaplace’sequationinadisk,leadingto
Poison’s formula, a representation of the solution as an integral, which we
eventually interpret in terms of Green’s functions. The disk has the advantage
thatwedonotneedspecialfunctionstosolvetheeigenvalueproblem.
ConsidertheDirichletprobleminadiskofradiusa>0:
Inplanepolarcoordinates,
wehavethefollowingproblemforu=u(r,θ):
Theboundary∂Uisthecircler=a,whereastheboundariesforthevariablesr,θ
alsoincluder=0,θ=0,2π.Atthediskcenter,r=0,weexcludenonphysical
solutionsbyinsistingthatsolutionsremainboundedasr→0+.Theboundariesθ
=0,2πrepresentthesamelineinthedisk,acrosswhichthesolutionshouldbe
as smooth as elsewhere in the domain. These boundaries are accommodated by
makingthesolution2πperiodicinθ.Similarly,theboundaryfunctionfistreated
asa2π-periodicfunctionofθ.
Let
SubstitutingintothePDE(8.5),weobtain
whence,separatingrfromθ,
WethenhaveaneigenvalueproblemforH,inwhichtheboundaryconditionis
thatH(θ)is2πperiodic:
ThecorrespondingequationforR(r)is
We can solve the eigenvalue problem (8.6), with the result H=H0 = A0/2 =
const.,forλ=0,and
Notethateacheigenvalueλn,n≥1,hastwoindependenteigenfunctions.Setting
λ=n2in(8.7),weget
Forn=0,thegeneralsolutionof(8.8)is
However, we seek solutions that are bounded as r → 0 so we set D0 = 0 and
consideronlythesolution
ThearbitrarymultipleC0willbeincorporatedintoA0.
Forn≥1,weseeksolutionsR(r)=rα.Substitutinginto(8.8),wefindα=
±n.However,r−nisunboundedattheorigin,soweretainonly
Again,thearbitrarycoefficientmultiplyingthissolutionwillbeincorporatedinto
Hn(θ).
Sofar,wehavesolutions
ThesefunctionsareharmonicintheballB(0,a),andtheyreducetofunctionsof
θaloneforr=a.
Weformaseries
:
andsetr=atosatisfytheboundaryconditionu(a,θ)=f(θ).Thus,
Consequently,anAn,anBnareFouriercoefficientsofthe2π-periodicfunctionf:
Ifwesubstitutethecoefficientsgivenby(8.10)backintotheseries(8.9),weget
thesolutionuintermsofthedata,andwecansumtheseries,justaswesummed
theseriestogettheDirichletkernelinprovingpointwiseconvergence.Thus,
Aftersomemanipulationofthesumofthegeometricseries
weobtainPoisson’sformulaforthesolutionoftheDirichletprobleminadisk:
ThisintegralhastheformofaconvolutionproductofthePoissonkernel
withtheboundarydataf(ψ)=u(a,ψ).Theformulareducestothemean-value
propertyofharmonicfunctionswhenr=0.Inthespecialcasef≡1,wehave
thesolutionu=1,fromwhichweconcludethat
Notethatyoucouldalsohaveguessedthisbyintegratingtheseriestermbyterm.
Just as for fundamental solutions, which are singular integral kernels, the
Poissonkernel,P(r,θ−ϕ)issingularattheveryplacethefunctionu(r,θ)isto
beevaluatedontheboundary:r=a,θ=ϕ. The singularity is needed for the
convolutiontoconvergetotheboundarydata:forfcontinuous,
Figure8.3.GeometricinterpretationofPoisson’sformula.
A more geometric interpretation of Poisson’s formula generalizes to higher
dimensions.Considerpolarcoordinatesfor
Thenwehavea2−r2=|x′|2−|x|2,and|x′−x|2=r2+a2−2arcos(θ−
ϕ)(seeFig.8.3).Thus,
The Poisson kernel is an example of a Green’s function, which we study in
detailinthenextchapter.
PROBLEMS
1.Provetheweakmaximumprinciple(Theorem8.3,part1)usinganargument
similartotheproofusedforthisprinciplefortheheatequation(Theorem5.2).
2. Prove the weak maximum principle (Theorem 8.3, part 2) from the strong
form.
3. Consider Poisson’s equation on a bounded open set U ∈ Rn with Robin
boundaryconditions
(a) Prove that if α > 0, then the energy method can be used to show
uniquenessofsolutions
.
(b)Forα=0,showthatsolutionsareuniqueuptoaconstant.
(c) Design an example to show that uniqueness can fail if α < 0. (Hint:
Choosen=1.)
4.DerivePoisson’sformula(8.11)bysummingtheseriesforu(r,θ).Providethe
details.
5.InRnletVr=|B(0,r)|=α(n)rn,Sr=|∂B(0,r)|.Explainwhy
6.Supposeu∈C2(U)hasthemean-valueproperty:
For all x ∈ U,
for all r > 0 such that B(x, r) ⊂ U. Write a
carefulproofbycontradictionthatΔu=0inU.
7.SupposeU⊂Rnisopen,bounded,andconnected,and
satisfies
Provethatifg∈C(∂U),g(x)≥0forallx,andg(x)>0forsomex∈∂U,then
1.Pierre-SimonLaplace,1749–1827,mademanycontributionstomathematics,physics,andastronomy.
Simeon Denis Poisson, 1781–1840, was a mathematician and physicist known for his contributions to the
theoryofelectricityandmagnetism.
CHAPTERNINE
Green’sFunctionsandDistributions
Green’sfunctions,integralkernelsthatallowlinearboundaryvalueproblemsto
be expressed as integral equations, appear in many contexts.1 The theory and
construction of solutions for the basic linear PDE of mathematical physics use
Green’s functions or fundamental solutions. For example, in linear elasticity, a
Green’sfunctionisthedisplacementoftheelasticmaterial(suchasrubber)due
toapointforce,whereasinelectrostatics,theGreen’sfunctionrelatesanelectric
fieldtoapointcharge.NumericalanalystsuseGreen’sfunctionsasaconvenient
way to convert PDE into integral equations that can be solved with numerical
integration.
InthischapterweshowhowtheGreen’sfunctionisasolutionofaPDEonly
inageneralizedsense.Thissenseiselegantlyexpressedintermsofthetheoryof
distributions, which we introduce in this chapter as a separate topic, before
returning to the construction and properties of Green’s functions for the
Laplacianandotherlineardifferentialoperators.
9.1.BoundaryValueProblems
Wewouldliketostudyboundaryvalueproblems,suchasthefollowing:
whereU⊂Rnisopenandboundedwithsmoothboundary∂U.
From Section 8.2 we see how to solve Poisson’s equation on all space by
writing the solution as a convolution with the fundamental solution Φ (see
Theorem8.1):
The purpose of Green’s functions is to find a similar formula that takes the
boundary conditions into account. An immediate difficulty is that we have no
reasontosupposethat(9.2)willsatisfytheboundaryconditionu=gon∂U.The
ideaistofirstmodifythefundamentalsolutionsothataformulasimilarto(9.2)
yieldsasolutionoftheinhomogeneousPDEthatsatisfieshomogeneousboundary
conditions (i.e., with g ≡ 0), a construction we explain in Section 9.3. As a
secondstep,wefindasmoothsolutionofthehomogeneousPDE(i.e.,withf≡0)
thatsatisfiestheinhomogeneousboundaryconditionu=gon∂U.Finally,weput
thetwopiecestogether,relyingonlinearity,togiveasolutionoftheboundary
valueproblem(9.1).
Figure9.1.Areaofintegration.Thediagonallineisz=y.
Let’sstartwithproblem(9.1) in one dimension with homogeneous Dirichlet
boundaryconditions:
Wesolvethisproblembyintegratingtwice:
whereC,Dareconstantsofintegration.
Theboundaryconditionu(0)=0leadsimmediatelytoD=0.Theboundary
conditionu(1)=0thengivesanequationforC.Itisconvenienttosimplifythe
double integral by reversing the order of integration in the shaded region of
Figure9.1.
Then
Figure9.2.TheintegralkernelG(x,y).
Theboundaryconditionatx=1becomes
Substitutingbackinto(9.4)andrearrangingterms,wearriveat
Wecanwritethisformulainthecompactform
usingtheintegralkernel
ThegraphofG(x,y)forfixedx∈(0,1)isshowninFigure9.2.
Equation(9.5)definesanintegraloperatorG:
Thedifferentialoperator
operatesonfunctionsuinthespace
XofC2functionsthatsatisfythehomogeneousboundaryconditions.Theintegral
operatorG:C[0,1]→Xgoestheotherway;itactsoncontinuousfunctionsand
gives twice-differentiable functions that satisfy (9.3). In this sense, the integral
operator is the inverse of the differential operator. In fact, G is the Green’s
function for the boundary value problem, because (9.4) satisfies (9.3). Observe
that(9.5)hassomesimilarityto(9.2).
HerearesomepropertiesofG:[0,1]×[0,1]→R:
1.Gisnonnegative:G(x,y)≥0;
2.Gissymmetric:G(x,y)=G(y,x);
3.Giscontinuous;and
4.Gisdifferentiable,exceptonthediagonalx=y.Onthediagonal, hasa
jumpdiscontinuity:
wherethebracketnotation[F]meansthejumpinafunctionF,ordifference
betweentherightlimitandleftlimit.Whilethesepropertiesarespecifictothe
Green’sfunction(9.6)forproblem(9.3),theyhavetheircounterpartsfor
differentdifferentialoperatorsL.
Now
exceptonthediagonal,where
negativeslope.Wewrite
maybethoughttohaveinfinite
whereδ(x)istheDiracdeltafunction.Thefunctionδ(x)isameasurethatassigns
massoneatx=0andzeromasselsewhere.Weshalltreatδasadistribution,or
generalized function, which leads us to the theory of distributions, a useful
frameworkforconsideringPDEandGreen’sfunctions,suchasG(x,y).
It is also useful to relate the Green’s function to the fundamental solution
.Infact,ifwewrite
then
foreachy∈[0,1].Wewritethesuperscriptyinthefunctionϕ
(x) to indicate that y is a parameter. Moreover, since G satisfies homogeneous
boundaryconditions,ϕysatisfiestheboundaryconditions(foreachy∈[0,1]):
Thus, to modify the fundamental solution to obtain the Green’s function, we
constructforeachyasolutionu=ϕyofthehomogeneousequationu″(x)=0
thatsatisfiestheboundaryconditions(9.7).
9.2.TestFunctionsandDistributions
In the previous section we constructed a Green’s function that is not twice
differentiableintheclassicalsensebuthasageneralizedsecondderivativethatis
adeltafunction,anexampleofadistribution.Thespaceofdistributions,defined
inthissection,isusedtobroadenthenotionofsolutionsofPDE,especiallylinear
PDE. We discuss distributions in a variety of contexts in the next couple of
chapters,butthisnarrativebeginswithaspaceofsmoothfunctionsknownastest
functions. Test functions are instrumental in defining distributional solutions of
PDEbyusingintegrationbypartstotransferderivativesfromdistributionsonto
thetestfunctions.
9.2.1.TestFunctions
To study test functions, we introduce some new notation and terminology. For
simplicity,westartbyconsideringsmoothfunctionsonR.Let
denote
thespaceofC∞functionsϕwithcompactsupport,suppϕ.ThenDisthespaceof
testfunctions,withaspecificnotionofconvergencedefinedasfollows.
Wedenotethejthderivativeofϕbyϕ(j).Asequence{ϕn}convergestoϕinD
asn→∞if
1.thereisacompact(closedandbounded)subsetKofRsuchthatsuppϕn⊂K
foralln,andsuppϕ⊂K;and
2.
asn→∞,uniformlyonK,foreachj≥0:
Similarly,thespaceD(Rn)oftestfunctionsonRnisdefinedbyreplacingordinary
derivativesbypartialderivativesinthedefinitionofconvergence.
Example1.(Atestfunction)Let’sconsideranexampleofatestfunctioninRn
thatwewillrefertorepeatedly.
Let
where
dxischosensothat
Toseethatηisatestfunction,wenotethatithascompactsupport{x:|x|≤1},
andithascontinuousderivativesofallorders,evenwherethedefinitionissplit
at|x|=1.Forexample,withn=1,thederivativesapproachzeroatx=1.To
see this, observe that every derivative is the product of a rational function and
theexponential
;theexponentialdominatestherationalfunctionasx→
1,sendingthederivativestozero.
Itissometimesusefultorescaleηusingaparameterϵ>0:
Thenthesupportisscaledbyϵ,whileleavingtheintegralunchanged:suppηϵ=
{x:|x|≤ϵ},and
.Thefunctionηϵiscalledamollifier,becauseitcan
be used to smooth rough functions. The next lemma shows us how this works
usingtheconvolutionproduct.
Lemma9.1.Letf∈C(Rn),anddefine,forϵ>0,fϵ(x)=ηϵfϵ∗f(x).Thenfϵ∈C∞,
andfϵ(x)→f(x)forallx,asϵ→0.
Let
denotethespaceoflocallyintegrablefunctionsonRn:
Then fϵ is defined and is C∞ for
. In that case, it takes a little measure
theory (the Lebesgue-dominated convergence theorem) to prove that fϵ → f
almost everywhere (see Evans [12], Appendix). Thus, the convolution of the
mollifierηϵwiththeonly-once-differentiablefunctionf(x)providesaninfinitely
smoothfunctionfϵ(x).
9.2.2.Distributions
NowthatwehaveestablishedpropertiesofthespaceDoftestfunctions,weare
readytodefinedistributions.EachdistributionfisafunctiononD,meaningthat
f(ϕ)isanumberforeachtestfunctionϕ.
Specifically, the space of distributions D′ is defined to be the space of
continuous linear functionals on the space D of test functions. That is, f ∈ D′
meansf:D→R,andfhasthefollowingproperties:
1.fislinear:f(aϕ1+bϕ2)=af(ϕ1)+bf(ϕ2)foreacha,b∈R,ϕ1,ϕ2∈D;and
2.fiscontinuous:ϕn→ϕinDimpliesf(ϕn)→f(ϕ)(asasequenceofnumbers).
The space of continuous linear functionals on a given topological space X is
calledthedualspaceofXandistypicallydenotedX′.Thus,D′isthedualspace
ofD.Followingtheusualcustom,wedenotef(ϕ)by(f,ϕ).Thenotationshould
notbeconfusedwiththeL2innerproductofintegrablefunctions.
Not every functional is a distribution. For example, if we define f:D → by
f(ϕ)=ϕ(0)2,thenfisacontinuousfunctional,butsinceitisnonlinear,itisnota
distribution.
Example 2. (Four distributions) Here we provide four examples of
distributions.ForeachjinD′,j=1,…,4,testfunctionsϕ∈Dareassignedto
numbers(j,ϕ):
1.(f1,ϕ)=fRϕ(x)dx.
2.
3.(f3,ϕ)=(δ,ϕ)≡ϕ(0).
4.(f4,ϕ)=ϕ′(0).
We leave to the problems verification that these examples are well defined,
linear,andcontinuous.
Many distributions are associated with locally integrable functions. The first
distribution f1 is associated with , which is not integrable on R, but it is in
.Then
.
Theseconddistributionf2isassociatedwiththeHeavisidefunction:
whichisin
and
.
Moregenerally,wehavethefollowinglemma.
Lemma9.2.If
,then definesadistributiong∈D′by
Proof.Linearityofgfollowsfromtheformulainthelemma;continuityfollows
fromtheformulaandanestimate:
asn→∞,wheresuppϕ,suppϕn⊂K.
Wesayadistributionfisregularifithasan
representative :
Thus, regular distributions can be thought of as functions. For example, f2 is a
regulardistribution;itcanbeidentifiedwiththeHeavisidefunction.
However, not every distribution defines an
function. A distribution f is
calledsingularifitisnotregular.Therearemanysingulardistributions;eventhe
deltafunctionf3=δissingular,aswenowshow.Itwillthenfollowthatf4in
Example2isalsosingular.
Lemma9.3.Thedeltafunctionisasingulardistribution.
Proof.Supposeδisregular,sothatthereis
suchthat
Now define a one-parameter family of test functions with parameter ϵ,
specifically,ϕϵ(x)=η(x/ϵ):
Wecalculatetheeffectofδonϕϵintwoways.First,fromthedefinitionofδ:
Second,usingtheassumption(9.9):
However,then
Since the latter integral approaches zero as ϵ → 0, we can choose ϵ > 0 small
enoughthattheintegralontherightislessthane−1,contradictingtheinequality.
We next discuss several general properties of distributions that are useful in
the analysis of PDE, both specific ones (such as Laplace’s equation) and in the
generaltheoryofPDE,includingbothlinearandnonlinearequations.
ConvergenceofDistributions
WeobservedinChapter5thattheheatkernelΦ(x,t)convergestoδ(x)ast→
0+. Here we make this convergence precise by defining what it means for a
sequenceofdistributionstoconvergetoalimitingdistribution.
Let
beasequenceofdistributions,andletf∈D′(Rn).Wesayfk
→f in D′in the sense of distributions if (fk, ϕ) → (f, ϕ) as k → ∞, for all ϕ ∈
D(Rn).
Example3.(Sequenceofdistributions)Letn=1,anddefine,fork=1,2,…:
Then,forϕ∈D(R),
Thus,fk→δinthesenseofdistributionsask→∞.
Example 4. (Convergence of heat kernel Φ(x, t) as t → 0) Let Φ(x, t) =
.ThenΦ(x,t)→δ(x)ast→0+means(Φ(x,t),ϕ(x))→ϕ(0)
ast→0+.
DistributionalDerivatives
Let f ∈ C1(Rn). Then f is differentiable, and each partial derivative is locally
integrableandthereforedefinesadistribution.Wehave
This calculation suggests that for any distribution f we define its distributional
derivative∂f/∂xitobeadistributiongivenby
Theneverydistributionisdifferentiableinthesenseofdistributions,andhencehas
derivativesofallorders.Itisnothardtoshowthat∂f/∂xiisindeedadistribution
bycheckingdirectlythatitiscontinuousandlinear.
Example 5. (Distributional derivative with n = 1) Consider the Heaviside
function H(x). As we saw earlier, H is in , and it acts on test functions by
.Thus,foranytestfunctionϕ,
Therefore,
We can also differentiate the δ function directly from the definition of
distributionalderivative:
Infact,thisisrelatedtoexamplef4inExample2.
Asanotherexample,letf(x)=|x|.Thenbecause
isan function,itdefinesadistribution.Infact,f′=2H−1;consequently,f″
= 2δ. Here we have used the fact that differentiation is a linear operation on
distributions,justasitisondifferentiablefunctions.
Translationbyy∈RnThedeltafunctionδ(x)isadistributionthatplacesunit
massatx=0.Toplacethemassatapointy,wedefineanewdistributionwith
the notation δ(x − y). In general, we can translate any distribution f by a
constantyasfollows.Wedefinethetranslationoff∈D′,byy∈Rn,asanew
distributiong∈D′,forwhich
whereϕ(−y)∈Disthetestfunctiondefinedbyϕ(−y)(x)=ϕ(x−y).Toseethat
thismakessense,wesimplycheckthatitisconsistentforany
:
Thenwecanassociatethedistributiongwiththetranslationf(x+y)ofthe
functionf(x)byy.
Anexamplewherethistranslationisusefulistheδfunction.Weoftenwrite
δ(x−y)tomeanthedistributionδ,translatedby−y,anditiscommontoleave
xintheargumentsofboththedistributionandthetestfunction:
Tomultiplyadistributionf∈D′byafunctionc∈C∞,wedefinecf∈D′by
notingthatcϕisatestfunction.Again,thisdefinitionismotivatedbythecaseof
a regular distribution f, in which case the formula makes sense interpreted as
integrals.
The next two examples illustrate the connection between properties of
distributionsanddifferentialequations.
Example6.(MultiplicationbyaC∞function)Considerc(x)=x,f=δinR.
Then
sothatxδ=0.Sinceδ=H′,wecaninterpretxδ=0assayingthaty(x)=H(x)
isadistributionsolutionofthedifferentialequation
Infact,thegeneraldistributionsolutionofthisequation(whichissingularatx=
0)is
for arbitrary constants a, b. Thus, we have a two-parameter family of
distributional solutions for the first-order equation. This highlights a danger in
enlarging the space of functions in which to define solutions, namely, that
uniqueness of solutions may be lost in the larger space. Uniqueness can be
restoredbyaddingsuitableboundaryorinitialconditions.
Example 7. (Application to shock wave solutions of conservation laws)
Shockwaves,introducedinChapter2,areinfactdistributionalsolutionsofPDE,
whichweillustrateinthisexampleforthescalarconservationlaw
inwhichf:R→RisagivenC1function.Whenweinterpret(9.10)inthesense
ofdistributions,itallowsustogivemeaningtoshockwavesolutions,whichare
discontinuous. The equation simply states that if u ∈ D′(R2) and f(u) can be
interpreted as a distribution, then the combination of distributional derivatives
ontheleft-handsideoftheequationshouldbezerooneverytestfunction.Thus,
foratestfunctionϕ(x,t):
In this way, we can define distribution solutions of differential equations, even
nonlinearequations.Toseethatthisinterpretationhassomesubstance,consider
ajumpdiscontinuity
whereu±andsareconstants.Then
WerewritethesefunctionsusingtheHeavisidefunction:
Thus,
andH′=δ.Equation(9.10)becomes
inthesenseofdistributions.Butthentheconstant−s(u+−u−)+f(u+)−f(u−)
mustbezero:
This equation is the Rankine-Hugoniot condition for shock wave solutions of
(9.10).Itisusefulbecauseitrelatestheshockspeedstotheone-sidedlimitsu±
of the solution on either side of the discontinuity. We will derive it for more
generaldiscontinuitiesinChapter13onscalarconservationlaws.
DistributionsonOpenSubsetsofRn
Itisstraightforwardtoformulatetheaboveideasfordistributionsonsubsetsof
Rn.Thekeystepistodefinetestfunctionsappropriatelyandtouseintegrationby
parts. Let U ⊂ Rn be open. Then we can define the space
of test
functionsonU(i.e.,theC∞functionsonUthathavecompactsupportinU).Then
D′(U)isthespaceofcontinuouslinearfunctionalsonD(U).Nowletu:U→R
be in
. Then u defines a distribution in D′(U). In particular, u is
differentiableinthesenseofdistributions:
We say u has a weak derivative
if the distributional derivative is a regular
distribution. Recall that this means it is represented by a locally integrable
function:
. In this way, we distinguish between the weak derivative
andadistributionalderivative.
9.3.Green’sFunctions
In this section we return to Green’s functions, first giving a general framework
within the theory of distributions and then showing how this applies to the
Laplacian. The primary use of distributions here is related to the delta function
and the notion of fundamental solution for a differential operator. Green’s
functionsprovideameanstoinvertthedifferentialoperatorforboundaryvalue
problems.
9.3.1.GeneralFramework
ConsideralinearpartialdifferentialoperatorL(L=−Δforexample),thatacts
on functions u : Rn → R. As we introduced for one dimension in Section 9.1,
Green’sfunctionsenableustoprovideanintegralrepresentationforsolutionsof
boundaryvalueproblems
HerefisagivenfunctiononU,andgisagivenfunctionon∂U,butnotethatU⊂
Rn may be unbounded. The term Bu represents a linear combination of u and
derivativesofuoflowerorderthantheorderofthepartialdifferentialoperator
L. Associated with L we have the fundamental solution Φ(x, y), which is
requiredtosatisfy
inthesenseofdistributions.Toseewhythisishelpful,considerΦtobelocally
integrable in y for each x ∈ Rn, and let f ∈ D(Rn). Then (using the L2 inner
productnotation,sinceΦandfareintegrable)
satisfiesLv=finRnbecause
Thatis,thefundamentalsolutionisthekeytosolvingtheinhomogeneousPDE.
However,(9.12)doesnotgenerallysatisfytheboundaryconditionBv=g.To
satisfytheboundarycondition,weaddasolutionwofthehomogeneousequation
Lw=0sothatu(x)=v(x)+w(x)satisfiestheboundaryconditionBu=g.But
thenw must satisfy the boundary condition Bw = g − Bv. Thus, provided we
have the fundamental solution Φ(x, y), the boundary value problem (9.11) is
reducedtosolvingtheproblem
Theboundaryconditionforwisabitclumsy;itreliesonapplyingtheboundary
operator to the integral (9.12) and restricting it to the boundary. The way to
express this more smoothly is to introduce the Green’s function for the
differentialoperatorLwithboundaryoperatorB.Forclarity,weconsiderLand
Btohaveindependentvariable
,andwelet
variable,whichistreatedasaparameterfornow.
beasecondindependent
TheGreen’sfunctionG=G(x,y)isdefinedasthesolutionoftheproblem
foreach
.
WeconstructGusingthefundamentalsolution,bydefiningafunctionϕy(x):
Thenϕy(x)satisfies
Nowwecanexpressthesolutionof(9.11)asasumu=v+w,wherev(x)=∫U
G(x,y)f(y)dy,andw(x)satisfies
This formulation is useful even when we do not have the Green’s function
explicitly,sinceestimatesontheGreen’sfunctioncanbeusedtoobtainestimates
on the solution u. In special cases, we can complete the solution as an explicit
formula by finding the Green’s function and solving (9.15) for w(x). We next
demonstratethisforPoisson’sequation,whereL=−Δ.
9.3.2.Green’sFunctionsfortheLaplacian
We apply the ideas just developed to the Dirichlet boundary value problem for
Poisson’sequation:
HereU is open and bounded, with piecewise smooth boundary ∂U;and f,gare
continuousonU,∂U,respectively.Intermsoftheprevioussection,L=−Δ,and
theboundaryoperatorBistheidentity:Bu=u.
Thefundamentalsolutionfor−ΔisafunctionΦ(x−y)ofx−y,sincethe
Laplacian Δ is translation invariant. (More generally, the fundamental solution
for any constant-coefficient PDE operator L will be a function of x − y.) The
Green’sfunctionG(x,y)=Φ(x−y)−ϕy(x)isthenexpressedintermsofthe
solutionoftheboundaryvalueproblem
In particular, ϕy is a harmonic function in U. In the proof of the following
theorem,weusethesymmetrypropertyϕy(x)=ϕx(y),whichfollowssinceΦ(z)
isanevenfunctionofz.Withthisconstruction,wecanwriteaformulaforthe
solutionoftheboundaryvalueproblem(9.16).
Theorem9.4.If
solves(9.16),then
Proof.Asexplainedabove,thefirstintegralin(9.17)solvestheinhomogeneous
PDE, with homogeneous boundary condition, so it remains to prove that the
second integral is harmonic and satisfies the given boundary condition. The
secondintegralisharmonicinx,sinceG(x,y)isharmonicinx∈Uforeachy∈
∂U, and the second term involves differentiating and integrating only with
respecttotheparametery∈∂U.Toverifytheboundarycondition,weeffectively
show that the normal derivative acts like a delta function as x approaches the
boundary. To do this, the proof uses the Divergence Theorem to recover the
boundarydata.
RecallthatG(x,y)hasthesamesingularityaty=xasdoesthefundamental
solution.IntheverificationofthesolutionofPoisson’sequationonallofspace
(Section8.2),usingthefundamentalsolution,weexcludedasmallballaroundx
∈Uandintegratedonthedomain
WedothesamethingheretoaccommodatethesingularityinG(x,y).Tostart,
weworkwithΦandafunction
(notnecessarilyasolution).Then,using
Green’sidentity,wehave
SinceΦ(x−y)isharmonicawayfromy=x,thefirsttermontheleft-handside
iszero.Ontheright-handsidetherearetwoterms,buttherearealsotwoparts
oftheboundary,namely, ∂Uand ∂B(x, ϵ).Wenowshowthatthecontributions
from∂B(x,ϵ)approachzeroasϵ→0.Let
AswiththesolutionofPoisson’sequation,Iϵ→u(x)asϵ→0(seeSection8.2).
Similarly,let
Weassumeinthetheoremthatuiscontinuouslydifferentiable.SinceΦ(x−y)is
constanton∂B(x,ϵ),proportionaltolnϵforn=2,andtoϵ−(n−2)forn>2,we
concludethat|Jϵ|→0asϵ→0.
Lettingϵ→0,wenowobtain
Thenextstepistouseasimilarargument,inwhichwereplaceΦ(x−y)by
theharmonicfunctionϕx(y)inthecalculation.Sinceϕxhasnosingularityatx=
y,andϕx(y)=Φ(x−y)fory∈∂U,wehave
Rearranging,andusingG(x,y)=Φ(x−y)−ϕx(y),weobtain
Finally, when u is a solution of (9.16) we obtain (9.17), and the proof is
complete.
9.3.3.TheMethodofImages
Theorem 9.4 gives a formula for the solution of (9.11) that relies on Green’s
functions. Here we show how the Green’s function for the Laplacian can be
calculatedfromthefundamentalsolutioninspecialcaseswhenthedomainUhas
symmetry.Thegeneralideaistoconstructimagepoints outsideUforeachx∈
Usuchthatthefunction
exactlycancelsΦ(x−y)ontheboundary∂U,
for some scale factor C, possibly depending on x. Then the new function is
harmonic with respect to y ∈ U (since the Laplacian is invariant under scaling
andtranslationbyaconstant),sowecanset
.
Figure9.3.MethodofimagestoderiveGreen’sfunctionforahalf-plane.
Wedemonstratethisconstruction,whichisknownasthemethodofimages,
in two examples. In the first example U is a half-space, and in the second
exampleUisaball.
Example8.(Methodofimages(half-space))LetU={x∈Rn:xn>0}.Forx
=(x1,…,xn)∈U,definetheimage
(seeFig.9.3),andlet
Since
,itfollowsthat
isharmonicwithrespecttoy∈U(forx∈U).
ThenG(x,y)willbeestablishedastheGreen’sfunctionifwecanshowG(x,y)=
0foryn=0,xn>0.Foryn=0,wehave
Therefore,
,asneeded.
Example9.(Methodofimages(unitball))ConsiderU=B(0,1),theunitball
inRn.Here,theimage includesascaling,inadditiontoreflectionthroughthe
boundary.(SeeFig.9.4.)Wedefine
,andlet
(Notethat
.)ToprovethatG(x,y)=0for x∈U, y∈∂U, we need to
showthatif0<|x|<1,
,and|y|=1,then
Figure9.4.MethodofimagestoderiveGreen’sfunctionfortheunitball.
Butthisfollowsbecause
Consequently, G(x, y) given by (9.18) is the Green’s function for the unit ball
withDirichletboundaryconditions.
In Chapters 4–9 we have considered all three canonical second-order linear
constant-coefficient PDE, emphasizing explicit solutions. In the next two
chapters, we introduce more theoretical approaches to general linear elliptic
equations.
PROBLEMS
1.Considertheboundaryvalueproblem
inwhichf:[0,1]→Risagivencontinuousfunction.
(a)Provethatthereisnosolutionunless
(b)Assuming(9.19),provethatsolutionsareuniqueuptoaconstant.Inother
words,ifu,varetwosolutions,then
forsomeconstantC.
(c)Writethesolutionu(x)intheform
writeaformulafortheNeumannfunctionN(x,y).
2. Write an explicit formula for ϕy(x) for
givenby(9.6),toverifythatϕy(x)islinearforeachy.
, where G is
3. Calculate the fundamental solution u ∈ D′(R) for the differential operator
,wherec∈Rand c≠0,satisfyinglim|x| → ∞u(x) = 0. That is, solve
.(Note:Thesignofcwillaffectyoursolution.)
4.Computeg∗fforthefunctions
andgraphtheconvolutionproductg∗f.
5. Verify that the examples of distributions f1,f2,f3,f4 in Example2 satisfy the
conditionsthatdefineadistribution.
6.Provethatηϵ→δinthesenseofdistributions,asϵ→0.(See(9.8).)
7.(a)Let
.Definethedistributionalderivative
.
(b) Let H : R → R be the Heaviside function. Prove that u(x, y) = H(y)
satisfies
inthesenseofdistributions.
8.Letk>0.
(a)ProvethatGk(x,y)=Cke−k|x−y|isafundamentalsolutionfortheequation
forsomeCk.FindaformulaforCk.
(b)FindtheGreen’sfunctionfortheboundaryvalueproblem
9.FindafundamentalsolutionΦ(x)dependingonlyonr=|x|,x∈R3,forthe
equation
satisfyinglimx→∞Φ(x)=0.Insteadofhavingadeltafunction(ontheright-hand
sideoftheequationdefiningΦ),usethesourcecondition
(Hint:Useachangeofvariableu=rv.)
10. Let
, x = (x1, x2) ∈ R2. For what values of α is u in L2(B),
whereB={x:|x|<1}?Explainyouranswer.
11.Considerf∈D′(R),g∈C∞(R).
(a)Derivetheformula(gf)′=gf′+g′finthesenseofdistributions.
(b)Henceprovethat
12.Letu∈C(U)satisfythemean-valuepropertyinadomainU⊂Rn:
providedB(x,r)⊂U.Letuϵ=ηϵ∗uinUϵ={x∈U:dist(x,∂U)>ϵ},where
dist(x,S)denotestheshortestdistancebetweenapointxandasetS.Thenuϵ∈
C∞(Uϵ). Prove the surprising result that uϵ=u on Uϵ. Consequently, u ∈ C∞(U)
andisharmonicin U. In particular, harmonic functions are C∞!(Hint:Youwill
needachangeofvariablesintheformulaforuϵtowritetheintegraloverB(x,ϵ)
in polar coordinates (dx = rn−1drdS) to take advantage of this version of the
mean-valueproperty.)
1. George Green, 1793–1841, was a self-taught British mathematician and physicist who made
fundamental contributions to the theory of electricity and magnetism. Several theorems and functions
relatedtothesetopicsnowcarryhisname.
CHAPTERTEN
FunctionSpaces
In this short chapter, we introduce function spaces that are used extensively in
theanalysisofpartialdifferentialequations.
10.1.BasicInequalitiesandDefinitions
Much of the theory of PDE relies on a variety of estimates, like the energy
estimatesweencounteredinChapter5fortheheatequation.Estimatesareused
toestablishallaspectsofwell-posednessandregularity.PDEestimatesroutinely
usetheinequalitiesthatweintroduceandproveinthissection,includingsome
basicinequalitiesforfunctionspaces,inparticular,LpspacesandSobolevspaces.
10.1.1.InequalitiesonR
We begin with several inequalities between numbers, which form a basis for
inequalitiesandestimatesoffunctions.
Cauchyinequality.Byrearranging(a−b)2≥0,wearriveattheinequality
Sometimesitisusefultoweightthetermsdifferently,asintheϵ>0versionof
theCauchyinequality:
Proofof(10.1).ApplyCauchy’sinequalityto
.
Young’s inequality. This relation is a different generalization of the Cauchy
inequality.Forp>1,defineqby
;wesayqisdualtop.Then
ThisinequalityistheCauchyinequalitywhenp=q=2.
Proof.Minimize
.WeleavethedetailstoProblem2.
10.1.2.FunctionSpacesandInequalitiesonFunctions
Nowwearepreparedtodefinethefunctionspacesweshalluse.Foreachspace,
wedefineanorm,andaninnerproductwherepossible.
For1≤p<∞,defineLp(U)tobethespaceof(measurable)functionswhose
pthpowerisintegrableoverU⊂∫URn:|u|pdx<∞.WedefineanormonLpby
AsdiscussedforL2spacesinSection7.2,thisdefinesanormonlyiffunctionsthat
areequalalmosteverywhereareconsideredequivalent.ThenLpisdefinedtobe
thespaceofequivalenceclasses,withthenormofanequivalenceclassdefinedas
here, in which u is any element in the equivalence class. Sometimes we
abbreviatethesubscriptandwrite||u||pfortheLpnorm.
The space L∞(U) is defined as the space of (measurable) functions that are
essentially bounded over U, with norm given by the essential supremum (see
AppendixB):
Of the following three defining properties of a norm, only the triangle
inequality requires proof, as the first two follow directly from the definition of
norminLp:
1.||u||≥0,withequalityonlyforu=0;
2.||αu||=|α|||u||forallα∈R;and
3.||u+v||≤||u||+||v||(thetriangleinequality).
The proof of the triangle inequality employs a further inequality involving
integralsoffunctions.
Lemma10.1.(Hölder’sinequality)Letu∈Lp(U),v∈Lq(U),where1<p,q<∞
aredual:
.Then
Proof. Since Hölder’s inequality is homogeneous, it is enough (and simpler to
write)ifwetake||u||p=1=||v||q.ThenweapplyYoung’sinequalitytou(x)v(x)
andintegrateoverU:
Thefollowingcasep=q=2isimportantenoughtohaveitsownname.
Theorem10.2.(TheCauchy-Schwartzinequality)Letu,v∈L2(U).Then
NowwecanprovethetriangleinequalityfortheLpnorm.
Proofofthetriangleinequality.Letu,v∈Lp(U).Then
Thetriangleinequalitynowfollowsbydividingby||u+v||p−1.
RecallthatL2isspecialbecauseithasaninnerproduct
consistent with the L2 norm ||f ||L (U) = (f, f)1/2. For the inner product (as for
Hölder’s inequality), we allow for the possibility of complex-valued functions.
However,fromnowonunlessotherwisestated,functionswillberealvalued.
2
It is sometimes helpful to think of Lp functions in terms of Fourier series. In
thespecificcaseofL2[0,π],wecanexplicitlyandeasilymaketheconnectionto
thespaceℓ2ofsequencesthataresquare-summable:
withinnerproductandnormdefinedby:
respectively,wherex={xj},y={yj}.Infact,Cauchysequencesinℓ2converge,
and their limits are in ℓ2. Thus, the linear inner product space ℓ2 is complete,
makingitaHilbertspace.
ToestablishtheconnectionbetweenL2[0,π]andℓ2,letu∈L2[0,π].Then
where(.,.)istheL2innerproduct,andthesinefunctions
sinjx,j=1,2,
…formacompleteorthonormalset.Nowletbj=(u,uj).ThenBessel’sinequality
(7.12)impliesthatthesequence{bj}isinℓ2,andmoreover,Parseval’sidentityis
.Moreconcisely,
AfancywaytosaythisisthatthemappingJ:L2[0,π]→ℓ2givenbyJ(u)={bj}
isanisometricisomorphism.
Integrable functions are not necessarily continuous or even bounded. In the
following examples, we examine the constraints between the nature of a
singularity and the dimension of the space and the exponent p in order for the
functiontobeinLp.Thefirstexampleexaminesthekindofsingularityallowedin
Lpfunctions.Thesecondexampleismoresophisticatedandshowsthatintegrable
functionscanbeverysingular.
Example1.(Asingularfunction)LetU=B(0,1)⊂Rn.Considerthefunction
inwhichβ>0issomeconstant,andu(0)=0(Infact,u(0)doesnothavetobe
specified,sinceLpfunctionsneedonlybedefinedalmosteverywhere.)
To work out the values of n,β,p for which we have u ∈ Lp(U), we need to
understand when up is integrable. Note that u has a singularity at x = 0. To
resolvethisissue,wecalculate
,andfindpreciselywhenitisfinite.For
example,forn=1,
ifandonlyifβp<1.Nowwehaveto
workouttheeffectofchangingnonthiscondition.ButinRnwehave
andthisisfinitepreciselywhentheinnerintegralisfinite,namely,when(asin
then=1case)thepowerof1/rislessthanunity:βp−n+1<1:
Example2.(Averysingularintegrablefunction)Let
beanenumeration
ofthe(countable)setofrationalsintheinterval[0,1].Define
where 0 < β < 1. Then the Monotone Convergence Theorem (Appendix B)
implies that the series converges in L1[0, 1], but u is unbounded in every
neighborhoodofeverypoint.Hence,althoughuisLebesgueintegrable,notonly
isitnotRiemannintegrable,itischronicallyunbounded!
10.2.Multi-IndexNotation
Tofurtherdiscussfunctionspaces,weusemulti-indexnotation.Thisnotationis
alsoaconvenientwaytorepresentPDEofarbitraryorderandwithunspecified
coefficients. A multi-index,α = α1 … αn, is a sequence of nonnegative integers.
Wewritethemulti-indexlengthas|α|=α1+…+αn.Ifx=(x1,…,xn)∈Rn,
wewrite
,and
adifferentialoperatoroforder|α|.Beawarethatforx∈Rn,thenotation|x|=
is still the Euclidean norm. The multi-index length |α| is reserved for
multi-indices.
Multi-index notation is useful for writing a polynomial of degree k in n
variablesxas∑|α|≤kaαxα=0,withconstantcoefficientsaα∈R.Correspondingly,
thenotationallowsustowriteageneralquasilinearPDEoforderkas
for theoretical purposes, where the coefficients aα and right-hand side f are
functions of u, derivatives of u, and of x ∈ Rn. In this form it is convenient to
place assumptions on the coefficients (such as ellipticity of the PDE) to capture
wholeclassesofequations.
IfuhasaweakderivativeDαu,then
forallϕ∈D(U),thespaceoftestfunctions.
Although multi-index notation is useful in some general contexts, such as in
thenextsection,itisoftensimplertousemoreconventionalnotation,suchas
inplaceofD010inR3.EvenasimplePDElikethewaveequationuxx−uyy−uzz
=0looksuglyinmulti-indexnotation:
10.3.SobolevSpacesWk,p(U)
Whenstudyingsolutionsuofkth-orderPDE,weneedderivativesofuoforderup
tok.SobolevspacesWk,p(U)consistoffunctionswithweakderivativesuptoorder
k,butwiththeadditionalrequirementthatthederivativesareinLp;thatis,their
pthpowersareintegrable.Thus,Wk,p(U)isdefinedtobethespaceoffunctionsu
suchthatDαu∈Lp(U)forallαsuchthat|α|≤k.Sobolevspacesareusedinthe
theoryofPDE,forexample,ellipticPDEinChapter11.
ThenorminWk,p(U)isdefinedfor1≤p<∞by
Forp=∞,wedefine
Just as we did for Lp spaces, we can ask when a function with an algebraic
singularityisinWk,p.
Example3.(Anothersingularfunction)Considertheexampleu(x)=|x|−β,x
∈U=B(0,1),x≠0(seeExample1).Then
whichhasasingularitylike
that
.ReferringbacktoExample1,inLp,wededuce
ifandonlyif(β+1)p<n.Thus,u∈W1,p(U)ifandonlyif
When we study elliptic equations with homogeneous Dirichlet boundary
conditionsinthenextchapter,theboundaryconditionsarebuiltintotheSobolev
spaces as follows. The space
is the completion of
in the Wk,p(U)
norm. That is, every element of
is the limit of a Cauchy sequence of
smoothfunctionswithcompactsupportinU.Inthissensewecanthinkof
as the space of Wk,p functions that are zero on the boundary ∂U. This notion is
madeprecisewithtracetheorems,whicharedevelopedinEvans[12].
In the important case of p = 2, we write Hk(U) = Wk,2(U). This space is a
Hilbertspace,inthatithasaninnerproductandiscomplete(withrespecttothe
norm defined by the inner product; see AppendixB). The inner product (·,·) on
Hk(U)isgivenby
andthenormis
.WegenerallyworkwithH1(U),forwhichthe
innerproductis
Moreover,weusethespace
withthesameH1norm.
Sobolevspacesarethenaturalenvironmentinwhichtostudygeneralelliptic
andparabolicPDE.Inthenextchapterwegiveaflavorofthetheoryofelliptic
equations.AmoreextensiveintroductiontothesubjectisgiveninEvans[12].
PROBLEMS
1.LetXbeavectorspacewithnorm||·||.
(a)Provethatforallu,v∈X,
(b)Showthatthefunctionf:X→Rgivenbyf(u)=||u||iscontinuousbut
notlinear.
2.ProveYoung’sinequality.
3.UseHölder’sinequalitytoprovethefollowing.
(a)ThegeneralizationofHölder’sinequalitytothreefunctionsu∈Lp(U),v∈
Lq(U),w∈Lr(U),withp−1+q−1+r−1=1:
(b)Ifp≤q≤rand1/q=λ/p+(1−λ)/r,thenforu∈Lr(U),
4.Thepartialderivativein(10.4)isnotafunctionofralone,sothecalculation
wedidinExample1doesnotapplydirectly.Usingcoordinatesx=rω,|ω|=1,
dotheintegralthatcompletestheargumenttocharacterizewhenu∈W1,p(U).
5.Letu(α)(x)=|x|(sin|x|)α,x∈R3.Findthepreciserangeofα∈Rinwhichu(α)
∈H1(B),whereB={x∈R3,|x|<1}.
CHAPTERELEVEN
EllipticTheorywithSobolevSpaces
WeusePoisson’sequationasastartingpointtoprovetheexistenceofsolutions
of a boundary value problem in an appropriate Sobolev space. Then we show
howasimilarapproachcanbeusedforgenerallinearsecond-orderellipticPDE.
The structure of the more general results and their proofs provides insight into
thetechniquesattheheartofthemoderntheoryofellipticPDE[12].
11.1.Poisson’sEquation
Our previous approach to Poisson’s equation involved finding the Green’s
function.AgainweletU⊂Rnbeopenandbounded,letf∈L2(U),andconsider
theboundaryvalueproblem
The approach of finding the Green’s function explicitly works only for special
choicesofU.InsteadofrelyingontheshapeofU,wecanusefunctionalanalysis
toestablishtheexistenceofasolutionindirectly.Theremainderofthissectionis
devotedtousingthisapproachtoprovethefollowingexistenceanduniqueness
theorem.
Theorem11.1.Foreachf∈L2(U)thereisauniqueweaksolutionof(11.1).
In this statement, note that f ∈ L2 is no longer required to be smooth.
Moreover,thetheoremreferstoweaksolutionsratherthanclassicalsolutions.A
weaksolutionof(11.1)isafunction
suchthat
forall
satisfies(11.1),wesaythatuisaclassicalsolution
of(11.1).Inthiscase,usatisfies(11.2)forevery
,asiseasilycheckedby
integrationbyparts.Sincefunctionsin
areapproximatedintheH1normby
smooth functions (see Section 10.3), so (11.2) makes sense as a definition of
weaksolution.
Weaksolutionshavetheadvantagethattheyrequirelessregularity(oneweak
derivative rather than two classical derivatives), and moreover, we can look at
weaksolutionsinaspaceoffunctionsthathasaninnerproductandiscomplete,
whichisnotpossiblewithsmoothfunctions.
Since the proof of Theorem 11.1 involves several subsidiary results, we
outlinethestepsthatmakeuptheproof.Thetheoremestablishesbothexistence
and uniqueness of weak solutions of (11.1) satisfying (11.2). In this section we
prove this result using the Riesz Representation Theorem. We first identify the
left-hand side of (11.2) as an inner product in
, thereby defining an
equivalent norm for
. Second we verify that the right-hand side defines a
bounded linear functional on
. We accomplish both using the Poincaré
inequality. Then the conclusion of the Riesz Representation theorem is the
existenceanduniquenessof
satisfying(11.2).Thiscompletestheproof
ofthetheorem.
11.1.1.ThePoincaréInequality
The Poincaré inequality is an example of an estimate in a function space that
allows a norm of a function to be estimated by the norm of a derivative of the
function.ThespecificestimateinthePoincaréinequalityboundstheL2normof
an functionbytheL2normofitsderivative,acrucialstepinestablishingthe
left-handsideof(11.2)asaninnerproduct.
Lemma 11.2. (The Poincaré inequality) Let U ⊂ Rn be open and bounded. There
existsaconstantC,dependingonlyonU,suchthat
forall
.
Proof. We use the approximation idea here for the first time. We prove the
lemmafor
andthenarguebytakinglimitsofsequencesthatitholdsfor
.TheproofissimpleinonedimensionandisverysimilarinRn.LetU=
(a,b)⊂R,andsuppose
.Thenu(a)=0=u(b).Nowwecalculate
Dividingby||u||L (a,b)completestheproofinonedimension.
2
Figure11.1.ThedomainUboundedbytwohyperplanes.
In higher dimensions, we simply integrate by parts in one of the variables,
say,x1.So,supposeUliesbetweenthehyperplanes{x1=±M},forsomeM>0,
as illustrated in Figure 11.1. (This suggests that U could be unbounded, and
indeed the Poincaré inequality is often stated for domains bounded in some
direction,butnotnecessarilyboundedinRn.) Mimicking the calculation above,
let
.Then
The second inequality relies on the Cauchy-Schwartz inequality with C = 2M.
Thisprovestheinequalityfor
.
Nowlet
.(Suchasequenceexistsfromthedefinition
of
asthecompletionof
intheH1norm.)Then||um||L (U)→||u||L (U),
and||Dum||L (U)→||Du||L (U),fromwhichthelemmafollows.
2
2
2
11.1.2.AnEquivalentNormon
Wedefinethenewnorm
2
wherethesubscript1,2referstoonederivativeofuinLpwithp=2.Thenew
normisusefulbecausetheweakformulation(11.2)onlyinvolvesthegradienton
the left-hand side, which will now be the inner product corresponding to this
norm.ForboundedUand
,thePoincaréinequalityimpliesthenewnorm
isequivalenttothe norm:
Thelatterinequalityinvolvesasmallamountofmanipulation:
Becausethenormsareequivalent,weconcludethat
isaHilbertspacewith
thenewinnerproductgivenbytheleft-handsideof(11.2).
ToapplytheRieszRepresentationTheorem(Section11.1.3),weneedtoshow
that f ∈ L2(U) defines a bounded linear functional on
. This allows us to
equatetheright-handsideoftheweakformulation(11.2)withaboundedlinear
functional.
Lemma11.3.LetU⊂Rnbeopenandbounded,andletf∈L2(U).Thenthelinear
functional
,definedby
isbounded,meaningthereisaconstantK>0suchthat
Proof.Let
.Then
HenceFisbounded.
11.1.3.TheRieszRepresentationTheorem
TocompletetheproofofTheorem11.1,weshowthatforeachf∈L2(U)thereis
a unique
such that (11.2) holds; that is, (u, v)1,2 = F(v) for every
. This follows from the following Riesz Representation Theorem for a
generalHilbertspaceXwithinnerproduct(·,·).WeonlyconsiderHilbertspaces
overthereals,buttheresultcanalsobestatedforcomplexHilbertspaces.
Theorem11.4.(RieszRepresentationTheorem)LetXbeaHilbertspace,andletF:
X→RbeaboundedlinearfunctionalonX.Thenthereexistsauniqueu∈Xsuch
that
forallv∈X.
Proof.TheproofinvolvesthenullspaceNofF:
First, let’s dispense with the trivial case, in which N = X. Then only u = 0
satisfies(11.3).
Now we consider N ≠ X. In this case, (11.3) implies u ∉ N. To see this,
supposeu∈N,andletv=uin(11.3).Then0=F(u)=(u,u),whichimpliesu
=0.Butthen(11.3)failsforanyv∉N.Thus,weseeku∉N.
We complete the proof in essentially three steps. First, we show there is a
nonzeroz∈XthatisorthogonaltoN:(z,v)=0forallv∈N.Putanotherway,
F(v)=(z,v)forallv∈N,sozisthecorrectchoiceforu,exceptthatifu=z,
then(11.3)willnotbesatisfiedingeneralforvinthecomplementofN.Inthe
secondstep,weadjustztogetthecorrectubynotingthatthecomplementofN
isonedimensional(soNhascodimensionone).ThisfollowssincetherangeofF
isonedimensional.Thus,theorthogonalcomplementN⊥ofNisonedimensional
and hence is spanned by z. Therefore, F(αz) must range through all of R as α
rangesthroughR,sinceF(z)≠0.So,weshouldbeabletofindutosatisfy(11.3)
byscalingz:u=αz,forsomeα∈R.Infact,settingv=u=αzin(11.3),we
cancalculatewhatαmustbeforthisspecialcase,leadingto
ThenF(u)=||u||2.Inthethirdstepoftheproof,weshowthat(11.3)holdsfor
thischoiceofu.
Tofindz∈N⊥,weusestandardfunctionalanalysisarguments.Firstweshow
thatNisclosed.Suppose{un}⊂Nandun→uasn→∞.Nisclosedifu∈N.But
as n → ∞, using linearity and then boundedness of F. Therefore, F(u) = 0,
meaningu∈N.
SinceNisnottheentirespace,thereisx∈Xsuchthatx∉N.ButNisclosed,
sothedistancefromxtoN(definedinproblem12,Chapter9)ispositive:
Wenowprovethatthereisw∈Nsuchthatdist(x,N)=||x−w||.Indeed,there
is a sequence {wn} ⊂ N such that ||x − wn|| → dist(x, N). It takes a tricky
calculation to show that {wn} is a Cauchy sequence. First, we appeal to the
parallelogramlaw
whichisverifiedbyexpressingthenormsintermsoftheinnerproduct.Nowwe
set
intheparallelogramlaw:
Asm,n → ∞, the left-hand side of this identity approaches 2d2. However, the
first term on the right is no smaller than 2d2, and so the second term must
approachzero:||wn−wm||→0asm,n→∞.Hence{wn}isaCauchysequence,
and is therefore convergent to an element w∈N,sinceX is complete and N is
closed.Moreover,||x−w||≤||x−wn||+||wn−w||→dasn→∞.Thus,||x
−w||≤d,whichimplies||x−w||=d,sincew∈N,andd=infy∈N||x−y||.
Nowletz=x−w.Weshowthatz∈N⊥.Lety∈N.Thenforanyλ∈R,w
+λy∈N,so
Ifwechooseλ=(z,y)/||y||2,thentheinequalitybecomes
whichimplies(z,y)=0.Sincey∈Nisarbitrary,wehavez∈N⊥.
Next we prove that ugivenby(11.4)satisfies(11.3).Sinceu ∉ N, we have
F(u)≠0.Moreover,usinglinearityofF,weobservethat
Thus,sinceu∈N⊥,
But this leads immediately to (11.3), since u was constructed to satisfy F(u) =
||u||2.
Itremainstoproveuniquenessofu∈Xsatisfying(11.3).Supposethereare
twovaluesofu,sayu=uk,k=1,2,bothofwhichsatisfy(11.3).Then(uk,v)=
F(v),k=1,2,fromwhichweget(u1−u2,v)=0forallv∈X.Henceu1=u2.
ThiscompletestheproofoftheRieszRepresentationTheorem.
TheproofofTheorem11.1isnowcomplete.
11.2.LinearSecond-OrderEllipticEquations
In this section we prove Theorem 11.1 for more general second-order linear
ellipticpartialdifferentialequations.Firstweframetheboundaryvalueproblems
thatwewillconsider.LetU⊂Rnbeopenandbounded,andlet
.Define
Note that we suppress the independent variable in u, but retain it in the
coefficients to emphasize that they are functions of x. We will see that L is
associated with a symmetric operator in which the leading-order (i.e., secondorder) terms are written in divergence form. A more general form of a linear
second-orderlineardifferentialoperator(withleading-ordertermsindivergence
form),isgivenby
whichisnonsymmetricwhenthecoefficientsbiarenotallzero.
Thecoefficientsaij,caregivenL∞functionsonU.Withoutlossofgenerality,
we can assume that the aijs are symmetric in ij: aij = aji. We place additional
conditionsonthecoefficientsaswegoalong.Generalizingproblem(11.1),welet
f∈L2(U)andconsidertheboundaryvalueproblem
Todefineweaksolutions,wemultiplyLu=fbyatestfunctionandintegrateby
partsoverU.Thus,aweaksolutionof(11.7)isafunction
suchthat
where
AsinPoisson’sequation,boundarytermsfromintegrationbypartsareallzero,
duetothechoiceofu=0astheboundarycondition.Notethatinthesymmetric
case bi(x) ≡ 0, i = 1, …, n, the middle term is absent, and indeed then B is
symmetric:
Inthesymmetriccase,treatedinSection11.2.1,werelateB[u,v]totheinner
product on
, and to do so, we require L to be elliptic. In fact, we shall
requireLtobeuniformlyelliptic,meaningthatthereisθ>0suchthat
forallx∈U,ξ∈Rn.
11.2.1.ExistenceandUniquenessofSolutionsintheSymmetricCase
Here we give the existence and uniqueness theorem for solutions of problem
(11.7)whenLissymmetric.Animmediatedifficultyisthatthezeroth-orderterm
c(x)uisnotcontrolled,inthesensethatunlesswemakeassumptionsaboutc(x),
the corresponding term in the bilinear functional (11.9) is not bounded. An
analogous issue arises in the finite-dimensional case, where the operators are
squarematrices.ConsiderasymmetricmatrixAwithpositiveeigenvalues,say,λ
≥θ>0,andagivenc∈R.Thenweconsidertheequation
for a given vector F, where L = A + cI. (I is the indentity matrix.) To obtain
existence and uniqueness, we require that c not be an eigenvalue of −A. This
wouldbeguaranteedifweknowc>0.ForthePDEcase,westatethisslightly
differently,toaccommodatethedependenceofcoefficientsonx∈U.Intermsof
thematrixproblem,werequirethatthereexistsaγ>0suchthatforallμ>γ
andanyF,thereisauniquesolutionoftheequation
Of course, this is merely the condition that c + γ be larger than the largest
eigenvalueof−A.
InthePDEversion,tomakethebilinearfunctionalnonnegativewhenv=u,
we add a large enough number μ to c(x) so that the final term in (11.9)
resemblesaweightedL2normwhenv=u.Effectivelythisguaranteesthatμis
awayfromeigenvaluesofthePDEoperatorL.
Theorem11.5.LetLbethesymmetricoperatorgivenby(11.5).Thereexistsγ∈R
suchthatforallμ>γandanyf∈L2(U),thereisauniqueweaksolution
of
theproblem
Toprovethistheorem,wewillusetheRieszRepresentationTheorem,butto
establishasuitableinnerproductintermsofB[u,v],weneedtwokeyestimates
provided by the following lemma. One estimate establishes that B is bounded,
andthesecondestimateallowsustomodifyB[u,v]inordertodefineanorm.
Lemma11.6.Thereareconstantsα>0,β > 0, γ ≥ 0 such that for all u, v in
,
1.
,and
2.
.
ProofofLemma11.6.Toproveestimate1,weworkdirectlywithB:
For estimate 2, we use ellipticity, replacing ξi by uxi in the definition (11.10).
FirstnotethatthePoincaréinequality(Lemma11.2)implies
withK=1+C.Thenwehave
Thisprovesthelemma,withγ=||c||L
,β=θ/K.
(U)
∞
ProofofTheorem11.5.WemodifyB[u,v]inLemma11.6tobe
ThenBμissymmetricandsatisfies
Thus,forμ > γ,Bμ[u,u]≥ 0, and Bμ[u,u]= 0 only for u = 0. Consequently,
Bμ[u, v]defines an inner product on
. The Riesz Representation Theorem
(Theorem11.4)completestheproof.
11.2.2.TheNonsymmetricCase
WhenLisnonsymmetric,B[u,v]cannotdefineaninnerproduct,becauseitisnot
symmetric in u, v. Nonetheless, a natural generalization of the Riesz
Representation Theorem can be formulated to cover the situation. That is, the
functional F is represented by an element
as before, and there is a
unique u satisfying (11.8). The proof of this generalization, the Lax-Milgram
Theorem,usestheRieszRepresentationTheoreminadifferent,moresubtleway.
ToovercomethelackofsymmetryinB[u,v],westartbyfixingusothatB[u,v]
defines a bounded linear functional, as a function of v. Then the Riesz
RepresentationTheoremgivesanelementwdependinglinearlyonusothatB[u,
v]=(w,v)forallv.Thesubtlepartoftheproofinvolvesshowingthatucanbe
varied so that w = f, the representative of the functional F(v) used in the
symmetriccase.Thatis,weneedtoshowthatthelinearmappingu→wisonto;
uniquenessfollowsbyshowingitisone-to-one.
Theorem11.7.(Lax-MilgramTheorem)LetHbeaHilbertspacewithinnerproduct
(·, ·) and norm || · ||. Let B : H × H → R be a bilinear functional such that the
followingpropertieshold
1.Bisbounded,meaning|B[u,v]|≤α||u||||v||forallu,v∈H(forsomeconstantα
>0);and
2.thereexistsβ>0suchthatB[u,u]≥β||u||2forallu∈H.
ThenforanyboundedlinearfunctionalF:H→R,thereexistsauniqueu∈Hsuch
that
Proof.Letu∈H.Thenv↦B[u,v]definesaboundedlinearfunctionalonH,by
property1ofthetheorem.Therefore,bytheRieszRepresentationTheorem,there
existsauniquew∈Hsuchthat
Let’sdefinethemappingA:H→HbyAu=w.Then
Hereishowtherestoftheargumentgoes.SinceFisaboundedlinearfunctional,
theRieszRepresentationTheoremimpliesthereisarepresentativew∈HforF:
SupposewecanshowthatwisintherangeofA.Thenthereisau∈Hsuchthat
Au=w.ButthisimpliesF(v)=(w,v)=(Au,v)=B[u,v]forallv∈H,which
completestheproofofexistence.UniquenessfollowsifweshowthatAisone-toone.Ofcourse,wehavenocontroloverw,sotoshowitisintherangeofA,we
mustshowthattherangeofAisallofH.Wedothisinseveralsteps.
First,notethatAislinear.Itisalsoeasytoseethatproperty1impliesthatA
isbounded:
Dividingby||Aw||,weobtain||Aw||≤α||w||forallw∈H.
NextweprovethatAisone-to-one.Fromproperty2,wehave
Nowdivideby||u||toobtain||Au||≥β||u||.SupposeAu1=Au2.ThenA(u1−
u2)=0.Thus,
Hence,u1−u2=0.ThisprovesthatAisone-to-one.
NextweshowthatAisontoH.ThekeyistoshowthattherangeR(A)ofAis
closed:itcontainsthelimitsofallCauchysequencesinR(A).
Let {un} be a Cauchy sequence in R(A). Since H is complete, the sequence
convergestosomeu∈H.Nowun=Awnforsomewn∈H,foreachn.Weneedto
showthat{wn} is a Cauchy sequence, since then it converges to an element w,
andweuseboundednessofAtoshowthatAw=u.Butwehave||un−um||=
||A(wn−wm)||≥||wn−wm||.Since{un}isaCauchysequence,sois{wn}.Letw
=limn→∞wn.Then
Thus,u∈R(A),provingthatR(A)isclosed.
Now we know from the proof of the Riesz Representation Theorem that if
R(A)≠H,thensinceR(A)isclosed,thereisu∈R(A)⊥withu≠0.Butthen
whichimpliesu=0.Thus,R(A)=H.
WeusetheconstructionandpropertiesofAtoprovetheexistenceofu∈H
satisfying(11.11). Let F : H → R be a bounded linear functional. By the Riesz
RepresentationTheorem,thereexistsauniquew∈HsuchthatF(v)=(w,v)for
allv∈H.Butwehavejustgonetoalotoftroubletoprovethateveryelementof
HisalsoinR(A).Inparticular,thereisauniqueu∈HsuchthatAu=w.Putting
thisalltogether,foranyv∈H,
asrequired.
Uniquenessofusatisfying(11.11)followsnaturally.Letu1,u2betwovalues
ofusatisfying(11.11).Then
forallv∈H.Inparticular,lettingv=u1−u2,weobtain0=B[v,v]≥β||v||2.
Thus,v=u1−u2=0.ThiscompletestheproofoftheLax-MilgramTheorem.
Now, to turn the Lax-Milgram Theorem into an existence and uniqueness
theoremforellipticequations,wehavetoverifythehypothesesofthetheorem
whenB is the bilinear functional associated with the elliptic partial differential
operator. This is only interesting in the nonsymmetric case, since the Riesz
Representation Theorem covers the symmetric case. But in the nonsymmetric
case, we have to work a bit harder to prove property 2 of the Lax-Milgram
Theorem. Specifically, we wish to prove the two estimates 1 and 2 of Lemma
11.6,butthistimeforBgivenby(11.9)withbinotidenticallyzero.Estimate1is
straightforward; we leave it as an exercise. Estimate 2, however, requires a bit
moreingenuity.Let’sproceedmuchaswedidinLemma11.6,usingellipticityto
establish
Thedifferenceisthatnowweneedtoestimatethemiddleterm.Thisisachieved
withYoung’sinequality:
Now the first term is incorporated into the left-hand side of (11.12); this will
change θ > 0, but keep θ positive provided we choose ϵ > 0 small enough.
Similarly, the second term in (11.13) is incorporated into the final term in
(11.12).Aftersomemanipulationoftheconstants,estimate2ofLemma11.6is
proved. These properties are the key to proving Theorem 11.5 for the general
case,whichwenowstate.
Theorem11.8.LetLbethePDEoperatorgivenby(11.6).Thereexistsγ∈Rsuch
thatforallμ>γandanyf∈L2(U),thereisauniqueweaksolution
ofthe
problem
PROBLEMS
1.LetU⊂Rnbeaboundedopenset,andletu(x)=1,x∈U.Provethatu∈
H1(U),but
.(Hint:UseproofbycontradictionandPoincaré’sinequality.)
2.LetA(x)=(aij(x))bethematrixofcoefficientsoftheprincipalpartofalinear
second-order elliptic partial differential operator L. Prove that if L is uniformly
elliptic on U, with parameter θ given in (11.10), then for each x ∈ U, the
eigenvaluesλofA(x)areboundedbelowbyθ:λ≥θ.
3.ConsidertheordinarydifferentialoperatorLu(x)=u″(x)+cu(x).Then,with
U=(0,1),weshouldbeabletosolve
forlargeenoughμ.Findγ>0forwhichTheorem11.8holdstruebyfindingthe
eigenvaluesof
.
4.Letu=u(x,y),Lu=−xuxx+(2+y)uxy−2uyy.Characterizetheregionin
R2inwhichLisuniformlyelliptic.
5. Find the smallest c ∈ R for which L given by Lu = −(xuxx + uxy + uyy) is
uniformlyellipticontheset{(x,y)∈R2:x>c+ϵ}foreveryϵ>0.
6. Expand the identity ||u + v||2 = (u + v, u + v). Then use the triangle
inequalitytoprovetheCauchy-Schwarzinequality(u,v)≤||u||||v||.
CHAPTERTWELVE
TravelingWaveSolutionsofPDE
Wehaveseeninearlierchaptershowthemethodofseparationofvariablescan
reducePDEtoODE.ThistechniqueworksmosteffectivelyonlinearPDE.Inthis
chapter we describe the analysis of traveling wave solutions of nonlinear PDE,
whichinvolvesODE.
Thereisanoverallpatterntothetechniqueofthischapter.ForeachPDEwith
an unknown function u(x, t), −∞ < x < ∞, we consider traveling wave
solutions
, in which the parameter s is the wave speed, to be
determinedinthecourseoftheanalysis.Substituting
intothePDEyieldsan
autonomous ODE for
, ξ = x − st. The ODE will have equilibria, and
traveling waves correspond to solutions that connect those equilibria, either to
oneanotherortothemselves,becausethesolutions
thatweseekapproachan
equilibriumasξ→±∞.
Eachsectionofthechapterisdevotedtotheanalysisoftravelingwavesfora
different equation. We begin in Section12.1 with Burgers’ equation, for which
the nonlinear analysis is simplest. Burgers’ equation is central to the study of
nonlinear convection-diffusion equations. In Section 12.2 we consider the KdV
equation,athird-orderequationfamousforhavingsolitarywavesolutions,which
are traveling waves that we calculate explicitly. Section 12.3 is devoted to
Fisher’s equation, a model for population growth and dispersal, and in Section
12.4 we consider special traveling waves for the bistable equation, which has
connectionstobinarymixturesinmaterialscienceandcomplexfluids.
12.1.Burgers’Equation
Burgers’equation
whereϵ > 0 is a constant, is a prototypical equation that includes a nonlinear
transportterm,andsmalldissipation.Laterwesolvetheinitialvalueproblemfor
Burgers’ equation using a special change of variable known as the Cole-Hopf
transformation, but here we consider only traveling wave solutions u(x, t) =
with speed s (to be determined) that connect constants u+ and u−.
Dropping the tilde, the function u(ξ), ξ = (x − st)/ϵ should satisfy boundary
conditionsatinfinity:
Substitutingu=u(ξ),ξ=(x−st)/ϵ,into(12.1),weinevitablyreducethe
PDEtoasecond-orderODE,butalsotheparameterϵcancels,whichiswhywe
scaledxandtbyϵ.IntegratingtheODEfromξ=−∞(notethat
),we
obtainthefirst-orderautonomousODE
From(12.2),(12.3),weseethatu±areequilibriaof(12.3).Thus,
s(u+
−u−)=0,fromwhichwededucethateitheru+=u−ortheparameterss,u±
arerelatedby
Of course, u ≡ u− solves (12.3) for all s but satisfies the boundary conditions
only if u+ = u−. This would be a very uninteresting traveling wave! However,
the constant solution u ≡ u− is the only solution satisfying the ODE and
boundaryconditionsifu−=u+.(Seeproblem1.) For the KdV equation of the
nextsection,thecorrespondingstatementwouldbeincorrect.
Equation (12.3) and its solutions can be represented on the u-axis (a onedimensionalphaseportrait)orusingdirectionfieldsinthe(ξ,u)plane.Asϵ→
0+,thetravelingwavesolutionsconvergetoadiscontinuousfunctionthatisin
fact a shock wave solution of the inviscid Burgers equation. Moreover, as is
immediately evident from the one-dimensional phase portrait, solutions exist if
andonlyif
which relates the wave speed s to the characteristic speed u+ and u− ahead of
and behind the shock wave, respectively. We leave the derivation of this
conditionasapartofProblem1.
12.2.TheKorteweg-deVriesEquation
TheKdVequation
isfamousbecauseoftheexistenceofsolitarywaves,whicharespecialtraveling
wave solutions with a remarkable property. As a faster wave catches up to a
slower one, the waves merge, and then the faster wave emerges ahead of the
slowerwave,unchangedexceptthatitisshiftedforwardinspacefromwhereit
might have been if the interaction had been through linear superposition. The
slower wave is shifted backward. This property is remarkable because the
equationisnonlinear,soyoumightexpectanevenmorecomplicatedinteraction
betweencollidingwaves.
The KdV equation also has the special property of possessing an infinite
numberofinvariants.Thesearespatialintegralsthatdependonthesolutionand
areconstantintime.ThefirsttwoinvariantsfortheKdVequationareassociated
withmomentumandkineticenergy,inwhichuisinterpretedasavelocity:
Weleaveittoproblem2toverifythatthesequantitiesareinvariantsinthesense
that they are constant in time if u(x, t) is a solution that decays sufficiently
rapidlyasx→±∞.ThispropertyoftheKdVequationisrelatedtotheinverse
scatteringtransforminwhichthesolutionu(x,t)servestoscatterwavesthrough
a linear equation with coefficients depending on u(x, t). An accessible
introductiontothistopiccanbefoundintheclassictextbyWhitham[46].
Inthissectionwefindthesolitarywavesandotherperiodictravelingwaves,
butfirstlet’sbrieflydiscussthedispersivenatureoftheequation.Ifwelinearize
(12.5)aboutaconstant,say,u=1,wegetthelinearizedKdVequation
Thisisachievedbysettingu(x,t)=1+ϵv(x,t)andretainingonlytermsthat
are linear in ϵ. Solutions of (12.7) can be found by separation of variables. We
seek solutions with a single spatial Fourier mode eiζx and time-dependent
coefficient φ(t): v(x, t) = φ(t)eiζx. The parameter ζ is the wave number and is
relatedtotheperiodLbyζ=2π/L,sinceeiθis2πperiodicinθ.Substitutinginto
(12.7),wegettheODEφ′+iζφ−γiζ3φ=0.Consequently,
.
Thus,wecanwritev(x,t)intheform
whereλ = λ(ζ) measures the time-dependent response at the wave number ζ;
λ(ζ)isgivenbythedispersionrelation
The first term (−iζ) corresponds to a traveling wave solution of the linear
transportequation(withspeedc=1)ofChapter1,inwhichγ=0.Thesecond
term is dispersive in the sense that the solution
is a
traveling wave, but it has speed 1 − γ ζ2 that depends on the wave number.
Thus,waveswithdifferentspatialfrequencytravelwithdifferentspeeds.
Tocharacterizesolitarywaves,weconsidertravelingwavesolutionsof(12.5)
oftheformu(x,t)=v(x−st),wheresisthewavespeed,andwesupposethat
v(ξ),ξ=x−stapproacheszeroatξ=±∞:
Figure12.1.Thephaseplanefor(12.8).
Substituting into the PDE and integrating once (assuming that derivatives of v
alsoapproachzeroasz→±∞),weobtainthesecond-orderODE
Ifwemultiplythisequationbyv′andintegrate,wefindthatthequantity
is constant on solutions v(ξ). Moreover, if we define this quantity to be the
Hamiltonian:
then,writingH=H(p,q),p=v′,q=v,wehavep′=−Hq,q′=Hp,whichis
the Hamiltonian structure of (12.8). As observed above, the Hamiltonian is
conservedalongtrajectories:
Thereadershouldverifythesestatementsusing(12.8)and(12.9).Itfollowsthat
trajectoriesinthephaseplanearelevelcurvesofH(v′,v),showninFigure12.1
fors>0.Inthisfigure,theclosedcurvesintherighthalf-planecorrespondto
periodicsolutions,calledcnoidalwaves. These are periodic traveling waves that
are important in the study of dispersive equations like the KdV equation. They
are named after the cnoidal function, a special elliptic function that gives the
shapeofthesolutionsasfunctionsofξ=x−st.
The trajectory in Figure 12.1 joining the origin to itself and enclosing the
periodictrajectoriesiscalledahomoclinicorbit.Thecorrespondingtravelingwave
isthesolitarywave.Wecanderivetheformulaforthissolutionbyintegratingthe
equation,butsincethesolutionisknown,let’ssimplywriteitdown:
Becausethespeedsisproportionaltotheamplitudea,solitarywaveswithlarger
amplitudemovefaster.Thus,ifalargesolitarywaveisbehindasmallerone,the
large one will catch up. This leads to the nonlinear interaction of two solitary
waves mentioned earlier in this section. Solitary waves and the connection to
integral invariants and inverse scattering are discussed in papers by Gardner et
al.[17]andbyMiura[36].
12.3.Fisher’sEquation
Fisher’sequation[14]isaversionofthelogisticequationthatincludesdiffusion
(see Section 1.3). While population growth can be modeled by the logistic
equation(eithertheODEoradifferenceequation),populationsinmanycontexts
tend to spread out or even invade nearby territory. This spatial dependence of
population is modeled simply in Fisher’s equation by diffusion. In this context,
diffusionarisesbecauseofthetendencyofapopulationu=u(x,y,t),tomigrate.
AccordingtoFick’slaw,thepopulationfluxisproportionalto−∇u.(Notethat
Fick’slawissimilartoFourier’slawofheatflow.)Combiningdiffusionwiththe
logisticpopulationmodel,weobtainthePDE
inwhichtheconstantsd,α,umaxareallpositive.Here,theLaplacianΔ=
andthepopulationu=u(x,y,t)isnonnegative.
,
Weseektravelingwavesolutionsu=u(x−st).Theseareplanewavesinthe
sense that u depends spatially only on x and is independent of the transverse
variabley.Thus,levelcurvesofuateachtimetarelinesparalleltothey-axis,
propagating (in the direction of the x-axis) with speed s. Since u is typically a
populationorconcentration,wewantu≥0,butwepursuetheanalysisinitially
withoutthisrestriction.
Letd=α=umax = 1 in Fisher’s equation. (This covers the general case as
showninproblem3asanexerciseinrescalingthevariables.)Thentakingu=
u(x,t)tobeindependentofy,wehavethesemilinearparabolicequationinone
spacevariablexandtimet:
Fortravelingwavesolutions,wesubstituteu=u(x−st),resultingintheODE
Thistime,wecannotintegratetheequationtoreducetheorder.However,justas
fortravelingwavesolutionsoftheKdVequation,thissecond-orderODEcanbe
studiedasafirst-ordersystem
If we set s = 0, the traveling wave does not travel at all, and ξ = x. These
stationarysolutionsareoscillationsinapotentialwell
(theintegral
of u(1 − u)) with a minimum at u = 0. The corresponding spatially periodic
solutionsof(12.11)arecnoidalwaves,justasfortheKdVequation.Noticehow
similar the Hamiltonian
is to the Hamiltonian for the KdV
travelingwaves.Thephaseportraitfors=0islikewiseamirrorimageofFigure
12.1.Thesaddlepointisat(u,u′)=(1,0),andthelimitoftheperiodicsolutions
around the center at the origin is a wave that approaches u = 1 as x → ±∞,
withasinglenegativeminimum.Ofcourse,noneofthesesolutionsisphysicalif
urepresentsapopulation,sincetheyallhaveu<0inaninterval.
Tofindtravelingwaveswithu>0,weareledtoinvestigatenonzerovalues
ofs.Equilibria(forwhichu′=v′=0)are(u,v)=(0,0)and(u,v)=(1,0),
andarethusindependentofs.Theirnatureisdeterminedfromthelinearization,
inwhichwetaketheJacobianofthevectorfieldG(u,v)=(v,−sv−u(1−u))
andcalculateeigenvalues.Fors>0,thiscalculationshowsthat(u,v)=(1,0)is
asaddlepoint(i.e.,havingrealeigenvaluesofoppositesign).Theorigin(u,v)=
(0, 0) is a stable spiral for 0 < s < 2 (complex conjugate eigenvalues
with negative real part), and a stable node for s > 2 (both
eigenvaluesnegative).
Thespeeds,whenpositive,actsasadampingparameter.From(12.12)wesee
that d/dξ H(u′, u) ≤ 0, so that the energy H decreases along trajectories. By
analogy with damped simple harmonic motion, for small s > 0, we expect
oscillations to be underdamped. Correspondingly, traveling waves oscillate
aroundu=0andarethereforeunphysical,buttheydecaytou=0.Valuesofs
>0sufficientlylargegiveoverdamping.Infact,fors≥2theeigenvaluesofthe
equilibriumatu=u′=0arerealandnegative.Travelingwavesolutionsthat
aremonotonicfromu=0tou=1existforalls≥2.Thesearethephysically
relevanttravelingwavesolutionsofFisher’sequation.Fors>2,thiswasproved
by Aronson and Weinberger [5]. The proof is quite technical in order to show
that the solution curve leaving the saddle point at (1, 0) does not cross the
verticalaxisu=0whens>2.Stabilityofthetravelingwaveasasolutionof
thePDE(12.11)wasanalyzedbyFifeandMcLeod[13].
12.4.TheBistableEquation
Thebistableequation
inwhichfhasthegraphshowninFigure12.2,isasimplemodelfortransitions
betweentwostablestates.Thiskindofmodelhasbecomecommoninstudiesof
phasetransitionsinsolids,withapplicationsinmaterialsscience.
The space-independent equation u′(t) = f(u) has stable equilibria at u = 0
andu=1;theequilibriumu=aisunstable.Travelingwavesbetweenu=0
andu = 1 are somewhat easier to analyze than for Fisher’s equation, since the
equilibria correspond to saddle points and the issue of keeping the solution
positive does not arise. However, the traveling waves occur only at isolated
speeds s. Traveling wave solutions u(x, t) = v(x − st) of (12.13) satisfy the
second-orderequation
Asintheprevioussections,weanalyzethissecond-orderequationbywritingit
asafirst-ordersystem:
Thethreezeroes0<a<1offareequilibriaoftheODE;theoutsideequilibria
atu=0,u=1aresaddlepoints,andthemiddleequilibriumatu=aisanode
fors≠0,asisseenfromtheeigenvaluesofthelinearization,calculatedbelow.
Fors=0,themiddlezeroisacenter,asshowninFigure12.3a.
LetG(v,w)=(w,−sw−f(v)).ThevectorfieldGhasJacobian(seeAppendix
Cforthedefinition)
Figure12.2.Thebistablefunction(12.13).
Figure12.3.Thephaseplanefortravelingwavesolutionsofthebistable
equation,assuming(12.16).(a)s=0;(b)s<0near0;(c)smallers<0.
Attheoutsideequilibria,f′(v)<0,andtheeigenvaluesofdGarereal,givenby
Thecorrespondingeigenvectorsare
TheunstablemanifoldMU is the solution curve leaving the saddle point, and
the stable manifold MS is the solution curve entering the saddle point. The
portionsofthesecurvesthatweconsiderarelabeledinFigure12.3.Notethatin
Figure12.3a,justasfortheKdVequationtravelingwaves,thesaddlepointatthe
originhascoincidentstableandunstablemanifolds,givingthehomoclinicorbit
shown.InFigures12.3b,cthereareorbitsfromthemiddleequilibriumatv=a
to the origin and to the equilibrium at v = 1. These orbits correspond to
travelingwavesapproximatingLaxshocks,forwhichtheLaxentropyconditionis
satisfied;seeChapter13,Section13.1.4.
Thetangentstotheinvariantmanifoldsattheequilibriaaretheeigenvectorsr
±ofthelinearizationof(12.15).(SeeAppendixC.)Thesolutionsrepresentedby
the invariant manifolds are asymptotic to eλ±ξ r± as ξ → ∓∞, so that the
eigenvectors are tangent to the invariant manifolds at the equilibria. Therefore,
each invariant manifold has the same slope as its associated eigenvector,
specifically,λ±.
Ourobjectiveistofindavalueofthespeedsforwhichatrajectoryjoinsthe
twosaddlepoints.Suchatrajectorywillcorrespondtoatravelingwavefromu
= 0 to u = 1 and is called a heteroclinic orbit (as opposed to the homoclinic
orbits we found for solitary wave solutions of the KdV equation). Since we are
joining saddle points in the phase portrait, we can expect heteroclinic orbits to
occuronlyforisolatedvaluesofs.Weproveanexistenceresultforheteroclinic
orbits,showingthatthereisavalueofsforwhichthereisaheteroclinicorbit.
Whenf(u)isgivenbyacubicfunctionf(u)=u(1−u)(u−a),thevalueofsand
theheteroclinicorbitcanbecalculatedexplicitly.
Multiplying(12.14)byv′,weobtaintheidentity
where
, and
. Thus, E(v, v′) increases along
trajectoriesifs<0,decreasesifs>0,andisconstantforstationarywaves(s=
0).Thephaseportraitfors=0isthestartingpointforouranalysis;trajectories
arelevelcurvesofE(v,v′),showninFigure12.3.Inthisfigureweassume
toensurethatthecurveE(v,v′)=0crossesthev-axisatapoint(v0,0)witha<
v0 < 1, as shown in the figure. Consequently, there is a homoclinic orbit
connectingtheorigintoitself,asshowninFigure12.3a.
The heteroclinic orbits we seek join the saddle point at the origin to the
saddle point at (1, 0). We prove the existence of such an orbit for some s < 0
usingashootingargumentonthestableandunstablemanifoldsshowninFigure
12.3.
Theorem12.1.Supposefsatisfies(12.16).Thenthereisaspeeds<0forwhichthe
bistable equation (12.13) has a traveling wave solution u = u(x − st) satisfying
u(−∞)=0,u(∞)=1.
Proof. As we vary s ≤ 0 in the ODE system (12.15), the stable and unstable
manifoldsmove,andwewishtoshowthatforsomes<0,theunstablemanifold
fromtheorigincoincideswiththestablemanifoldentering(1,0).Todoso,we
employakindofshootingargument,inwhichthestableandunstablemanifolds
shoot inward toward the line L : v = a, and we measure the height w of the
intersections.AsshowninFigure12.3bc,letpointP:(v,w)=(a,p(s)) denote
the intersection of MU with L, and let point Q : (v, w) = (a, q(s)) denote the
intersectionofMSwithL.Itisstraightforwardtoarguethattheintersectionswith
Laredefinedforalls<0,forexample,byconsideringthecurvew=−f(v)/s,
on which w′ = 0 (see (12.15)), so that the trajectories have a maximum or
minimumwheretheyintersectthiscurve.
For s = 0 (see Fig. 12.3a), P is below Q : p(0) < q(0), due to (12.16).
Therefore,bycontinuity,fors<0nears=0,PisstillbelowQ:p(s)<q(s)(see
Fig.12.3b),andweseekasmallervalueofs<0forwhichPliesaboveQ:
as shown in Figure 12.3c. Then by continuity, an intermediate value of s < 0
exists for which the two points coincide, and there is a heteroclinic orbit along
thecoincidingmanifolds.Thiswillcompletetheproof.
Weshow(12.17)forsmallenoughs<0inthreesteps:(1)Weestablishthat
q(s)<q(0)foralls<0.(2)Wechooseβ>0sothatβa>q(0).(3)Finallywe
showthatforthisvalueofβ,wecanchooses<0smallenoughsothatp(s)>
βa.Theseinequalitiestogetherprove(12.17)forsomesmallenoughs<0.
Let’sshowthatMSfors<0liesbelowthecurveMSfors=0.Thiswillimply
Toprove(12.18),weconsidertheportionofthecurveMSwithitsgraphdenoted
w=ws(v),a≤v≤1,foreachs≤0.Thenq(s)=ws(a),andfromtheearlier
discussion,
Consequently,since
satisfy
, the slopes of the tangent to MS at the equilibrium
Thus,ws(v) < w0(v), for v < 1 near v = 1. Next we show that this inequality
holdsallthewaytotheverticallineL:v=a.
Fromthechainruleand(12.15),wehave
Supposeasvdecreasesfromv=1,thereisafirstvalueofv=v∗<1atwhich
the stable manifold MS with s < 0 crosses the stable manifold MS with s = 0.
Then
thelatterinequalityfollowingfromws(v)<w0(v),v∗<v<1.
v ,
Subtracting(12.19)withs=0from(12.19)withs<0,wethenhaveatv=
∗
However, the two sides of this equality have opposite signs, providing a
contradiction.Thus,MSwiths<0liesbelowMSwiths=0,atleastdowntov
=a.
Now we consider the unstable manifold MU emanating from the origin. We
usethenotation
,0≤v≤aforthevaluesofwalongMU.Then
.
Letβ>0bechosensothat
Then,from(12.18),wehave
foralls<0.
Thevectorfielddefinedby(12.15)hasslopew′/v′ateachpointinv-wplane.
Specifically,ateachpointonthelinew=βv,theslopeofthevectorfieldis
Thus,sincef(v)/visboundedfor0≤v≤a,wecanchooses<0smallenough
thatthisslopeisgreaterthanβ.Thenthetrajectoriescrossthelinew=βvfrom
belowtoabove.Inparticular,theunstablemanifoldMUmustlieabovethisline,
atleastuntilitcrossestheverticallineL,wherev=a,w=p(s).Consequently,
p(s)>βa>q(s).Thisverifies(12.17)forsufficientlynegativesandestablishes
theexistenceofaheteroclinicorbit,asclaimed.Asimilarargumentcanbeused
toshowthereisatrajectoryforsomes>0thatjoinsu=1tou=0.
Example 1. (Traveling waves) Consider traveling waves for (12.13) with a
cubicfunctionf(u)=u(1−u)(u−a),with0<a<1.Forthisspecialcasewe
canseeks∈Rforwhichthereisaninvariantparabolaw=kv(1−v)forthe
ODEsystem(12.15)thatpassesthroughthetwosaddlepointsatv=0,1.Then
the traveling wave solution will have speed s and corresponds to a trajectory
lyingontheparabola.
Toobtaintheinvariantparabola,letw=kv(1−v).Thenfrom(12.15),
implies
Therefore,
Thus,−2k2=−1,whichleadsto
,sincewewanttheparabolatolieinthe
upperhalf-planewhen0<v<1.Equatingtheconstanttermsleadsto
Consequently, s < 0 for
, and s > 0 for
. The traveling wave is
stationarywhen
ands=0.Inthiscase,theODEsystemisHamiltonian(see
(12.14)withs=0,whichkillsthedampingterm),withheteroclinicorbitsfrom
(0,0)to(1,0)intheupperhalf-planeandfrom(1,0)to(0,0)inthelowerhalfplane,correspondingtotheparabola
.
In this chapter we have seen how to use phase plane techniques of systems of
ODE to study smooth traveling wave solutions of several important PDE. In the
next chapter we study shock wave solutions of scalar conservation laws. Shock
wavesarediscontinuoustravelingwaves,requiringnewtoolsfortheiranalysis.
PROBLEMS
1.(a)Sketchtheone-dimensionalphaseportraitfor(12.2)–(12.4).Thisshouldbe
theu-axis,withequilibriau±marked,andarrowsindicatingthesignofu′.
(b)Showthatthereisanonconstantsolutionof(12.3)satisfying(12.2)ifand
onlyifu+<u−.
(c)Infact,thesolutioninpart(b)canbefoundexplicitly,sincetheright-hand
sideof(12.3)isquadratic,sothatseparatingvariablesu,ξandusingpartial
fractionsyieldsthesolution.Findthesolutionintheform
byfindingconstantsa,b,casfunctionsofu±.
(d)Plotorsketchthesolutionasagraphu=u(ξ)whenu−=2andu+=
−1.Whatisthewavespeed?
2. Prove that the two integrals of (12.6) are invariants of the KdV equation
(12.5),andfindη(dependingonγ)sothattheintegral
isalsoaninvariant.
3.Scalex,t,uin(12.10)sothatthethreeconstantsd,umax,αcanbesettounity.
4. Use a computer program to sketch phase portraits for the ODE for traveling
wave solutions of Fisher’s equation, for speeds s = 0, 1, 3. The phase portraits
canalsobedrawnbyhandwiththeaidofthenullclinecurves,whereu′=0(i.e.,
theu-axis,wherethevectorfieldisvertical,andhencetrajectoriescrossingitare
too),orv′=0(wherethevectorfieldishorizontal).
5. In the example of the bistable equation(12.13) with cubic f(u), determine a
formula for the traveling wave u = v(x − st), using the invariant parabolic
manifoldw=kv(v−1)foundinExample1.(Notethatw=v′.)
6.ConsidertheKdV-Burgersequation
whereα,βarepositiveconstants.
(a)FormulateanODEfortravelingwavesolutionsu(x,t)=v(η),η=x−st,
withboundaryconditions
(b)Provethatthereissuchasolutionifandonlyif
(c) Find all values of the parameters α, β for which the traveling wave is
monotonic (i.e., find a necessary and sufficient condition for v(η) to be a
decreasingfunctionofη).
7.Letf:R→RbeagivenC1function.Showthattravelingwavesolutionsu=
u(x−st)ofthePDE
aregivenimplicitlybytheequation
wherea,barerealconstants.
CHAPTERTHIRTEEN
ScalarConservationLaws
The theory of scalar conservation laws was developed in the 1950s by many
people,includingKruzkov[31],Lax[32],andOleinik[37].Thestartingpointis
the method of characteristics, treated in detail in Chapter 3. Recall that for a
scalarequation
the characteristic speed is f′(u), and characteristics are curves in the x-t plane
definedbytheODE
ifuisasolutionofthePDE.However,aswesawinChapter3,smoothsolutions
of nonlinear equations typically develop singularities after a finite time. This is
themotivationforconsideringweaksolutions.
We begin this chapter with a detailed analysis of the inviscid Burgers
equation,whichhasaquadraticfluxfunction—thesimplestconvexflux.Wethen
sketchhowthetheoryisgeneralizedtoscalarequationsinonespacedimension
with general convex flux functions. Finally, we analyze examples of equations
withnonconvexfluxesandequationswithtwospacevariables.
13.1.TheInviscidBurgersEquation
InthissectionwediscusstheinviscidBurgersequationinsomedetail,motivating
thetheoryofscalarconservationlaws,includingthedefinitionofweaksolutions
ofinitialvalueproblemsandtheanalysisofshocksandrarefactions.
13.1.1.ScaleInvarianceandRarefactionWaves
InthefollowinginitialvalueproblemfortheinviscidBurgersequation,theinitial
conditionhasajumpdiscontinuityseparatingtwoconstantvalues:
Thiskindofinitialvalueproblem,withastepfunctioninitialcondition,iscalled
aRiemannproblem.Riemannproblemsplayacentralroleinthetheoryofweak
solutions, partly because they have the property of scale invariance in the
followingsense.Ifwescalexandtbyaconstanta>0anddefinenewvariables
then
so the PDE is unchanged by the change of variables. The initial condition
dependsonlyonthesignofx,soitdoesnotchangeeither.Thus,problem(13.1)
issaidtobescaleinvariant.
Animmediateconsequenceofscaleinvarianceistheriskofmultiplesolutions
ofproblem(13.1).Letu(x,t)beasolutionof(13.1).Thenv(x,t;a)=u(ax,at)is
alsoasolutionof(13.1)foranya>0.Sincewewantuniquenessofsolutionsof
(13.1)(partofthewell-posednessconditionforinitialvalueproblems),weneed
v(x,t;a)=u(x,t).Itfollows(seeProblem1)that
forsomefunction
. Then u(x, t) is constant on rays x = ct through the origin in the x-t
plane.When iscontinuous,thistypeofsolutioniscalledacenteredrarefaction
wave.Laterwealsoconsidershockwavesolutionsof(13.1),whichcorrespondto
discontinuousfunctions .
13.1.2.CenteredRarefactionWaves
Continuous, piecewise smooth (nonconstant) scale-invariant solutions u(x,t)=
ofthePDEin(13.1)arecenteredrarefactionwaves.Then
implies
. One possibility for this equation would be
,butthis
leads only to the constant solution. The other alternative gives the solution we
seek:
SincethePDEistranslationinvariant(i.e.,itdoesnotchangeifxortistranslated
byaconstant),rarefactionwavescanbetranslatedsothattheyarecenteredon
points(x0,t0)otherthantheorigin:
InFigure13.1 we show the solution of the Riemann problem (13.1) with uL=
−1, uR = 1, in which the solution is constant to the left and right of the
rarefaction wave. Continuity of the solution implies that the derivative ux is
discontinuous at the leading and trailing edges of the rarefaction, but this is
consistentwiththePDE,sincethesepointstravelwithcharacteristicspeedu.A
general result concerning the propagation of discontinuities in derivatives of
solutionsisprovidedinthenextchapter(seeSection14.2.5).
Figure13.1.ArarefactionwavesolutionoftheinviscidBurgersequation.
13.1.3.ShockWaves
In Section 9.2.2, Example 7, we discussed discontinuous solutions of a scalar
conservationlawasanapplicationofdistributions,includingaderivationofthe
Rankine-Hugoniot condition for constant-speed shock waves. Here we give a
moregeneralderivationoftheRankine-Hugoniotcondition.However,ifallshock
waveswereallowedinweaksolutions,thensomeRiemannproblemswouldhave
multipleweaksolutionsforthesameconstantsuL,uR.Torectifythislackofwellposedness,weintroducethenotionoftheadmissibilityofshocks.
Consider a solution u(x, t) of Burgers’ equation (13.1) with a jump
discontinuityonacurvex=γ(t)withγaC1function.WeassumethatuisC1
and satisfies the PDE (13.1), except on x = γ (t). Since u has a jump
discontinuity,ithasone-sidedlimitsu±(t)=limx→γ(t)±u(x,t)=u(γ(t)±,t).
InChapter 9, Example 7, we showed how to treat a discontinuous function
withajumpdiscontinuityasadistributionalsolutionofaconservationlaw.We
takethisupagaininthenextchapteronsystemsofconservationlaws.Herewe
give a more direct treatment going back to the balance law formulation to see
whatittakesforutosatisfytheintegralformoftheequation.
Consider points x = a, x = b with a < γ (t) < b, for t in an interval, as
showninFigure13.2.ThebalancelawexpressesthePDEintheform
This equation can be derived much as was done for the traffic flow equation
(2.16)ofChapter2.Toevaluatetheleft-handside,webreakuptheintegral:
Figure13.2.Ashockwavex=γ(t).
Nowwecancarryoutthedifferentiationtoget
NextweusethePDEintheintegrals:
Thecalculationnowreducesto
Sincex=γ(t)isadiscontinuityforu,wehaveu+(t)≠u−(t),so
whichistheRankine-Hugoniotjumpcondition.Theleft-handsideisthespeedof
thediscontinuity,orshockwave;theconditionstatesthattheshockspeedγ′(t)is
the average of u across the shock (i.e., γ′(t) is the average of the characteristic
speedsoneachsideoftheshock).
Sincetheclassofpossiblesolutionsnowincludespiecewisesmoothfunctions,
we lose uniqueness of solutions for some initial value problems. This issue is
familiarfromsimplercontexts.Forexample,ifweconsideronlyrealsolutionsof
x3 = 1, then there is a unique solution x = 1. However, when we widen the
admissible class of numbers to include complex numbers, then there are three
solutions.
In the next example we show a one-parameter family of weak solutions for
thesameinitialvalueproblem.Theexampleissuggestiveofhowuniquenesscan
berecovered.
Example1.(Nonuniqueweaksolutions)ConsidertheRiemannproblem(13.1)
withuL=−1,uR=1:
TherarefactionwavesolutionisshowninFigure13.1.However,thereisalsoa
one-parameterfamilyofweaksolutions.InFigure13.3werepresentonemember
ofthisfamily,withaparticularvalueoftheparameterα∈(0,1].Theformula
forthesolutionis
Inthesesolutionsnoticethatforα>0,thereisastationaryshockonthet-axis,
and characteristics leave the shock on both sides. Since u is constant on
characteristics, the solution adjacent to the shock is not determined from the
initialdata.Wesaythatcausalityfailstohold.Toavoidthislackofcausality,we
selectthesolutionwithα=0,forwhichthesolutioniscontinuous,consistingof
thecenteredrarefactionwavewhosegraphappearsinFigure13.1,butnoshock
forms.
Figure13.3.One-parameterfamilyofsolutionsofproblem(13.3).
13.1.4.TheLaxEntropyCondition
Example 1 shows that to have unique solutions of initial value problems, we
cannot allow all solutions with shocks that only satisfy the Rankine-Hugoniot
condition (13.2). In many circumstances a unique solution is selected if we
impose the additional condition that characteristics must impinge on a
discontinuity from both sides. This is consistent with causality, as the
characteristicscarryinformationaboutthesolutionforwardintimefrominitial
and boundary conditions. The condition on characteristics is known as the Lax
entropycondition.Formulatedinthisway,itappliestodiscontinuoussolutionsof
any scalar conservation law ut + f(u)x = 0, and it also generalizes to
discontinuitiesinhigherdimensions.
WecanwritetheLaxconditionasapairofinequalities.Considerashockx=
γ (t), represented in Figure 13.3, with left and right limits u−(t) and u+(t),
respectively.ThentheLaxentropyconditionstatesthat
Inparticular,thesolutionmustjumpdownfromu−tou+.
By contrast, in Example 1 the solutions u with α > 0 jump up at the
discontinuity.Thus,theonlyacceptablesolutionisthecontinuousone,forwhich
α = 0; the others have shocks that satisfy the Rankine-Hugoniot condition but
nottheLaxentropycondition.
Example2.(SolutionoftheRiemannproblem(13.1))Wecannowsolvethe
Riemann problem (13.1) for Burgers’ equation for all choices of uL, uR. The
solution consists of a single wave with the constants u = uL to the left of the
waveandu=uRtotheright.IfuL<uR,thesinglewaveisacenteredrarefaction
wave,whereasifuL>uR,thenthesinglewaveisashockx=st,s= (uL+uR).
Inthenextexample,wecomputeashockwaveasafreeboundary,showing
how information from the left and right determines the location of the shock.
This property of locating the shock based on information from both sides is
anotherreasonfortheLaxcondition—informationflowsintotheshockfromthe
left and the right. Since the information (carried by characteristics) varies
continuouslywiththeinitialdata,theLaxconditionensuresthatsolutionsvary
continuously with the data. In this sense, shocks satisfying the Lax entropy
conditionaresaidtobestable.
Example 3. (Piecewise-constant initital data) We can use rarefactions and
shocks to solve initial value problems with piecewise-constant initial data by
solving the Riemann problem and then solving interaction problems when
individualwavescollide.Asanexample,weconsidertheinitialvalueproblem
ThesolutionstructureisshowninFigure13.4.Toconstructthesolutionstepby
step,firstconsidert>0small.Therewillbeararefactioncenteredatx=t=0
joiningregionsx<0whereu=0andx>twhereu=1.Theconstantsu=
1totheleftofu=0arejoinedbyashockwaveoriginatingatx=1,t=0and
havingconstantspeed .Theshockwavethereforeliesonthecurve
.
Nowtheleadingedgeoftherarefactionwave,specifically,thecharacteristicx
=t,collideswiththeshockwave
,forwhichx=2.Thesolution
is continued with a curved shock x=γ(t), since the limit from the left comes
fromtherarefactionwaveandisnotconstant,whereasthelimitfromtheright
continues to be u = 0. The speed is related to the left and right limits by the
Rankine-Hugoniotcondition,whichtakestheform
Theshockwaveisattachedtothecollisionpoint,sowehavetheinitialcondition
Thesolutionoftheinitialvalueproblemforγ(t)is
.Thisisthecurved
portion of the shock shown in Figure13.4. Notice that γ(t) has speed γ′(t) =
, which approaches zero as t → ∞. Consequently, the shock strength
decreases,buttheshockpersistsforalltime.
Figure13.4.SolutionoftheRiemannproblem(13.4).
13.2.ScalarConservationLaws
MuchofwhatwehaveanalyzedfortheinviscidBurgersequationappliestothe
generalscalarequationinonespacevariableandtime
inwhichthefluxf:R→RisC2.WeconsidertheCauchyproblem,withinitial
data
For some properties, it is convenient to write the equation as a nonlinear
transportequation
wherec(u)=f′(u)isthecharacteristicspeed.Weshowhowtheconstructionof
rarefactionwavesandshocksgeneralizesto(13.5).
Firstobservethatuisconstantoncharacteristics
,since
=0,
and that therefore characteristics x = x(t) are straight lines. Consequently, the
solutionoftheCauchyproblemisexpressedimplicitlybytheequation
Using the balance-law version of (13.5), the Rankine-Hugoniot jump
conditionforadiscontinuoussolution,suchasthatshowninFigure13.2,is
where
is the shock speed. Notice that the shock speed is the slope of the
chordinthegraphoffjoiningthepoints(u±,f(u±)),asinFigure13.5.Thefigure
provides a useful interpretation when considering the Lax entropy condition,
becausethisslopeiseasilycomparedtothecharacteristicspeedsc(u±)=f′(u±),
whichareslopesofthetangentstothegraphoffatthesesamepoints.TheLax
entropy condition states that characteristics should enter the shock from each
side.(Seeγ(t)inFig.13.4foranexample.)Thismeans
Shocksthatsatisfythisconditionaresaidtobeadmissible.
Centeredrarefactionwaves
arespecifiedbytheimplicitequation
areconstantoncharacteristicsandthus
Sincex/tisincreasingfromlefttorightinthex-tplane,thisformulamakessense
only in intervals of in which
is increasing as a function of x/t.
Therefore,
inaninterval,whichistherangeof .
Figure13.5.Shockspeedsastheslopeofachordforaconvexfluxf.
13.2.1.TrafficFlowModel
ThemodelfortrafficflowintroducedinChapter2hastheform
In this equation, ρ(x,t) ≥ 0 represents the traffic density at a location x on a
highway at time t, and the traffic flux is given by a function f(ρ) = ρv(ρ) in
which v = v(ρ) models how velocity v changes with traffic density ρ. The
simplestv(ρ)islinear:v(ρ)=vm(1−ρ/ρm),wherevm,ρmdenotethemaximum
speedandmaximumtrafficdensity,respectively.Then(13.9)hasaquadraticflux
andcanbeconvertedintotheinviscidBurgersequationwithasimplechangeof
variables.
Moregenerally,formsforv(ρ)canbebasedonrealtrafficdata,andtheresult
neednotbelinear.However,v′(ρ)≤0isareasonableassumption,andf(ρ)=
ρv(ρ)istypicallyaconcavefunction:f″(ρ)<0,implyingthatthecharacteristic
speedisadecreasingfunctionofdensityρ.
Notethatthecharacteristicspeedf′(ρ)=ρv′(ρ)+v(ρ)ispositiveforρnear
zero,butitisnegativenearmaximumdensity,wherev(ρ)=0.Thus,according
to the model (13.9), information about the traffic may flow forward, in the
direction of the traffic motion at low density, or backward at high density. We
can illustrate both cases with a classic example, in which vehicles approach a
trafficlight.
Example4.(Trafficlightprobleminthetrafficflowmodel)Alineoftraffic
withuniformdensity approachesatrafficlightlocatedatx=0.Supposeyou
areatrafficengineer,andyourjobistodesignthetrafficlightsequencetomake
thelightworkefficiently.
Let’ssupposethelightneedstostayredforatimet1>0,toreleasetrafficon
theotherroadsattheintersection.ThenyouneedtofindthelengthoftimeT>
0forthegreenlight.Asweshallsee,thetrafficbuildsupbehindtheredlight,
andthengraduallyreleaseswhenthelightturnsgreen.Thechallengeistofind
anexpressionforshortesttimeTatwhichthetrafficdensityreturnsto
.We
findTasafunctionoftheparametersinthemodel: ,ρm,vm,t1.
Firstnotethatwhenthelightturnsred,thedensityimmediatelytotheright
ofx=0isρ=0,andbehind,
.Subsequently,sincecarsarestoppedatthe
red light, the density immediately to the left of x = 0 is maximal, ρ = ρm.
Correspondingly,thereisastationaryshockatthelight,thatis,onewithspeed
zero.Notethatthismakessense,sincethelightpresumablydoesnotmove,and
butalsobecauseitisconsistentwiththeRankine-Hugoniotcondition.
Sincethereisnothinginthemodelthatallowscarstoslowastheyapproach
thelineofstationarycars,ashockformsbetweenthemovingcars(whichhave
density
) and the stationary cars (with density ρ=ρm). Consequently, the
shockformsandpropagatesbackwardasmorecarsjointhequeuewaitingatthe
light.ThespeedisgivenbytheRankine-Hugoniotcondition
sincef(ρm)=0.
Nowsupposethelightturnsgreenattimet1.Thisreleasescarsattheheadof
the queue to start moving. Since there are no cars ahead of the lead car, the
model dictates that it immediately travels with maximum speed vm,
correspondingtoρ=0.Infact,att=t1,thesolutionnearx=0constitutesa
Riemann initial value problem, with ρ = ρm,x < 0, and ρ = 0, x > 0. This
jump down gives rise to a centered rarefaction wave, as shown in Figure 13.6.
The rarefaction interacts with the backwards-propagating shock at some time t2
> t1, increasing its speed. Finally at time t3 > t2, the shock, labeled γ in the
figure, crosses x = 0, after which the traffic density is back to crossing the
trafficlightatx=0.TheintervalT=t3−t1istheminimumtimethatshould
besetforthegreenlighttoreleaseallthevehiclesthatwereheldupbythered
light.Noticethat,asinExample3,theshockpersistsforalltime,butthestrength
decaysastgetslarger.
13.2.2.NonconvexFlux
TheclassicscalarconservationlawwithanonconvexfluxistheBuckley-Leverett
equation,formulatedinthe1940sasasimplemodelfortheflowofoilandwater
in an oil reservoir. It is based on Darcy’s law relating pressure gradients in the
fluidstotheirvelocities.Aftersomesimplification,theequationtakestheform
In this equation, u = u(x, t) is the saturation of oil at a location x in the
reservoir, meaning the fraction of volume of pore space occupied by oil. Then,
assumingalltheporespaceisoccupiedbyeitherwateroroil,thesaturationof
wateris1−u.Thefunctionsλo,λwarerelativepermeabilitiesofoilandwater,
respectively,dependingontheirsaturations;theyarebothtypicallytakentobe
positive, increasing, and convex functions. The positive constant vT is the total
velocity(i.e.,thesum)ofthetwofluids.Weleaveitasanexercise(seeProblem
3)toshowthatf(u)in(13.10)ispositive,increasing,butnonconvexwithasingle
inflection point, when λo(u) = kou2 and λw(1 − u) = kw(1 − u)2, for positive
constantsko,kw.
Figure13.6.Solutionofthetrafficlightproblem.
Rarefactionwavesolutions
of(13.5)withanonconvexfluxf(u)area
littletricky,becausethecharacteristicspeed
hastoincreasewithx/t.
Moreover, because is obtained from the equation x/t = f′(u), f′ must be
invertible.Consequently,f″≠0throughouttherarefactionwave.
To show the role of nonconvex flux functions, we choose the simpler
nonconvexfunctionf(u)=u3,forwhichwecanmakeexplicitcalculations.Thus,
weconsidershocksandrarefactionwavesfortheequation
ut+(u3)x=0.
In the specific case f(u) = u3, the characteristic speed is f′(u) = 3u2, so that
rarefaction waves u(x, t) = (x/t) centered at x = t = 0 are given by
.Thus,rarefactionwaveshavepositivespeedandjoinconstantvalues
u=u±,providedu+<u−≤0or0≤u−<u+(whereasusualu−istotheleft
ofthewaveandu+istotheright).
Forconstantu−>0,thevaluesofu+<u−forwhichthepiecewise-constant
function
is an admissible shock for some speed s require the chord from u− to u+ to lie
abovethegraphoff.Thechordhasslopes=u2++u+u−+u2−andbecomes
tangenttofintwoways,correspondingtotwodifferentvaluesofu+.Whenu+
=−u−/2,thechordistangentatu+;whenu+=−2u−,thechordistangentat
u−. By comparing the slopes
of the tangents at u± to the shock speed, we
observethattheLaxentropycondition(13.8)becomes−u−/2<u+<u−when
u−>0.
Havingdescribedrarefactionwavesandadmissibleshockwaves,wecansolve
theRiemannproblem
SincethePDEisunchangedbychangingthesignofu,wecantakeuL>0and
solve (13.11) for different values of uR. From the discussion of shocks and
rarefactionsinExample2,weseethatthesolutionwillbeasingleshockwaveif
−uL/2<uR<uLandararefactionwaveifuR>uL.However,foruR<−uL/2,a
singlewavecannotsolvetheproblem,andweneedanewconstruction,ashockrarefaction. This solution includes a shock wave from uL to −uL/2, with speed
,thecharacteristicspeedatu=−uL/2.Butthenwecanjoin−uL/2touRby
ararefactionwave,sinceuR<−uL/2.Thesolutionthenisashockconnectedto
the rarefaction wave, hence the name shock-rarefaction. We illustrate the
solutionsinFigure13.7.
Figure13.7.SolutionoftheRiemannproblem(13.11).S:shock;R:rarefaction;
SR:shock-rarefaction.
13.3.TheLaxEntropyConditionRevisited
The Lax entropy condition can be motivated by considering changes in entropy
acrossshockwavesingasdynamics.Butwecanalsotreatentropyasanabstract
quantity, a significant notion in the modern theory of conservation laws,
includingtheexistenceandpropertiesofsolutionsofsystemsofequations.Inthis
section we relate the Lax entropy condition to this mathematical notion of
entropy.
Considertheconservationlaw
inwhichf:R→RisaC2strictlyconvexfunction:f″(u)>0forallu∈R.Letη:
R→RbeaC2strictlyconvexfunction:η″(u)>0forallu∈R,calledaconvex
entropyfunction.Wedefineanentropyfluxq:R→Rby
Thesepairsoffunctionsareassociatedwithanadditionalconservationlawthatis
satisfied by smooth solutions. We verify that ηt + qx = 0, using (13.12) and
(13.13):
Note that (13.13) also makes sense when η is only piecewise smooth, that is,
continuousandpiecewiseC1(forexample,piecewiselinear).Inthatcase,wesay
ηisaconvexentropyfunctionifitisaconvexfunctionintheweakersense:
forallx<y,0≤θ≤1.
TheobjectiveofthissectionistorelatetheLaxentropycondition(13.8)for
shockwavestotheinequality
To simplify the calculation of the inequality (13.14), we consider a piecewiseconstantshockwavesolutionof(13.12):
Herethespeed
andthesolutiononeachsideoftheshockareconstant.
Recallthatforaconvexfluxf(u),theLaxentropycondition
isequivalenttorequiringthattheshockjumpdown:
Inequality(13.14)istobeinterpretedinthesenseofdistributions:
foralltestfunctionsϕ∈C∞(R×R)withcompactsupport,satisfyingϕ(x,t)≥
0.Fortheshockwave(13.15),letη±=η(u±),q±=q(u±).Wefind
To verify this inequality from (13.17), write η(u) and q(u) in terms of the
HeavisidefunctionH(x),andusethefactthatthedistributionalderivativeH′=
δ,theDiracdeltafunction,anonnegativedistribution.Then
ifandonlyif(13.18)holds.
We say the shock satisfies the entropy inequality if we get strict inequality in
(13.18):
The point of the next theorem is to show that if the shock (13.15) satisfies the
Lax entropy condition, then not only is the inequality (13.18) satisfied for any
convexentropyη,butinfactthesharperstrictinequality(13.19)holds.
Theorem13.1.LetfbeaC2strictly convex flux, and let η be a C2strictlyconvex
entropy with entropy flux q. If the shock wave (13.15) satisfies the Lax entropy
condition,thenitsatisfiestheentropyinequality(13.19).
Proof.Supposetheshockwave(13.15)satisfiestheLaxentropycondition.Then
u−>u+.Letη:R→RbeC2,withη″>0.Wedefinefunctions
ThenE(u−)=0.TheproofwillbecompletewhenweshowthatE(u+)<0.To
doso,wedifferentiatetheRankine-Hugoniotcondition
aswellasE(u).Differentiating(13.20),wehave
Hence,foru<u−,andusing(13.13),
since
and
(by convexity of η) for u < u−.
Moreover,E′(u−)=0.Consequently,E(u+)<0foru−>u+.
We state the converse to Theorem 13.1 slightly differently, as we use
piecewiselinearconvexentropiesintheproof,whereasintheprevioustheorem
theentropywasassumedtobeC2.
Theorem13.2.Letf:R→RbeaC2strictlyconvexflux.Iftheshockwave(13.15)
satisfies the entropy inequality for all convex entropies η, then it satisfies the Lax
entropycondition(13.16).
Proof.Sincetheshockisassumedtosatisfytheentropyinequalityforallconvex
entropies, we can choose an entropy to give us the result. Suppose for a
contradictionthatu−<u+.Letk∈(u−,u+),anddefine
Thenfrom(13.13),wehave
Therefore,sinceu−<k<u+,
Substitutingintotheentropyinequality(13.19)gives
Thisisaninequalitybetweenchordsconnectedto(u−, f(u−)) in the graph of f.
Butsinceu−<k<u+,theinequalitycontradictstheassumptionthatfisstrictly
convex.Therefore, u−>u+, since in the case u− = u+, the entropy inequality
(13.19)failsalso.
It is helpful to sketch the graph of the convex function f and the various
chords referred to in the proofs above, to understand how the proofs work.
Incidentally,asimilarcalculation(essentiallyexchangingu+andu−)showsthat
ifu−>u+,then
foru+<k<u−,whichisconsistentwiththeconvexityoff.
Itisworthnotingthatthespecificentropies(13.21)areacrucialconstructin
the existence theory of Kruzkov [31] for the Cauchy problem for scalar
conservationlaws.Ratherthangoingmoredeeplyintotheexistencetheory,we
return in the next section to some other issues relating to nonconvex flux
functions.
13.4.UndercompressiveShocks
Inthissectionwereturntotheexample
of a scalar conservation law with concave-convex flux f(u) = u3. Shock waves
(13.15)satisfytheLaxentropyconditiononlyfor
,ifu−>0.Fora
convex flux function, all other shock waves are expansive in the sense that
characteristics leave the shock on both sides. In contrast, for a nonconvex flux,
thereareshockwavesforwhichthecharacteristicsentertheshockononeside
and leave on the other side. For (13.22), we found earlier (by comparing the
shock speed to characteristic speeds) that these shocks have
whenu−>0.
Inthemid-1990s[24]itwasrealizedthatsuchshockwavescanbelimitsof
travelingwavesolutionsofequationsthatincludehigher-orderderivatives.This
was after a decade of research on such undercompressive shocks in systems of
equations,occuringasdynamicphasetransitions[1,25,40,42]inelasticsolids
and gases, as transition waves in oil recovery models [41, 23] and in
magnetohydrodynamics[47].
To justify the shocks that fail the Lax entropy condition, we seek traveling
wavesolutionsofthemodifiedKdV-Burgersequation,
Thetermmodifiedisusedbecausetheu2nonlinearityofBurgers’equation,andof
the KdV equation, is modified to u3. The names Burgers and Korteweg-deVries
(KdV)areassociatedwiththedissipativeanddispersivetermsontheright-hand
side, respectively. The parameters ϵ and γ are positive constants. We choose a
positive coefficient for the dispersive (third-order) term; the consequences of
reversingthesignofthedispersivetermareleftasproblem12.
Thetravelingwavesweseekhavethesameformasinthepreviouschapter:
, ξ = (x − st)/ϵ, satisfying far-field boundary conditions
u±.
Sincetheequationisunchangedbythetransformationu→−u,wecanrestrict
attentiontotravelingwaveswithu−>0.
AsinChapter12,weconvertthePDEintoanODEandintegrateonce,with
theresult(inwhichwehavedroppedthebarsonu)
Recallthatweanalyzedasimilarequationwhenseekingtravelingwavesforthe
bistable equation (12.13). As in the analysis of that equation, we write the
second-orderODEasafirst-ordersystem:
Let F(u, v) = (v, −γ v + (u3 − u3−) − s(u − u−)). The vector field F has
Jacobian
For u− > 0, and −2u− < u+ < −u−/2, let
. Then there are
threeequilibria(u,v)=(u,0),withu=u±andu=u0satisfyingu+<u0<u−,
and
Attheoutsideequilibria,u=u±,weobservethats<3u2;theeigenvaluesofdF
arerealandofoppositesigns,givenby
Thatis,theequilibriaaresaddlepoints.Themiddleequilibriumatu=u0 has
, so that it is a stable spiral (if the eigenvalues are complex) or a stable
node.
Much as we did for the bistable equation, we seek heteroclinic orbits (u(ξ),
v(ξ)) that are saddle-saddle connections from (u−, 0) to (u+, 0) by finding
parametervaluesforwhichthereisaninvariantparabolicmanifold
through these two equilibria. Since u+ < u−, we must have k > 0 to get a
travelingwavethatfollowstheparabolafromu−tou+andhenceisdecreasing:v
=u′<0.
Writing
(13.24)usingv=v(u),
and rewriting the system
weseethat(aftercancelingfactors(u−u−)(u−u+))
Equatingcoefficientsgives
Using(13.25),weget
Inparticular,themiddleequilibriumdependsonlyonγ,sothatthechordwith
slope s pivots on this point as u− is varied. But then for u0 to be the middle
equilibrium, we must have
. At the threshold, u+ = u0. In
summary,wehavethefollowingtheorem.
Theorem13.3.[24] For each γ > 0, ϵ > 0 and for each
traveling wave
of (13.23) satisfying
,withspeed
.
there is a
and
Asϵ → 0+, the traveling wave u(x,t) of the theorem approaches the shock
wave solution (13.15) of (13.22). This solution is undercompressive in the sense
that characteristics pass through the shock from left to right, meaning that the
shockissubsoniconbothsides.As
,wehave
,andthechord
becomestangentatu+=u0=−u−/2.
Undercompressive shocks have considerable influence on the structure of
solutionsofthescalarequation(13.22).TheRiemanninitialvalueproblem
hasweaksolutionsthataremonotonicforallchoicesofuL,uRifonlyLaxshocks
areused,butitcanbenonmonotonicifundercompressiveshocksareconsidered.
Forexample,considerγ>0tobeset,letuL=1,and
.
ThenthesolutionoftheRiemannproblem(13.26)consistsoftwoshocks,eachof
whichhasatravelingwavesolutionof(13.23):
Here
,
. The slower shock is
undercompressive,andthefasteroneisaLaxshock.Becauseofthestructureof
characteristics in this solution, we observe that small disturbances ahead of the
faster wave are overtaken and absorbed by that wave, whereas small
disturbancesbehindtheslowerwavepassthroughthewaveandareabsorbedby
thefasterwave.Detailsofthistwo-shocksolutionandotherscanbefoundinthe
articlebyJacobsetal.[24]andinthebookbyLeFloch[34].
13.5.The(Viscous)BurgersEquation
InthissectionweconsidertheviscousBurgersequation
in which the viscosity ϵ > 0 is a constant. The equation couples nonlinear
transport to linear diffusion. To solve the Cauchy problem (13.28) with initial
datau0∈L1(R),
we convert the PDE into the heat equation with a clever change of dependent
variable known as the Cole-Hopf transformation [9, 21]. To motivate the
transformation,let
andintegratetheresultingequation
withrespecttox,takingtheconstantofintegrationtobezero:
Nowu=wx,sothetermswithspatialderivativescanbecombinedandwritten
as
suggestinganintegratingfactor
Nowmultiply(13.30)by
Thus,since
toget
,wehavetheheatequationforz:
andtheinitialcondition(13.29)becomesaninitialconditionforz:
Wecansolvethisinitialvalueproblemforz(x,t),usingthefundamentalsolution
fortheheatequation(see Section5.1). To exploit the exponential in the initial
condition(13.32),let
.Then
solves(13.31),(13.32).
TorecoverthecorrespondingsolutionofBurgers’equation,wehave
whichistheCole-Hopftransformation.
Of course, we could have started with the transformation (13.33) and
substitutedintoBurgers’equation(13.28), leading to the heat equation (13.31)
aftersometediouscalculation,butitwouldnothavebeensomuchfun.
Finally,wehavethesolutionoftheinitialvalueproblem(13.28),(13.29):
This explicit formula for the solution of the Cauchy problem is explored in
variouswaysinWhitham’sclassicbookonwaves[46].Inparticular,withsome
careful asymptotic analysis it is possible to take the singular limit ϵ → 0+, to
recover Lax’s solution of the Cauchy problem for the inviscid Burgers equation
[12].
13.6.MultidimensionalConservationLaws
Inthisshortsectionweconsiderscalarconservationlawsintwodimensions:
Themethodofcharacteristicsgiveslines
thus,x=f′(u)t+x0,y=g′(u)t+y0,onwhichu(x,y,t)isconstant.Thus,the
Cauchyproblem,withinitialcondition
hasthesolutionu(x,y,t)givenimplicitlyby
AcalculationduetoConway[10]andreportedbyMajda[35]showsthatshock
formationiseasilydescribedasacriterionontheinitialdata.Ifwedifferentiate
(13.35)withrespecttoxandy,wegetequationsfortheevolutionof∇u=(ux,
uy).Let(v,w)≡∇u.Thenalongacharacteristic,
where
isdifferentiationalongthecharacteristic.
Nowletq(t)=f″(u)v(t)+g″(u)w(t).Notethatq(t)=∇.(f′(u),g′(u)).Sinceu
is constant along the characteristic, we have
. But using
(13.36),wefind
TheinitialconditionforthisODEis
Thesolutionis
Consequently,q(t)blowsupinfinitetimet∗=−1/q0if q0<0.Thatis,shock
formationtakesplaceinfinitetimefromsmoothinitialdataif
atsomepoint(x,y).
Example 5. (Shock formation in the avalanche model) The avalanche
example(3.17)givesaninterestingtwistonthiscalculation.ThePDE(havingset
S=1forsimplicity)isoftheform(13.35),exceptthatf=yudependsexplicitly
ony:
Differentiatingwithrespecttox,yleadstothesystem
forthegradient(v,w)=∇u.Noticethatthissystemresembles(13.36),butthe
linear term makes the dynamics work rather differently. Trajectories for system
(13.37)arerepresentedbycurvesinthev-wplanethatwefindbywritingsystem
(13.37)as
Thisisanequationforw2asafunctionofv,whichcanbesolvedwiththeresult
thatthetrajectoriesareallconicsections
where c ∈ R is a parameter, the constant of integration. Equation (13.38)
representsellipsesifc > 0, hyperbolas if c < 0, and a parabola if c = 0. The
trajectoriesareshowninFigure13.8.Aremarkablepropertyofsystem(13.37)is
thatthesolutionscanbewrittenexplicitly:
Consequently, ∇u = (v, w) becomes unbounded in finite time if q(t) has a
positivezero.Thisisaconditionontheinitialgradient(v0,w0).Theresultisthat
all the unbounded trajectories shown in the figure correspond to initial
conditionsthatresultinfinite-timeshockformation.
Thesolution(13.39)isobtainedusingODEtricks.Letv=zw.Thenv′=z′w
+zw′.Using(13.37)toeliminatev′andw′,wefindz′−z2=0.Thus,z=v/w
=(c−t)−1,wherec=w0/v0fromtheinitialconditiont=0.Therefore,w=(c
−t)v,andthefirstequationin(13.37)becomes v′=2v2(t−c),whichcanbe
solvedbyseparatingvariables,leadingtotheexpressionforv(t)in(13.39).The
expressionforw(t)isthengivenbyw(t)=(c−t)v(t).
Figure13.8.Phaseportraitandtrajectoriesforsystem(13.37).
13.6.1.ShocksinTwoDimensions
The Rankine-Hugoniot jump condition in multiple dimensions is derived using
theDivergenceTheorem.Inonespacedimensionashockcurveisapointateach
time,acrosswhichthesolutionjumps,butweusedadirectcalculationtoderive
the jump condition. In two space dimensions a shock is a space-time surface, a
curveateachtime,acrosswhichthesolutionjumps,andwhichevolvesintime.
Equation(13.35)isinconservationform,sothatwecanapplytheDivergence
Theorem in x-y-t space. Suppose a weak solution is piecewise C1, with a jump
discontinuityacrossasurfaceϕ(x,y,t)=0.Thenormaltothesurfaceis∇ϕ=
(ϕx,ϕy,ϕt).Sincewewantthesurfacetointersecteachconstant-timeplaneina
curve, we assume that the surface can be parameterized in terms of t and a
combination of the spatial variables. Rotating if necessary, we can assume
, so that
, and the normal is
. Now
considerthat(13.35)canbewrittenas
Thenthejumpconditionstatesthatthenormalcomponentofthevectorfield(u,
f(u),g(u))iscontinuousacrossthesurface,andhencehasnojump:
where
.Specifically,thisbecomes
Correspondingly, the Lax entropy condition is formulated just as for the onedimensionalcase,wherethesinglespacevariableisnormaltotheshockcurveat
eachtime.Thedetailsofthefollowingcalculationarecoveredinsomegenerality
in Serre’s monograph [39]. Let u± be the limits from the right and left in x:
. The characteristic speeds in the normal direction at the
shockaregivenby
The Lax entropy condition compares the characteristic speeds with the shock
speedin(13.40):
This condition can be used to show stability of Lax shocks, meaning that for
nearby initial conditions, the Cauchy problem is well posed [39]. Stability of
shock waves for scalar equations and systems is a topic of ongoing research. In
the next chapter we discuss shocks, rarefactions, and the Riemann problem for
systemsofequationsinonespacedimensionandtime.
PROBLEMS
1.Differentiate u(ax,at)=C with respect to a. Solve the resulting linear firstorderPDEbythemethodofcharacteristicstoprovethatuisafunctionofx/t.
2.Showthatifu(x,t)iscontinuous;isC1oneithersideofacurveC;satisfiesthe
PDEateachpointnotonC;and ux,utarecontinuousupto Cbuthaveajump
discontinuityacrossCthenthecurveCisacharacteristic.
3.Supposeλo(u)=kou2andλw(1−u)=kw(1−u)2,withpositiveconstantsko,
kw in (13.10). Show that f(u) in (13.10) is positive, increasing but nonconvex,
withasingleinflectionpoint.
4.Showthatif
for all u,v, then f(u) is a quadratic polynomial. (Hint: Take v to be fixed, and
solvetheODEforf(u).)Explainthesignificanceofthisresult.
5.ForthesolutionofExample3,Figure13.4,dothefollowing.
(a)Sketchgraphsofthesolutionu(x,t)asafunctionof x for various times,
suchast=0,t=1,t=3.
(b)Writethesolutionoutasaformula,anddescribehowu(x,t)approaches
zeroast→∞.
(c)Istheconvergencetozeroast→∞uniform?Explainwithaproof.
6.Findallvaluesofthepositiveconstantsa,b,c,dforwhichthefunction
isaweaksolutionoftheinviscidBurgersequationfor−∞<x<∞,t≥0,and
satisfiestheLaxentropyconditionattheshock.(Hint:ThePDEandtheRankineHugoniotconditiongivethreeequationsfora,b,c,d.)
7.ConsiderthePDE
(a)WritetheRankine-Hugoniotconditionforashockwave
(b) Find all the shocks (13.41) with u− = 1 that satisfy the Lax entropy
condition.
(c)Letu−=1.Findallvaluesofu+forwhichthereisararefactionwave
andwriteaformulaforthefunction .
8. Use the implicit equation(13.7) or differentiate (13.5) to derive an ODE for
theevolutionofv=uxalongcharacteristics.BysolvingtheODE,findconditions
underwhichthefollowingaretrue.
(a)Thesolutionremainssmoothforallt>0.
(b) The solution develops a singularity in finite time, due to |v| → ∞.
Characterizethetimeatwhichthissingularityforms.
9. Complete the solution of the traffic light problem (Example 4), shown in
Figure13.6,bydoingthefollowing.
(a)Findaformulafortheshockcurveγ.
(b)Determinethetimet3asafunctionoftheotherparameters.
10.Trafficofuniformdensityisproceedingalongastraightsingle-lanehighway
atspeedv0.Attimet=0,thecaratlocationx=0slowssuddenlytohalfspeed
v0/2.Determinewhathappenssubsequently,writingformulasandsketchingthe
x-tplanetoshowthestructureofthesolution.
11.ConsiderthemodifiedKdV-Burgersequation(13.23)withϵ=1butwiththe
signofthedispersivetermreversed:
(a)Explainwhythisequationcannotbereducedto(13.23)withachangeof
variables,whereasforthequadraticnonlinearityoftheKdV-Burgersequation,
thesignofthedispersivetermcanbechanged(withoutchangingγ>0).
(b)WritetheODEfortravelingwavesolutionsu=u(x−st)asafirst-order
system.Considervaluesofu±forwhichtherearethreeequilibriau+<u0<u
−. Show that the only heteroclinic orbits join equilibria u± to u0, and the
correspondingshockwaves(13.15)satisfytheLaxentropycondition.
(c) Summarize what is different about this example from the modified KdVBurgersequationinSection13.4.
12.Letγ>0.SolvetheRiemannproblem(13.26)withuL=1,inthefollowing
twocases.
(a)When
.Yoursolutionshouldhaveararefactionwaveandan
undercompressiveshock.
(b)When
.Inthiscase,thesolutionisasingleshock.Showthat
itisaLaxshockandhasacorrespondingtravelingwavesolutionof(13.23).
Show that for
, the solution consists of two shocks
traveling at different speeds, an undercompressive shock and a faster Lax
shock.
13.FindthecurveΓshownastheheavysolidcurveinFigure13.8.Showthat,
togetherwiththepositivehalfofthew-axis,Γseparatesinitialconditionsv0,w0,
for which a shock forms, from initial conditions that do not lead to shock
formation. (Hint: You can solve this problem by examining when the quadratic
equationq(t)=0hasapositivesolution.)
CHAPTERFOURTEEN
SystemsofFirst-OrderHyperbolicPDE
Systems of hyperbolic conservation laws are fundamental to the study of
continuum mechanics: the dynamics of fluids and solids. Nonlinear systems
possess a richer structure than scalar equations or linear systems, because they
havemorethanonecharacteristicspeed,andwaveswithdifferentspeedsinteract
nonlinearly.
We begin the chapter with linear equations, for which the method of
characteristicscanbegeneralizedfromthescalarequationsofChapters3and13,
leading to a short-time existence theorem for classical solutions. Then we show
howweaksolutionsofnonlinearsystemsofconservationlawscanbeconstructed
fromshocksandrarefactionwaves.Alongthewayweshowapplicationstothepsystem,totheelasticstring,andtoshallowwaterwaves.
14.1.LinearSystemsofFirst-OrderPDE
Weconsiderlinearsystemsofequationsinonespacevariablex∈Randtimet>
0oftheform
whereU=U(x,t)∈Rnisthedependentvariable,andA=A(x,t)isann×n
matrix depending smoothly on its arguments. We shall consider the Cauchy
problem,forwhichweposeinitialconditions
System (14.1) is hyperbolic if A has n real eigenvalues λk, with n linearly
independent (right) eigenvectors rk, k = 1, 2, …, n. The system is strictly
hyperboliciftheneigenvaluesλk,k=1,2…narealldistinct.Theeigenvaluesof
Aarecalledcharacteristicspeeds,asinthescalarcase.Wewillalsoneedtheleft
eigenvectors lk, which are row vectors. We can assume the eigenvectors have
beenchosentosatisfy
14.1.1.LinearConstant-CoefficientSystems
We begin by considering linear constant-coefficient systems, in which A is a
constantmatrix.Inthiscase,thesystemadmitssolutionsintravelingwaveform
(muchlikethelineartransportequation):
SubstitutingUintosystem(14.1),wehave
sothatλ=λkisaneigenvalueofA,andv=rkisarighteigenvector.Sucha
solutionisatravelingwavewithspeedλk.
Tosolvethesystem(14.1)whenAisaconstantmatrix,wesimplydiagonalize
thematrix,orequivalently,decomposeUusingtheeigenvectors:
whereuk:R×R+→R.Substitutinginto(14.1)andusingArk=λkrk,weget
decoupledequationsfortheunknowncoefficientsuk:
Thus,uk(x,t)=φk(x−λkt)forscalarfunctionsφk,k=1,…,n,andwehavethe
generalsolutionofthesystem
The initial condition (14.2) is satisfied by setting t = 0 and using the left
eigenvectors:
Example1.(Thewaveequationasasystem)Asimpleexampleofaconstantcoefficient first-order system is provided by the wave equation in one space
dimensionandtime
WewritethePDEasasystembyletting
Thenwehave
andtheinitialconditionsw(x,0)=f(x),wt(x,0)=g(x)become
Thus,intermsofsystem(14.1),wehave
Thecharacteristicspeedsare±c,witheigenvectors
Thesolutionoftheinitialvalueproblemisthus
inwhichφ±arerelatedtof′,gbyφ++φ−=f′;φ+−φ−=g/c.
14.1.2.MethodofCharacteristicsforVariableCoefficients
Now let’s consider how the method of characteristics, which we used so
successfullyforscalarequations,canbeappliedtolinearsystemswhenA=A(x,
t)isnotaconstantmatrix.Forthispurpose,weagaindiagonalizethematrixA
usingtheeigenvectors.Thatis,wesubstitute(14.4)intosystem(14.1),butnow
theeigenvectorsridependonx,taswellasonthecoefficientsui:
Taking the scalar product of this equation with the left eigenvectors lj (see
(14.3)),weobtainasystemoftheform
in which the coefficients dij are continuous functions of x, t. Similarly,
decomposingtheinitialconditions(14.2)intoeigenvectorsofA(x,0),weobtain
initialconditionsforthecoefficientsui:
Characteristicsaredefined,asforscalarequations,by
Inparticular,througheachpoint(x,t),witht>0,therearencharacteristicsx
=xi(τ),i=1,2,…,n,passingthrough(x,t),eachwithadifferentspeed.Each
intersectsthex-axisatadifferentpoint
.Werecognizetheleft-handside
of the ith equation in (14.5) as dui/dτ (i.e., differentiation along the ith
characteristic). This enables us to convert equations (14.5) into integral
equations,byintegratingalongtheithcharacteristicfrom
to(x,t):
Nextwewritethissysteminvectorform:
where
,andK(u)(x,t) denotes the vector of integrals on the righthandsideof(14.6).
Solving the Cauchy problem is equivalent to finding a fixed point of the
mapping B : u → φ + K(u) in a suitable space of functions. The appropriate
settingisthecontractionmappingprinciple(seeAppendixA)intheBanachspaceX
=CB(R×[0,T],Rn)ofcontinuousboundedfunctionsfromR×[0,T]toRn,
withT>0tobechosen.Thisprincipleisusedtoproveexistenceofsolutionsof
initialvalueproblemsforsystemsofODE.Thecontexthereisnotmuchdifferent,
as we have reduced the Cauchy problem to a system of integral equations. The
norminXisthesupremumnormonboundedcontinuousfunctions:
We assume that the entries in the matrix A(x, t) are in X, and so are their
derivatives.ThenthecoefficientsdijareallinX,sincetheydependonderivatives
oftheeigenvectorsri(x,t),whichbystandardmatrixanalysishavederivativesin
X[28].Sinceforlinearequationsthecharacteristicspeedsdonotdependonthe
solutionu,thefunctionφisalsoindependentofu.(Fornonlinearequations,the
dependenceonuwouldbethroughthepoints .)
Foru,v∈X,wehavethatfor0≤t≤T,
Therefore,forT>0sufficientlysmall,Kisacontraction;hencethemappingBis
also.BytheBanachfixed-pointtheorem(seeAppendixA),Bhasauniquefixed
point in X. Thus, we have proved the following theorem for the initial value
problem
Theorem14.1.LetU0∈CB(R),andsupposethematrixA=A(x,t)in(14.7)has
entries in CB(R), together with their derivatives. Then there is T > 0 such that the
initialvalueproblem(14.7)hasauniquesolutionu∈CB(R×[0,T],Rn).
14.2.SystemsofHyperbolicConservationLaws
Inthissectionweconsidernonlinearsystemsofconservationlaws
in which F : Rn → Rn is a C2 function. For nonlinear systems we can obtain a
short-time existence and uniqueness theorem similar to Theorem 14.1 under
suitableassumptions.However,inthissectionwefocusonissuesarisingbecause
of nonlinearity—notably, shock waves and rarefaction waves—and show how
thesewavesappearinapplications.
We refer to (14.8) as an n × n system. We shall assume strict hyperbolicity,
meaningthattheJacobiandF(U),ann×nmatrixforeachU∈Rn,hasdistinct
realeigenvalues,namely,thecharacteristicspeeds
withcorrespondingleftandrighteigenvectorslj(U),rj(U),j=1,…,nsatisfying
(14.3)ateachU:
Strict hyperbolicity allows us to discuss unambiguously the kth characteristic
fieldbyreferringtothecharacteristicspeedλk.
Thenotionofgenuinenonlinearityforscalarequationshasacounterpartfor
systemsbyconsideringthecharacteristicfieldsseparately.Thekthcharacteristic
fieldisgenuinelynonlinearinasetΩ⊂Rnif
ThefieldislinearlydegenerateinΩif
Afunctionψ:Rn→Risak-Riemanninvariantif
Then for each k, there are, locally in U, n − 1 Riemann invariants whose
gradients are linearly independent at each U. This follows because (14.9) is a
scalarequationforψ(U).Theequationsaysthatwisconstantoncharacteristics,
whichareparalleltork(U).Consequently,eachRiemanninvariantisconstanton
rarefactioncurves,whichareintegralcurvesofthevectorfieldrk(U):U′=rk(U).
Asforscalarequations,theconstructionofrarefactionwavesiscloselyrelatedto
thepropertyofgenuinenonlinearity.
ARiemanninvariantψcanbespecifiedlocallynearagivenpointU0∈Rnby
defining its values on an n − 1 dimensional manifold M orthogonal to rk(U0),
say,
onM.Then
istangenttoMatU0,where∇Misthegradient
operator along the linear space tangent to M at U0. We define ψ near U0 by
solvingrk·∇ψ=0bythemethodofcharacteristics.Asetofn−1independent
Riemann invariants is given by choosing n − 1 functions on M with the
propertythatthen−1gradients
arelinearlyindependentatU0.Thenthe
gradients∇ψ(U)arelinearlyindependentforallUnearU0andareorthogonalto
rk(U).
For a 2 × 2 system (of n = 2 equations in 2 unknowns u,v), the Riemann
invariantscanbeusedasanalternativepairofvariables,withtheadvantagethat
they partially diagonalize the system. To be specific, let λ±(ψ−,ψ+) denote the
two characteristic speeds, written as functions of the Riemann invariants ψ±.
Thenthefollowingholds:
We leave verification of this property to Problem 7. This property means that
eachRiemanninvariantψ±isconstantoncharacteristicsx′=λ∓oftheopposite
family!
Itisworthcomparingthenotationandconstructionsforsystemsofhyperbolic
conservationlawswiththecorrespondingdevelopmentinSections13.1and13.2
on scalar conservation laws from the previous chapter. We shall use three
examples of systems of conservation laws to illustrate the ideas in this chapter,
beginningwiththeissueofhyperbolicity.
14.2.1.Thep-System
This2×2systemisderivedfromthequasilinearwaveequation:
inwhichσ:R→Risagivensmoothfunction.Theequation,derivedinSection
4.1.1,isamodelofnonlinearone-dimensionalelasticity,inwhichw=w(x,t)is
thedisplacement,andwxisthestrain(ordisplacementgradient),andσ=σ(wx)
isthestress(agivenfunctionofthestrain).
Atypicalstress-straincurveisshowninFigure14.1.Theverticalasymptoteat
wx = 0 corresponds to the infinite force needed to compress the material to a
point, and σ(1) = 0 corresponds to a uniform unstressed material with
displacementw(x,t)=x.
Towrite(14.11)asasystem,weset(asforthelinearwaveequation)u=wx
andv=wt,namelythestrainandvelocity,respectively:
Figure14.1.Atypicalnonlinearstress-strainlaw.
Thisisanexampleofsystem(14.8)inwhich
Tocheckhyperbolicity,wecalculate
whichhaseigenvaluesandeigenvectors
Thus,thesystemisstrictlyhyperbolicifandonlyifσ′(u)>0;inwords,stressis
astrictlyincreasingfunctionofstrain.Inmostcircumstancesitisappropriateto
consider such monotonic stress-strain laws, although strain-softening materials,
whichhavesomerangeofstrainsoverwhichthestressdecreases,areimportant
inmaterialsscience.Examplesincludecertainplastics(trystretchingathinstrip
of soft plastic to feel its strain softening) and some metal alloys used to make
smartmaterialsthatchangetheirstress-strainpropertiesincontrolledways.
Thesystemisgenuinelynonlinearinbothcharacteristicfieldsprovidedσ″(u)
≠0.Itislinearlydegenerateinbothfieldsifσ″(u)≡0,inwhichcaseσ(u)isan
affinefunctionofu,correspondingtoHooke’slaw.TheRiemanninvariantsare
satisfying ∇ψ± · r± = 0 and corresponding to
, respectively. Recall that the Riemann invariants allow us to
diagonalizethesystemofequations.(See(14.10).)
System (14.12) also describes one-dimensional isentropic gas dynamics in
Lagrangianvariables,inwhichurepresentsthespecificvolume(thereciprocalof
density),visonceagainvelocity,and−σ=pisthepressure.Thenamep-system
derivesfromthiscontextofgasdynamics.Furtherpropertiesofthep-systemare
discussedlaterinthischapterandinthebookbySmoller[43].
Figure14.2.Shallowwaterflow.
14.2.2.TheShallowWaterEquations
Forabodyofwateronaflathorizontalsurface,wewishtodescribethemotion
of the free surface as the water flows. The full equations of flow for a viscous
incompressiblefluid(suchaswater)aretheNavier-Stokesequations,discussedin
the next chapter. The shallow water equations can be derived by a scaling and
approximationargumentfromthefullequations,butherewegiveanabbreviated
andmoredirectroute.
Let’srestricttofluidmotioninaverticalplane(forexample,weassumethe
fluidismovingparalleltosidewalls,likeariverflowingparalleltoitsbanks),so
thatthevelocityandheightofthefluiddependonlyononehorizontalvariablex
andaverticalvariabley,asinFigure14.2.
As shown in the figure, let y = h(x,t) be the height of the fluid surface at
timet,andletu(x,t)denotethehorizontalvelocityofthelayeroffluid,which
wetaketobeindependentofdepth.Moreprecisely,u(x,t)representsthedepthaveraged horizontal velocity. Note that we are using Eulerian coordinates to
describe the flow, since (x, y) is a fixed physical location. In a Lagrangian
formulation, using a reference configuration, individual fluid particles are
labeled,andvelocityisdeterminedfromtheparticletrajectories.
Conservationofmassisexpressedbytheequation:
The pressure p in the fluid layer depends on depth, and we assume it is
hydrostatic, meaning that pressure increases linearly with depth and is
atmosphericpressurepaatthefreesurface:
Neglecting viscous forces, we can write the conservation of horizontal
momentum(anexpressionofNewton’ssecondlawF=ma)as
Substitutingforp,weobtain
Equations(14.13),(14.14)constitutea2×2systemoftheform(14.8),with
Tocheckhyperbolicity,wecalculate
whichhaseigenvaluesandeigenvectors
respectively.Thus,thesystemisstrictlyhyperbolicforh>0,andonefamilyof
characteristicstravelsfasterthanthefluidvelocity,whiletheotherfamilymoves
more slowly. The Riemann invariants
satisfy ∇ψ± · r± = 0.
TheseRiemanninvariantsfortheshallowwaterequationswillbeusefulwhenwe
solvetheclassicdam-breakproblem,whichwetreatlaterinSection14.3.
There is an extensive literature on shallow water waves and the related
equationsofshallowflow[2,15,22,46]usedtomodeltsunamisandavalanches.
Thehyperbolicstructureoftheequationsisimportantwhenfindingsolutionsin
theseapplications.
14.2.3.TheElasticStringEquations
Theequationsformotionofanelasticstringintwodimensionsformasystemof
two quasilinear wave equations, which we rewrite as a 4 × 4 system of firstorderconservationlaws.
As with one-dimensional elasticity, leading to the p-system, the reference
configuration is one dimensional, but now we allow deformations in two
dimensions.Thedisplacementorpositionvector,whichwewritehereasr=(r1,
r2)∈R2,isafunctionofx,t,withxinanintervalIrepresentingmaterialpoints
inthestring(seeFig.14.3).ForcesinthestringarerepresentedbythetensionT,
afunctionofx,tthatmeasuresthemagnitudeoftheinternalforceofthestring.
We assume the direction of this force is tangential to the string, based on the
simplifyingassumptionthatthestringexertsnoresistancetobending.Modeling
theforcebyelasticitymeansthatTisafunctionofthestrain,heregivenbythe
scalar|rx|.
Theequationsofmotionarethen
The density and cross-sectional area of the string, both taken to be constant
functions of x and t, are incorporated into the tension T. On the left of this
equationistheacceleration,andontheright,thetensionTisthemagnitudeofa
force whose direction is that of the unit tangent. The equation expresses
conservationoflinearmomentum,Newton’ssecondlawF=ma.
Figure14.3.Theelasticstring.(a)Referenceconfiguration,intervalI;(b)
physicalstring.
ThefunctionT is typically increasing and takes the form of the stress-strain
law shown in Figure 14.1. This makes sense, because for purely longitudinal
motion(alongthelengthofastraightstring),thestringequations(14.15)reduce
to the single wave equation(14.11), with T = σ. However, even if we assume
linearelasticitygivenbyHooke’slaw,
withelasticconstantE>1,theequationsarestillnonlinear,unlessthemotionis
purelylongitudinal.
Towrite(14.15)asasystemoffirst-orderequations,wedefinevariablesp,q,
u,vby
Then(14.15)isequivalentto
whichhastheform(14.8)whenwewrite
Tocheckhyperbolicity,wecomputedF(U):
inwhich
Theblockstructureofthematrix
impliesthattheeigenvaluesofdF(U)aresquarerootsofthetwoeigenvaluesof
B.Calculatingthelattereigenvalues,wefindthatthecharacteristicspeedsare
Consequently,theelasticstringequationsarestrictlyhyperbolicifandonlyif
In particular, if we assume the stress-free equilibrium state corresponds to the
normalizationT(1)=0,thenT(ξ)<0forξ<1,sothat(14.18)doesnothold
forξ < 1. This is related to the fact that since the string has no resistance to
bending, its motion under compression is not well behaved (the equations are
linearlyillposed).Consequently,weconsidertheelasticstringequationsonlyfor
stringsundertension(i.e.,withξ>1).
Hooke’s law (14.16) leads to a strictly hyperbolic system for ξ > 1. More
generally,weseethatif
then(14.18)issatisfiedforξ∈(1,ξmax),whereeitherξmax=∞orξmax<∞is
thefirstvalueofξ>1forwhich
Thenwehave
As we saw in Section 14.1.2, the characteristic speeds are associated with
transverse waves, whereas
are associated with longitudinal waves
(corresponding to solutions of the p-system, a connection with longitudinal
motion suggested above). Thus, longitudinal waves travel faster than transverse
wavesfor1<ξ<ξmax.
Thenotionsoflongitudinalandtransversewaveshavecounterpartsinthreedimensionalelasticity,forexample,inmodelingsubterranianearthquakewaves.
Longitudinalwavesarepressurewaves,whereastransversewavescorrespondto
shearwaves,inwhichmaterialisslidingoveritselftransversetothedirectionof
propagationofthewave.
We include Problems 2 and 6 at the end of the chapter to explore further
properties of the elastic string equations, and there is a nice paper on the
equationsbyAntman[4].Havingestablishedthreerepresentativeexamples,we
return to a general strictly hyperbolic system to discuss rarefaction waves and
thenshocks.
14.2.4.RarefactionWaves
Wereturntothegeneralsystem
inwhichF:Rn→RnisaC2function.Ourstandingassumptionisthatthesystem
isstrictlyhyperbolic,meaningthattherearendistinctrealcharacteristicspeeds,
the eigenvalues λk(U), k = 1, …, n of dF(U), with associated right/left
eigenvectorsrk/lksatisfying
For rarefaction waves, we consider characteristic fields that are genuinely
nonlinear,meaningvaluesofkandrangesforUforwhich
Genuine nonlinearity is essential for the existence of centered rarefaction waves,
continuousscale-invariantsolutions
,whereV:R→Rn.Substituting
thisforminto(14.19),wefind
Multiplyingbytandletting
,wethenhave
Hence,ξisaneigenvalueofdF(V),andV′isaneigenvector:
for some k = 1, …, n. Thus, V(ξ) lies on an integral curve of rk(U). (See
Appendix C.) Moreover, ξ and λk increase together, so the integral curve is
orientedsothatλk(U)increasesasUmovesalongthecurve.
Figure14.4.Centeredrarefactionwaveforthekthcharacteristicfield.
Thecorrespondingrays
inthex-tplanearek-characteristics:
. If we differentiate the first equation in (14.21) with respect to ξ and use the
secondequation,wefindadirectionalderivative:
Wecaninterpret(14.22)asdeterminingthemagnitudeoftheeigenvectorrk;it
also includes the choice of sign so that rk points in the direction of increasing
characteristicspeed.
We next relate the oriented integral curve to the rarefaction wave that
continuouslyconnectsaconstantU−toaconstantU+,asshowninFigure14.4.
In the figure, the straight lines are k-characteristics, and the curved line is a
characteristicofadifferentfield.Thesolutioniscontinuousevenattheedgesof
thefan,sincethetrailingedgeofthewaveisconstructedtohavespeedλk(U−),
andtheleadingedgehasspeedλk(U+).Thecharacteristicspeedhastoincrease
through the rarefaction wave. (In the figure, the slopes of the characteristics
decrease).WedefinetherarefactioncurveRk(U−)throughU−tobethehalfofthe
integralcurvethroughU−onwhichthecharacteristicspeedincreasesfromλk(U
−).ThenU− can be connected to any U+∈Rk(U−) with a rarefaction wave. In
particular,wehave
Note that characteristics of the other eigenvalues are not straight as they pass
through the rarefaction wave, as illustrated in the figure with a characteristic
movingleft,acceleratingasitpassesthroughthewave.
Example2.(Rarefactionwavesforthep-system)Let’sworkoutthedetailsfor
thep-system.Herewehave
Thus, the p-system is genuinely nonlinear (in both characteristic fields) if and
onlyifσ″(u)≠0.Inthiscase,wenormalizer±(u)bychoosinga=
,
sothat(14.22)issatisfied.Forararefactionwave,wehavefrom(14.21),
whichismoreconvenientlywrittenas
Thus,
Inparticular,forararefactionfrom(u−,v−)to(u+,v+),wemusthave
For the characteristic speed to increase through the rarefaction wave (see Fig.
14.4),wemustsatisfy(14.23),whichforthep-systembecomes
forubetweenu−,u+.
We get more information about the slope and curvature of the rarefaction
curvesbydifferentiating(14.24):
.Forexample,
ifσ″(u)>0,thenforaforwardrarefactionwave(i.e.,withpositivespeed),we
must have u+ > u−, whereas for a backward wave (with negative speed), we
haveu−>u+.Thecorrespondingvaluesofu+,v+lieontherarefactioncurvesR
±(u−,v−),shownintheu-vplaneinFigure14.5a.Ifσ″(u)<0,forexample,as
shown in Figure 14.1 near u = 1, then the configuration changes to that in
Figure14.5b.WeshallusetherarefactioncurvesshowninFigure14.5tosolve
theRiemannproblemafterwehavediscussedshockwavesandshockcurvesin
Section14.2.7.
14.2.5.PropagationofSingularities
The edges of a rarefaction wave have discontinuities in the derivative of the
solution, even though the function itself is continuous. By construction, these
singularities propagate along characteristics. Here we prove the more general
result for continuous solutions of (14.19) that jumps in the derivative lie on
characteristics.
Figure14.5.Rarefactioncurvesforthep-system.(a)σ″(u)>0;(b)σ″(u)<0.
Theorem14.2.ConsiderafunctionU:R×(0,∞)→Rnthatiscontinuousandis
C1exceptonaC1curveΓ:x=γ(t).Supposethatderivatives∂xU,∂tUhaveleftand
rightlimits∂xU(γ(t)±,t),∂tU(γ(t)±,t)thatvarycontinuouslyonΓ,andthatU(x,t)
satisfiesthePDE(14.19)for(x,t)notonΓ.Then
1.thecurveΓisacharacteristic:
forsomek=1,…,n,and
2.thejump[∂xU](t)=∂xU(γ(t)+,t)−∂xU(γ(t)−,t)isparalleltothecorresponding
righteigenvector:[∂xU](t)=η(t)rk(U(γ(t),t))forsomeη(t)∈R.
Proof.From(14.19)wehave
andbycontinuity,(14.25)holdswithUt=Ut(γ(t)±,t),Ux=Ux(γ(t)±,t).Thus,
Moreover,differentiatingtheidentity
wehave
where
,and
.Itfollowsthat
Substitutingfor[∂tU](t)from(14.26),wefind
Thus, isaneigenvalueofdF(U),and[∂xU]isacorrespondingeigenvector.The
theoremnowfollows.
We now discuss shock waves for hyperbolic systems, beginning with the
formationofshocksfor2×2systems.
14.2.6.ShockFormation
Forscalarconservationlaws,itisasimplecalculationtoidentifythemechanism
for formation of shocks, in which a spatial derivative blows up in finite time.
Therearemanyderivativesforsystems,andtheanalysisisnotsosimpleasitwas
forscalarequations.HereweshowthecalculationbyLax[33]forsystemsoftwo
conservationlawswithsmoothinitialdatathatareconstantoutsideaninterval.
Lax’s calculation was generalized to systems of n ≥ 3 equations by John [27]
under similar conditions on the initial data. Lax’s calculation relies on uniquely
defined Riemann invariants, but for n ≥ 3, each characteristic field has n−1
independent Riemann invariants. John’s analysis instead relies on a
decomposition of Ux into characteristic fields, thereby defining generalized
Riemanninvariants,whosepropagationandinteractioncanbetrackedmuchas
Laxdoeswiththecalculationinthissection.
Forn=2,weletU=(u,v),andwritesystem(14.19)as
Letλ(r,s),μ(r,s)bethecharacteristicspeeds,intermsoftheRiemanninvariants
r,s.Thensystem(14.27)ispartiallydiagonalized:
Herewehave∇r(u,v).rμ=0and∇s(u,v)·rλ=0.Mimickingthescalarcase,
wedifferentiate(14.28)withrespecttoxanddefinep=rx,q=sx:
Thedifficultyliesinthepqtermineachequation,representingtheinteractionof
waves.Let’sfocusontheequationforp.
Definedifferentiationalongtheλcharacteristicby
.Thislastequationyields
.Then
and
Thenwecanrewritethefirstequationin(14.29)as
Nowweuseanintegratingfactortocollapsetheleft-handside.Defineafunction
h(r,s)sothat
.Then
,since
. Consequently, if we let
z(t)=ehp,(14.30)becomes
Ifthecoefficientλre−hwereconstant,wecouldwritethesolutionoftheRiccati
equation(14.31)explicitly.However,ingeneralwehavelittleinformationabout
the dependence of this coefficient on the solution. Nonetheless, we can use a
comparison argument to show that solutions of (14.31) blow up in finite time,
providedthecoefficientisboundedandweassumegenuinenonlinearity.
ConsiderboundedsmoothC1initialconditionsfor(14.27).ThentheRiemann
invariants remain bounded, and as long as their derivatives stay bounded, the
solution of the initial value problem is defined and smooth, and h(r, s) is
bounded.
Weassumethattheλcharacteristicfieldisgenuinelynonlinear,sothatλr≠
0. Then the coefficient λre−h is bounded above and below on the same side of
zero.Let’stakeλr>0.ThenthereareconstantsA,Bsuchthat0<A<λre−h<
B.Consideraspecificλ-characteristicoriginatingfromx=x0att=0.Thenz(0)
=z0 is a specific number given by the initial data at x0. We can compare the
solutionz(t)of(14.31)withsolutionsoftheinitialvalueproblems
SinceA,Bareconstants,wehavesolutions
Now suppose z0 < 0. Then Y′(t) < z′(t) < X′(t) for t ≥ 0. To see this, first
observethatitistrueatt=0,sobycontinuityitremainstrueforashorttime.
ButthenY(t)<z(t)<X(t)<0,sotheinequalityonderivativespersists,andso
does the comparison of the solutions. Since X(t) → −∞ as t → −1/(Az0), we
concludethatz(t)→−∞also,ast→t∗,forsomet∗≤−1/(Az0).Letzm=min
z0<0,wheretheminimumistakenovertheinitialpointsx0.Thenthesolution
ofthesystem(14.27)hasderivativesthatblowupinfinitetimetm≤−1/(Azm).
Incontrast,wealsohaveconditionsunderwhichthederivativesofthesolution
staybounded.Ifλrz0>0,thenz(t)staysboundedanddecaystozeroast→∞.
Just as for scalar conservation laws, we presume that as a derivative of the
solutionbecomesinfinite,theweaksolutioncontinueswithashockwave.Next
weanalyzeshockwavesexplicitly.
14.2.7.WeakSolutionsandShockWaves
Hereweformalizethedefinitionofweaksolutionofthesystemofconservation
laws.Thedefinitionresemblesthatofweaksolutionsoflinearellipticequations
in that it is based on integration by parts, or equivalently, by distribution
solutionsthatareintegrable.
ItisconvenienttodefineweaksolutionsoftheCauchyproblem,consistingof
(14.19)withinitialcondition
To derive the appropriate weak formulation of the Cauchy problem, consider a
smooth solution U(x, t), with smooth initial data U0. To incorporate the initial
condition into the weak formulation, we define test functions φ ∈
tohavecompactsupportintheupperhalf-plane,includingthe
x-axis.Thatis,φcanbenonzeroinaboundedsubsetofthex-axis.Notethathere
weonlyrequirethetestfunctionstobeC1,butwecouldjustaswellconsiderC∞
testfunctions.
Multiply(14.19)byφandintegratebyparts,observingthatUt+Fx=div(F,
U),andtheoperationappliestoeachcomponentofthevector-valuedfunctions.
Thiscalculationleadstotherelation
inwhichthefinalintegraliszerounlessthesupportofφ(x,0)includespartof
thex-axis.
Motivatedbythiscalculation,wedefineafunctionU∈L∞(R×[0,∞)→Rn)
tobeaweaksolutionoftheCauchyproblem(14.19),(14.32)if(14.33)holdsfor
all
Note that the idea of integrating by parts to obtain an integral identity for
weak solutions, using test functions, was previously used for elliptic equations.
(See(11.8)inChapter11.)However,forsecond-orderequationsweassumedthat
the weak solution had a derivative in L2, whereas here we want to allow jump
discontinuities, whose derivatives are distributions but not in L2. Consequently,
we consider functions in L∞. This definition of weak solution of systems of
conservationlawsappliestoscalarequations,butwedidnotusethegeneralityof
thedefinitioninthepreviouschapter.
Nextweusethedefinitiontocharacterizeweaksolutionsof(14.19)thathave
jumpdiscontinuitiesacrossacurveΓ.
Theorem14.3.SupposethatU : R × (0, ∞) → Rn is a weak solution that is C1
exceptonaC1curveΓ:x=γ(t).SupposetheleftandrightlimitsU±(t)=U(γ(t)
±, t) vary continuously on Γ. Then these limits satisfy the Rankine-Hugoniot
conditions
Figure14.6.Domainofintegrationforashock.
Proof.LetφbeatestfunctionwithsupportΩint>0thatisdividedintotwo
disjoint sets Ω± by the curve Γ, as indicated in Figure14.6. Since U is a weak
solution,itsatisfies(14.33),whichwerewriteas
Now integrate by parts in Ω± separately, to put the derivatives back on U and
F(U).Inthisprocess,weareleftwithonlyboundaryterms,sinceUt+F(U)x=0
in each of Ω±. Since φ vanishes on the boundary of Ω± except on Γ, the only
contribution remaining comes from the integrals on Γ ∩ Ω. We need the unit
outwardnormalsonx=γ(t),whicharegivenby
on∂Ω±,respectively,where
.Asaresult,weobtainnequations,one
equationfromeachcomponentof(14.35):
Thatis,ifΓ∩Ω={x=γ(t):a<t<b},thenwehave
Consequently,sinceφisarbitraryandtheintegrandiscontinuous,weobtainthe
Rankine-Hugoniotconditions(14.34).
14.2.8. Analysis of the Rankine-Hugoniot Conditions Using Bifurcation
Theory
To understand the jump conditions more thoroughly, it is useful to consider
piecewise-constantsolutions,orshockwaves
inwhichsistheshockspeed.WesayUgivenby(14.36)isashockfromU−to
U+ifs,U±arerelatedbytheRankine-Hugoniotconditions(14.34):
Thissystemofnequationsin2n+1unknownshasoneeasyfamilyofsolutions,
the trivial solution: U+ = U−, s arbitrary. But this corresponds to constant
solutions of the PDE. We seek nontrivial solutions, for which there really is a
discontinuity.
Ourapproachtothisproblemderivesfrombifurcationtheory,theanalysisof
the branching of curves of solutions of nonlinear equations [8, 19]. To set this
approach up, we fix U− and let G(U+, s) denote the left-hand side of (14.37).
Thatis,G:Rn×R→Rnisdefinedby
Thenwehave
Herearetwobasicresultsofthetheory,statedinformally:
1.Bifurcationpointsareeigenvaluesofthelinearizedproblem.Nontrivialsolutions
canonlybranchoffthecurve{(U−,s):s∈R}oftrivialsolutionsat
eigenvaluess=λk(U−)ofthelinearizationdF(U−).
2.Bifurcationfromasimpleeigenvaluegivesacurveofsolutions.Iftheeigenvalues
=λk(U−)issimple,thenthereisacurveofnontrivialsolutionspassing
through(U,s)=(U−,λk(U−)).
The first result is a simple consequence of the Implicit Function Theorem.
Supposes0isnotaneigenvalue.ThendUG(U−,s0)=−s0I−dF(U−)isinvertible.
By the Implicit Function Theorem, there is a curve of solutions U = U(s) for s
nears0.Moreover,noothersolutionsexistinaneighborhoodof(U−,s0).ButU=
U−isasolutionforalls,sowemusthaveU(s)≡U−.
ThesecondresultisalsoaconsequenceoftheImplicitFunctionTheorem,but
it requires much more careful analysis. It appeared in various forms in the late
1960s and early 1970s, culminating in the 1971 paper of Crandall and
Rabinowitz[11].Thebasicideaoftheproofistosolveforn−1componentsof
Uasafunctionofsandasmallvariableϵrepresentingtheremainingcomponent
ϵrk(U−). This construction uses the Implicit Function Theorem in a
straightforwardwayonasystemofn−1equationsobtainedbyprojectingthe
original system onto the range of dUG(U−, λk(U−)). The problem is thereby
reduced to a single equation f(s, ϵ) = 0, called the bifurcation equation. This
reductionprocesstoarriveatthebifurcationequationisknownastheLyapunovSchmidt reduction. Locally, there are two curves of solutions of the bifurcation
equation.Zeroesoffare{(s,ϵ)=(s,0)},formingthecurveoftrivialsolutions,
andafunctions=ŝ(ϵ),givingthecurveofnontrivialsolutionscorrespondingto
shockwavesolutions.
Hereisthepreciseresult,whichwestatewithoutproof.
Theorem14.4.SupposeFisCm,m≥2,andthePDE(14.19)isstrictlyhyperbolicin
a neighborhood of U−. Then for each k = 1, …, n, there exist δ > 0 and Cm−1
functions
thatareCmawayfromzeroandaresuchthat
1.theproperty
2.theidentity
holds;
|ξ|<δholds;and
3.thereisaneighborhoodN⊂Rn×Rof(U−,λk(U−))suchthatif(U,s)∈N
satisfies(14.37),theneitherU=U−,or
δ).
and
forsomeξ∈(−δ,
We use the theorem to establish some properties of weak shock waves, in
which the jump in U is small. Following the calculation of Lax [32], we
differentiateidentity2twiceandsetξ=0.Differentiatingonce,wehave
sothat,settingξ=0andusingproperty1,weget
Wededucethat
,wherewehaverescaledtheparameterξsothatthe
scalarmultipleof rk(U−)isunity.(Recallthatrk(U−)isnormalizedby(14.22).)
Differentiating(14.38)again,andsettingξ=0,weget
where d2F(U−) : Rn × Rn → Rn is the bilinear operator involving second
derivativesofFthatappearsintheTaylorseriesexpansionofF(U)aboutU=U
−.(SeeAppendixAfortheTaylorseriesofascalarfunctionofseveralvariables;
thebilinearoperatorputsthesetermsintoavector-valuedfunction.)
Now we can eliminate the second derivatives of by projecting onto the
complementoftherangeof
).Thisismosteasilyaccomplishedby
takingtheinnerproduct(scalarproduct)oftheequationwiththelefteigenvector
ℓk(U−):
Thus,wehave
Toverifythefinalequality,wedifferentiatetheidentity
withrespecttoξandletξ=0.
From(14.39),wededucethat
Now suppose the kth characteristic field is genuinely nonlinear at U−, and
normalizerk(U−)asforrarefactionwaves(see(14.22)).Then,
andforξ
nearzero,wehave
if and only if ξ < 0. This inequality is highly significant in terms of the Lax
entropy condition that we arrive at in the next section, where we state the
condition for systems of conservation laws. The consequence is that for weak
shocks (in which |U+−U−| is small) only half of the branch of solutions, the
halfwiths<λk(U−),correspondstoshocksthatareadmissibleaccordingtothe
Laxentropycondition.
14.2.9.AdmissibilityofShockWavesandtheRiemannProblem
Much of the theory of weak solutions of hyperbolic conservation laws, for both
scalar equations and systems, is centered on understanding solutions of the
Riemannproblem.Thisistheinitialvalueproblem
whereUL,URaregivenconstants(inRn).Wewillbuildscale-invariantsolutions
of the Riemann problem from centered rarefaction waves and shock
waves,separatedbyregionswhereU(x,t)isconstant.Fornow,let’sfocusona
singleshockwave,joiningconstantsU−ontheleftandU+ontheright.
Just as for the inviscid Burgers equation, if we allow all shock waves
(satisfying the Rankine-Hugoniot conditions (14.34)), then solutions are not
unique. We correct this lack of uniqueness by imposing an additional condition
on shock waves, just as was done for scalar equations. The most fundamental
suchadmissibilityconditionistheLaxentropycondition.
Figure14.7.Ak-shockwithrepresentativecharacteristicssatisfyingtheLax
entropycondition.
WesaytheshockwavefromU−toU+withspeeds(see(14.36))isadmissible
accordingtotheLaxentropyconditionifthefollowinginequalitiesholdforsomek
=1,…,n:
Herewedefineλ0=−∞andλn+1=∞,sothattheseinequalitiesmakesense
for k = 1and k = n. The Lax entropy inequalities require that the kth
characteristics enter the shock from both sides, while all other characteristics
passthrough,eitherfromlefttoright(thefastercharacteristics),orfromleftto
right(theslowercharacteristics).ThisbehaviorisillustratedinFigure14.7.We
say that a shock satisfying (14.43) is a k-shock. Note that by continuity of the
characteristicspeeds,condition(14.43b)isredundantforweakshocks.Moreover,
from(14.41),halfoftheshockcurveofTheorem14.4consistsofk-shocks,and
theotherhalfviolatestheLaxentropycondition.
Example3.(Shockwavesforthep-system)TheRankine-Hugoniotconditions
forthep-system(seeSection14.2.1),are
Figure14.8.WavecurvesW±(u−,v−)forthep-system.
Thus,wehave
Note that the latter formula states that s2 is the slope of the chord joining the
points (u±, σ(u±)) in the graph of σ. Now let’s interpret the Lax entropy
inequalities (14.43). As for rarefaction waves, the sign of σ″ will make a
difference.Wefirstassumeσ″>0.Thenforforwardshockwaves,withs>0,
andk=2in(14.43a):
which implies u+ < u− for s > 0. Note that the other condition (14.43b),
, is automatically satisfied, since s > 0. For s < 0, we similarly
concludeu+>u−.
Nowlet’slookattheshockcurves,whicharecurvesofvaluesof(u+,v+)for
whichthereisanadmissibleshockforsomespeed.Weagaintakethecaseσ″>
0.Fors>0,eliminatingsfrom(14.44),weobtainacurveS+(u−,v−):
Fors<0,wegetthesameformula,butnowwehaveu+>u−.Thisdefinesthe
shockcurveS−(u−,v−)ofbackward(left-moving)shockwaves.Theshockcurves
meettherarefactioncurvesofFigure14.5at(u−,v−),formingawavecurveW±
(u−,v−)foreachcharacteristicfamily,definedby
Finally,wecansolvetheRiemannproblem(14.42)forthep-systemusingthe
wave curves shown in Figure 14.8. The solution of the Riemann problem is
representedinFigure14.9.Inthisfigure(uL,vL)isfixed,andweshowthespecific
combinationofwavesfor(uR,vR)ineachofthequadrantsseparatedbythewave
curves W±(uL, vL). For example, as shown in the figure, the combination S−R+
indicatesabackwardshock(withnegativespeed)from(uL,vL)toapoint(uM,vM)
∈S−(uL,vL),andaforwardrarefactionfrom(uM,vM)to(uR,vR)∈R+(uM,vM).The
correspondinggraphsofuandvareshowninFigure14.10.
Figure14.9.Riemannproblemsolutionforthep-system.Fourregions
correspondingtodifferentwavecombinationsarelabeledinblue.
Figure14.10.Graphsofu,v,thesolutionoftheRiemannproblemfor(uR,vR)in
theregionS−R+ofFigure14.9.
14.3.TheDam-BreakProblemUsingShallowWaterEquations
In this short section we solve an initial value problem for the shallow water
equations.Considerabodyofwaterofconstantdepthaboveahorizontalvalley
floorandtoonesideofadam.Weareconcernedwithwhathappenswhenthe
dam collapses instantaneously. We idealize the problem in a couple of ways.
First,thebodyofwaterhasinfiniteextent,andweassumethatthebehaviorafter
the dam breaks depends primarily on a single spatial variable x, along the
horizontal direction perpendicular to the dam. This scenario is shown in Figure
14.11. This is an example of a Riemann problem (14.42), but the system has
coincidentcharacteristicspeedswhenh=0,totherightofthedam.Whenthe
dambreaks,waterrushestotheright,drawingwaterfromthereservoir.Hence,
eventhoughthewatermovesonlytotheright,thedisturbanceisfeltatapoint
labeled x = x1(t) that travels backward into the reservoir (Fig. 14.11c). Of
primaryinterestinthisproblemisthelocationoftheleadingedgex=x0(t)of
thewaterflood,wherehgoestozero.
Figure14.11.Thedam-breakproblem.(a)Therarefactionwave;(b)the
reservoirbeforethedambreak(t=0);(c)thewaterafterthedambreaks(t>
0).
InFigure14.11 we show the rarefaction wave solution in the x-t plane, the
graphofh(x,t)beforethedambreaks(att=0),andthegraphofh(x,t)after
the dam breaks (for t > 0). To analyze the problem fully, we start with the
equationsandinitialcondition:
It is convenient to introduce the relative speed
(see Section 14.2.2), so
thatthecharacteristicsλ±=u±chavespeedcrelativetothefluidvelocityu.
As predicted by (14.10), the Riemann invariants w± = u∓ 2c diagonalize the
system
The curved characteristic shown in Figure 14.11a is associated with the faster
speedλ+.TheRiemanninvariantu+2cisconstantonthischaracteristic.
Therarefactionwaveisassociatedwiththeslowerspeedλ−,sothatu−2cis
constant on each characteristic, but w− = u + 2c is constant throughout the
wave.Totheleftofthewave,h=h0andu=0.Thus,theslowestcharacteristic
isx=x1(t)=−c0t,
.Sinceu+2cisconstant,wehaveu+2c=2c0.
Therefore,attheleadingedgeoftherarefactionwave,whereh=0(sothatc=
0),wehaveu+2c=u=2c0.Buttheleadingedgetravelswithcharacteristic
speedλ−=u−c=2c0.Weconcludethattheleadingedgeofthefloodwaters
islocatedat
.
14.4.Discussion
Thetheoryofsystemsofhyperbolicconservationlawsisnowquiteextensive.In
thischapterwehavefocusedontheconstructionofshocksandrarefactions,and
we have shown how these can be combined to solve Riemann problems. This
suggestsaneffectiveapproximationtosolutionsoftheCauchyproblem,inwhich
aninitialcondition
isposed,withU0anintegrableandboundedfunction.IfU0isapproximatedbya
piecewise-constantfunction,thenwhereverthereisajumpbetweenconstants,a
Riemann problem can be solved to get the exact solution for some short time
interval, until adjacent waves meet. If these waves are shocks, then the
interaction can be resolved by solving another Riemann problem. However, if
one or both waves is a rarefaction, then these can be approximated by a small
jump, or a staircase of small jumps, in which each jump satisfies the RankineHugoniot condition but not the Lax entropy condition. These jumps travel with
approximately characteristic speed (as in the rarefaction they replace), and on
meetingotherjumps,theygeneratefurtherRiemannproblems.Continuinginthis
way,anapproximatepiecewise-constantsolutionoftheCauchyproblemcanbe
generated. Justifying the continuation and proving that this method—known as
wave front tracking—converges requires detailed estimates of the interaction of
waves and the use of suitable spaces (such as BV, the space of functions of
boundedvariation).DetailsofthisapproacharegiveninthebookbyBressan[6].
A related method, involving approximations in a lattice of grid points and
using randomness to capture wave speeds approximately, is known as Glimm’s
random choice method. It is outlined in Glimm’s classic 1965 paper [18], in
whichheprovestheexistenceofsolutionsoftheCauchyproblemusingaseries
of profound insights into wave interactions and convergence of approximate
solutions.
In the next chapter, after a discussion of the equations of fluid mechanics, we
examine the structure of the one-dimensional Euler equations. The Euler
equations of inviscid compressible fluid motion form a system of hyperbolic
conservationlawspossessingshocksandrarefactions.
PROBLEMS
1.Checkwhethertheshallowwaterequationsaregenuinelynonlinear.
2.Provethatsystem(14.17)isstrictlyhyperbolicforallξ>1ifT(1)=0,T′(1)
>0,andT(ξ)isaconvexfunction:T″(ξ)≥0forξ≥1.Discussthepossibilities
ifT(1)=0,T′(1)>0,andTisconcave:T″(ξ)≤0forξ≥1.
3.Letσ(u)=u2,u>0inthep-system.Findexplicitformulasfortherarefaction
curves.Hencederiveformulas
forrarefactionwavesjoining(u−,v
−)=(1,0)to(u+,v+).Graphthefunctionsu,vagainstxfort=1.
4.ConsiderthesystemofPDE
(a)Showthatthesystem(14.47)withϵ=0isstrictlyhyperbolicforu≠0,v
≠0ifα=−1,butitfailstobehyperboliceverywhereinu,vifα=1.
(b) Assume that α = −1. For ϵ = 0, write the Rankine-Hugoniot jump
conditionsforashockwavesolution
(c)Assumethatα=−1.Forϵ=0,let(u−, v−)=(1,0).Findallpossible
values of (u+, v+) for which (14.48) satisfies the jump conditions for some
speeds.(Hint:Theanswershouldbetheunionofalineandahyperbola.)
(d) Assume that α = −1. For ϵ > 0, show that there is a traveling wave
solution(u,v)=(U,V)((x−st)/ϵ)ofsystem(14.47)satisfying
for some value of s (which you need to calculate), but that the corresponding
shock (ϵ → 0+) fails to satisfy the Lax entropy condition because only one
characteristic enters the shock from each side in the x-t plane, and one
characteristicleavesoneachside.
5.Writetheequation
inwhichz=u+ivisacomplexvariable,asasystemoftwoconservationlaws
forrealvariablesu,v.
(a) Find conditions on the function f : R → R under which the system is
strictly hyperbolic. Calculate the characteristic speeds and the corresponding
righteigenvectors.
(b) Show that one characteristic field is linearly degenerate and the other is
genuinelynonlinearprovidedf″(r)>0,r∈R.
(c)Observethatthecomplexformoftheequationisrotationallyinvariant,by
changingvariables:ζ=eiθz,whereθisconstant,andwritingthePDEforζ(x,
t).Howdoesthispropertyrelatetopartsa,b?
(d)Letf(r) = r2. For (u−,v−) = (1, 0), find all values of (u+,v+) ∈ R2 for
which there is a shock satisfying the Lax entropy condition, a contact
discontinuity, or centered rarefaction wave connecting (u−, v−) to (u+, v+).
ThispairofequationsisknownastheKeyfitz-Kranzersystem[29].
6.Fortheelasticstringequations(14.17),showthefollowing.
(a)Transversewavesarelinearlydegenerate.
(b)LongitudinalwavesaregenuinelynonlinearprovidedT″≠0.
7.(a)Provetheproperty(14.10)thatfor2×2systems,theRiemanninvariants
diagonalizethesystem(forsmoothsolutions).
(b) Verify this property for the p-system. (Hint: Begin by deducing from
(14.24)formulasfortheRiemanninvariants.)
8. Write out the details of the argument that gets you from (14.40) to (14.41)
and thence to the statement about the admissibility of weak shocks following
(14.41).
CHAPTERFIFTEEN
TheEquationsofFluidMechanics
An abundance of interesting and important PDE exists, many of which are
systemsofequations,becausephysicalsystemsoftenrelatedifferentquantitiesas
dependentvariables.ExamplesofphysicalsystemsaregiveninSerre’stext[39],
includingtheequationsofelectromagnetism(Maxwell’sequations),equationsof
three-dimensionalelasticity,andthoseofmagnetohydrodynamics.Inthischapter
we focus on the equations of fluid mechanics, specifically, the Navier-Stokes,
Stokes, and Euler equations. We discuss how these equations are related, the
contextsinwhichtheyareused,andsomeelementaryproperties.
15.1.TheNavier-StokesandStokesEquations
Themotionofafluidisdescribedbytheevolutionofphysicalquantities,suchas
density ρ, velocity u ∈ R3, pressure p, and temperature θ. We shall assume
constanttemperature,eventhoughvariabletemperaturecanbeverysignificant,
forexample,inunderstandingpatternsthatarevisiblewhenheatingwaterina
pot. Assuming constant temperature simplifies the equations quite a bit. The
equationsofmotioncanbederivedfromconservationlawsofmass,momentum,
andenergy,coupledtoconstitutivelaws.Detailscanbefoundinfluidmechanics
texts,suchastheonebyAcheson[2].
Let’s start with dimensional independent variables
and dimensional
dependentvariables
.Foranincompressiblefluidsuchaswater,thedensityρ
istakentobeconstant,althoughforstratifiedincompressiblefluids,variationsin
densitycanbeimportant.TheNavier-Stokesequationsofincompressiblefloware
abalancebetweeninertialtermsandforces:
where
is the acceleration due to gravity expressed as a vector in the
verticaldirection.Thus,thetermρgismassperunitvolumetimesacceleration.
In fact, the first equation in (15.1) expresses Newton’s law, force = mass ×
acceleration in each small volume of fluid, so that mass is replaced by density
(i.e.,massperunitvolume).Thesecondequation,knownastheincompressibility
condition, expresses conservation of mass for an incompressible fluid in which
thedensitydoesnotchange.Fromthisequation,theDivergenceTheoremimplies
thatvolumesoffluidarepreservedbytheflow:ifafluidvolumeexpandsinone
directionthenitcontractsinotherdirectionstocompensate.
We now introduce a typical length scale L and time scale T, together with
velocityscaleUandpressurescaleP.Thesescalesallowustonondimensionalize
the variables and express the equations in nondimensional form. The
nondimensionalvariablesdonothavethetildes:
Substitutinginto(15.1),wearriveat
Next,dividethefirstvectorequationbyρU2/LandchoosescalesforTandP:
ThenTisatypicaltimeforafluidparticletravelingatthetypicalvelocityUto
travel a typical length L, and the pressure scale P is chosen to simplify the
equationasfollows.WedefinetheReynoldsnumberReby
andarriveatthenondimensionalsystem
where
.TheReynoldsnumbermeasurestherelativeimportanceofinertial
terms,withdimensionρU2/L,andviscousforces,withdimensionμU/L2.
Since the velocity u has three components, the first equation of (15.2) is
reallythreescalarequations.Theyexpressconservationoflinearmomentum.The
termsontheleft-handsidearetheinertialoraccelerationterms,andtherighthand side represents the divergence of the forces, due to pressure, internal
friction (i.e., viscous forces) and the body force due to gravity. The
incompressibilitycondition∇.u=0isunchangedbythescaling.
Thenondimensionalformoftheequationsisapowerfultool.Ifanexperiment
isdoneonafluid,suchaswater,inalaboratory,theresultscanapplyinamuch
larger context, such as a river or an ocean, by calculating the appropriate
Reynoldsnumberfromtypicallengthandvelocityscales.Theresultswithwater
wouldalsoapplytoamuchmoreviscousfluid,suchashoney,againbyusingthe
Reynolds number to relate the two scenarios. Moreover, the Reynolds number
controls the relative importance of the various terms in the equations, so it is
importanttounderstandtwolimits,Re→0,andRe→∞.
InthelimitRe→0,correspondingtoveryslowflow(Usmall)orveryviscous
flow(μlarge),issomewhattricky.IfweweretosimplymultiplybyReandsetRe
= 0, we would recover Laplace’s equation. However, this is misleading, as the
flow should be driven by pressure gradients. If we rescale the pressure by the
Reynoldsnumberanddefineanewappropriatetimescale(ineffect,weareusing
adifferentnondimensionalization),werecovertheStokesequationsinthelimitRe
→0:
whichhavethevirtueofbeinglinear.Inthisderivation,wehaveassumedthat
thegravitationaltermsaresignificantlysmallerthanthetermsretained.Insome
contexts, the effect of gravity can be significant, even for slow flow. Stokes
equations are an important tool for studying the motion of small organisms in
viscousfluids,andforveryslowflows,suchaslavaandglacierflows.
15.2.TheEulerEquations
At the other extreme, letting Re → ∞, we immediately get the incompressible
versionoftheEulerequations:
in which viscosity is negligible. These equations model the flow of slightly
viscousfluids,suchasair,providedthatthepressuregradientsarenotsolargeas
tomakecompressibilitysignificant.
ThecompressibleEulerequationstaketheform
In these equations, the density ρ is variable (as it is in a gas, for example).
Thevariable
is the total energy (per unit volume) in terms of the
kinetic energy
and the potential (or stored or internal) energy e. The
variablesaregenerallyconsideredtobeρ,u,ande,withp=p(ρ,e)beinggiven
byanequationofstate,anothertermforaconstitutivelaw.Bothρandemustbe
nonnegative, and the case ρ = 0 is known as the vacuum state. The tensor
productu⊗ugivesamatrixwiththe(i,j)entrybeinguiuj.Thecorresponding
quadratic term in the Navier-Stokes equations (where ρ is constant) looks
different,becauseuisdivergencefree.
Let’sconsidertheone-dimensionalequations
inwhichu is now a scalar velocity (the component of u parallel to the x-axis).
Notice that these equations (and in fact system (15.4)) are in the form of a
system of conservation laws in which the conserved quantities are mass ρ,
momentum ρu, and energy E. There is a natural symmetry in this system
between left and right. Specifically, the equations are unchanged by the
transformations x → −x, u → −u. A similar symmetry holds for the wave
equation,bothlinearandquasilinear.
If we carry out the differentiations (assuming for the moment that the
variables are smooth functions of x,t), we can write the system in the form of
nonlinear transport equations, with the abbreviation dt = ∂t + u∂x for the
convectivederivative:
As in the previous chapter, hyperbolicity of this system depends on the
eigenvaluesofthecoefficientmatrix,whichinthiscaseis
Notethatincalculatingthiscoefficientmatrix,weusepx=pρρx+peex,andthe
spatialpartu∂xoftheconvectivederivative.
Tocalculatethecharacteristicspeeds,wefindeigenvaluesofA(ρ,u,e)from
thecharacteristicequation:
Thus, λ = u, or λ = u ± (pρ + ρ−2ppe)1/2, provided pρ + ρ−2ppe ≥ 0. The
vacuumstateρ=0isthereforesingularsinceρ−2→∞,andhyperbolicityforρ
>0requirespρ+ρ−2ppe>0.
Ifwewritethethreecharacteristicspeedsas
wherec=(pρ+ρ−2ppe)1/2,weobservethatcisthesoundspeed,meaningthe
speedofsmalldisturbances,relativetothefluidspeedu.Thecharacteristicspeed
λ0isthesameasthefluidspeedanddoesnotpropagatesmalldisturbances.Not
surprisingly,theλ0characteristicfieldislinearlydegenerate,asweshowbelow.
Since λ− < λ0 < λ+, 1,2,3-characteristics are associated with λ−, λ0, λ+
respectively.Thecorrespondingeigenvectorsare
respectively, and we can check genuine nonlinearity by calculating ∇λ · r. For
theλ±waves,wehave
Thus,genuinenonlinearityofthesecharacteristicfieldsdependsontheequation
of state p = p(ρ, e). In the isentropic case, p = p(ρ) is independent of e, and
genuine nonlinearity reduces to the condition p″(ρ) ≠ 0. The λ0 characteristic
fieldisindeedlinearlydegenerate,since∇λ0·r0≡0.
Thecharacterizationofshockwavesandrarefactionwavesforsystem(15.5)
iscomplicatedandisexplainedcarefullyandindetailinSerre’stext[39],vol.1,
Section 4.8. Here we sketch the steps involved in processing the RankineHugoniotjumpconditionsforshockwaves.Weconsiderashockwithspeedsand
leftandrightlimitsgivenbysubscripts:v±,ρ±,e±.Withthebracketnotation[v]
=v+−v−forjumps,theRankine-Hugoniotconditionsfor(15.5)are
Ifweletz=v−s,then[ρz]=0from(15.7a),andwecanletm=ρ±z±bethe
common value on each side of the shock. Then we are left with the two
conditions:
Withsomealgebra,theseconditionscanbecombinedintotheequation
sothat
Thus,eitherm=0,or
.
Ifm=0,andassumingthatρ±>0,wehavez±=0,sothatv±=s.Then
(15.8a) implies p+ = p−, and the Rankine-Hugoniot conditions are satisfied.
Theseshocks,movingwithcharacteristicspeed,thatis,thefluidparticlespeedv,
are contact discontinuities. Across such a wave, density jumps, but velocity and
pressurearecontinuous.
Ifm≠0,thenwehave
From(15.9)and
,itfollowsthat
Thus,ifρ±,e±,p±=p(ρ±,e±)satisfy(15.10),thentherearetworealsolutions
of(15.11)provided[p]/[ρ−1]<0.Ifwelabelthesetwosolutionsm1andm3,they
correspond to 1-shocks (m1 > 0 associated with 1-characteristics) and 3-shocks
(m3<0associatedwith3-characteristics).Thenthevelocitiesaregivenbyv±=
z±+s,withthespeedsbeingchosen.
Itisthenpossible(butintricate)togiveaparameterizationof1-shocksand3shocks satisfying the Lax entropy condition. The entropy condition is related to
thethermodynamicentropyS=S(ρ,e),whichsatisfiesthePDEρ2Sρ+pSe=0.
Thisentropyhastosatisfym[S]> 0 across the shock, with the result that fluid
particlesgainentropy(Sincreases)astheypassthroughtheshock.
In this chapter we have summarized some basic mathematical properties of the
keyequationsoffluidmechanics.Themathematicaltheoryoftheseequationsis
vast and is a very lively topic of current research. Moreover, the equations are
usedtoexplainallsortsofphenomenainvolvingfluidflow,fromtheswimming
ofsmallorganismstothepredictionofweatherpatternsandtheaerodynamicsof
airplanes. The interested reader will find the books of Chorin and Marsden [7]
and of Serre [39] useful introductions to some more of the mathematical
properties of these equations, whereas the text of Acheson [2] is an intuitive
treatment of the equations, including informal discussions of many applications
andthecalculationofphysicallymeaningfulsolutions.
PROBLEMS
1. Show that if ρ is constant in the Euler equations, then the equation for
conservation of momentum collapses to the corresponding equation in the
incompressibleEulerequations.ForsmoothsolutionsoftheincompressibleEuler
equations,deriveanenergyequalityandrelateittotheenergyequationforthe
compressiblecase.
2. Let u = (u, v, w), x = (x, y, z). Write the Navier-Stokes system (15.2) in
components(sothattherearefourscalarequations,and∇isreplacedbypartial
derivativeswithrespecttox,y,z).
3. Consider flow between two parallel horizontal plates held a distance h > 0
apart. Suppose the bottom plate is stationary and the top plate is moving
horizontallyatspeedU.UsetheNavier-Stokessystem,neglectinggravity,tofind
a simple steady flow (independent of time t) in which fluid particles move
paralleltotheplates.Assumetheplatesareatz=0,h,thereisnodependence
ony, and the top plate is moving to the right, parallel to the x-axis. Begin by
sketchingtheplatesandhowyouthinktheflowmightlook.Thenconsiderwhich
componentsofthevelocityu=(u,v,w)canbesettozero.Youcanthensolve
thereducedsetofequationsforthevelocityandpressure.
APPENDIXAMultivariableCalculus
Whenthedistinctionbetweenscalars(realorcomplex)andvectors(n-tuples)is
needed,weuseboldfaceforvectors.Thus,x=(x1,…xn)∈Rn.
WegenerallyusethenotationUtodenoteanopensubsetofRn.Then∂Uis
theboundaryofU,and
istheclosureofU.IfUisbounded,then is
closedandbounded,henceitiscompact.Forexample,iff:Rn→Risafunction,
thenfhassupportdefinedassupp
ifthefunctioniszerooutsideaboundedset.
;thus,fhascompactsupport
IfUhasaC1boundary∂U,thentheunitnormalν=ν(x)variescontinuously
with x ∈ ∂U. For example, the unit ball U = {x ∈ Rn : |x| < 1} has as its
boundarytheunitsphere∂U={x∈Rn:|x|=1},andunitoutwardnormalν(x)
=x.
Theopenballwithcenteratxandradiusr>0isB(x,r)={y∈Rn:|y−x|
<r}.Tocalculateωn,thesurfaceareaoftheunitsphere∂B(0,1)inRn,webegin
byintegratinge−π|x| overRn:
2
Usingthefactthatthesurfaceareaof∂B(0,r)isrn−1ωn,wehave
(The gamma function Γ is defined below.) Thus
. This expression
leadstothefamiliarcircumferenceoftheunitcircle:ω2=2π,andareaofthe
unitsphereinR3:ω3=4π.
LetαndenotethevolumeoftheunitballB(0,1)inRn.Itfollowsthat
Inparticular,α2=πistheareaoftheunitdisk,andα3=4π/3isthevolumeof
theunitballinR3.
TheintegralaveragesofafunctionfoverB(x,r)and∂B(x,r)aredefinedby
The chain rule gives formulas for differentiating the composition of two
functions. It takes various useful forms, depending on the number of variables
involved.Considerx∈Rnandtwofunctionsf:Rn→Randy:Rn→Rn,sothatf
=f(y)andy=y(x).Thenwehave
Nowconsiderafunctionwithanextravariablet∈R:F=F(t,y),andsupposey
=y(t).Thenitfollowsthat
Ifg:R→R and ξ : Rn → R with g = g(ξ) and ξ = ξ(x), then the following
holds:
Forexample,foratravelingwaveu(x,t)=f(x−ct)withwavespeedc,wehave
LetF:Rn→RbeC1.Then(awayfrompointswhere∇F(x)=0)theequation
F(x) = const defines a manifold M of dimension n − 1, sometimes called a
hypersurfaceoralevelsurfaceofF.Forexample,ifn=3,thenthemanifoldisa
two-dimensionalsurface.Letx0∈M.If∇F(x0)≠0thentheunitnormalν(x0)is
givenby
Letf:Rn→RbeC∞.TheTaylorseriesoffaboutx0∈Rnisgivenby
where
. This expression uses multi-index notation, which is
explainedinSection10.2.
Thegammafunctionisthefunction
.ThenΓ(1)=1,
,andtherecurrenceΓ(s+1)=sΓ(s)impliesthatforintegersn≥0,Γ(n+1)
=n!,whichuses0!=1,aconventionalsoimplicitintheTaylorseries.
TheInverseFunctionTheoremstatesthatifadifferentiablefunctionffromRn
toRn has an invertible Jacobian matrix df(x0) at a point x0, then the function
itself is invertible in a neighborhood of that point, and the inverse is as
differentiableasf.
TheoremA.1.(InverseFunctionTheorem)LetU⊂Rnbeopen,andsupposef:U→
RnisCkforsomek≥1.Letx0∈U,andy0=f(x0).SupposetheJacobianJ=det(df
(x0))isnonzero.ThenthereareopensetsV⊂UandW⊂Rn,withx0∈V,y0∈W,
suchthat:
1.f:V→Wisone-to-oneandonto,and
2.theinversef−1:W→VisCk.
The Implicit Function Theorem relates to solving simultaneous nonlinear
equations
near a known solution (x0, y0), where F : Rn × Rm → Rn is differentiable.
SimilarlytotheInverseFunctionTheorem,thefunctionisassumedtobelinearly
nondegenerateatthissolutioninthesensethattheJacobianwithrespecttoxis
nonzero.Theconclusionisthattheequationscanbesolveduniquelylocallyforx
as a function of y for y close to y0, with x(y0) = x0, and with x(y) as
differentiableasF.
TheoremA.2.(ImplicitFunctionTheorem)LetU⊂Rn×Rmbeopen,andsupposef
:U→RnisCkforsomek≥1.Supposethat
ThenthereareopensetsV⊂UandW⊂Rm,with(x0,y0)∈V,y0∈W,andaCk
function
suchthat:
1.
2.
;
forally∈W;and
3.if(x,y)∈VandF(x,y)=0,then
.
ThecontractionmappingprinciplecanbeusedtoprovetheInverseFunction
Theorem.Theprincipleisstatedhereinthebroadersettingofacompletemetric
space X with metric d. A mapping T : X → X is a contraction if there exists a
constantL∈(0,1)suchthat
The contraction mapping principle, also known as the Banach Fixed-Point
Theorem,statesthatif(X,d)isanonemptycompletemetricspacewithmetricd,
andT:X→Xisacontractionmapping,thenThasauniquefixedpointx∗∈X.
Thatis,T(x∗)=x∗.Furthermore,x∗=limn → ∞xn,wherex0 ∈ X is arbitrary,
andxn+1=T(xn),n≥0.
Green’sTheoremintheplanerelatesadoubleintegraloveraboundedopenset
U⊂R2toalineintegralalongthecurve∂U.
Theorem A.3. (Green’s Theorem in the plane) Suppose
continuousandisC1inU,andtheboundarycurveispiecewiseC1.Then
is
whereτistheunittangentinthecounterclockwisedirection,sdenotesarclength,and
dA=dxdyinCartesiancoordinatesistheareametricintheplane.
Stokes’TheoremissimilartoGreen’sTheoremintheplane,butitrelatesthe
lineintegralovertheclosedboundarycurve∂Sofatwo-dimensionalsurfaceS⊂
R3tothesurfaceintegraloverS:
whereF:R3→R3.Here,ν is the normal to Sconsistentwithτ and the righthandrule.
Green’sTheoremintheplaneisthetwo-dimensionalversionoftheDivergence
Theoremforavectorfield
,whereU⊂Rn.Thetheoremrelatesthe
netfluxofFthroughtheboundary∂UtothetotaldivergenceofFovertheentire
regionU:
Here,νistheunitoutwardnormal.
AmorefundamentalintegrationisthecomponentversionoftheDivergence
Theorem.Let
beafunctionin
.Thenforeachj=1,…,n,
Note that the Divergence Theorem and this result are equivalent. However, the
latterresultisadirectconsequenceoftheFundamentalTheoremofCalculus.
The Leibniz integral rule describes how to bring a partial derivative with
respect to one variable into the integral of a multivariable function when the
integralistakenwithrespecttotheothervariable:
To reverse the order of integration in a double integral, we use Fubini’s
Theorem.
TheoremA.4.(Fubini’sTheorem)f=f(x,y)becontinuousovertherectangleR=
{(x,y):a≤x≤b,c≤y≤d}.Then
Two elements x and y of a vector space X with an inner product (·, ·) are
orthogonalif(x,y)=0.AsetS⊂Xiscalledorthonormalifallx≠yinSare
orthogonaland(x,x)=1.
Afunctionf:Rn→RisconvexonRnifforallx,yinRnt∈(0,1),
fisstrictlyconvexiftheinequalityisstrict.
APPENDIXBAnalysis
TheLebesguemeasure|·|:M→[0,∞]isafunctiondefinedonthefamilyMof
LebesguemeasurablesubsetsofRn,whichincludesallopensubsets.ThefamilyM
isaσ-algebra,meaningthat0andRnareinM;complements,countableunions,
and intersections of members of M are also in M. The Lebesgue measure has
properties consistent with being a generalization of the idea of volume. The
measureofanyballisthevolumeoftheball;themeasureofdisjointunionsof
countable families of measurable sets is the sum of their measures; measurable
subsetsofsetsofmeasurezeroalsohavemeasurezero.Apropertyissaidtohold
almost everywhere, abbreviated as a.e., if the property holds except on a set of
measurezero.
Afunctionf:Rn→RismeasurableifforeveryopenS⊂R,theinverseimage
f−1(S)∈M.Thus,continuousfunctionsaremeasurable.Nonnegativemeasurable
functions are integrable, the integral being defined using approximation by
simplefunctions.Moregenerally,iffismeasurable,thenfisintegrableaslongas
thepositivef+andnegativef−partsoffcancomprisetheintegral:
whereoneoftheintegralsontheright-handsideisfinite.
Theessentialsupremumofameasurablefunctionfisdefinedby
Amongmanypropertiesofmeasurablefunctions,thefollowingtwotheorems
areespeciallyimportantforPDE.
TheoremB.1.(Monotone Convergence Theorem) If a sequence
functionsismonotonicallyincreasing:
of integrable
then
TheoremB.2.(DominatedConvergenceTheorem)Forasequence
ofintegrable
functionswithfk→fa.e.,ask→∞,and|fk|≤ga.e.,andforapositivemeasurable
functiongwith∫R gdx<∞,thefollowinglimitholds:
n
FurtherdetailsonLebesguemeasureandintegrablefunctionscanbefoundin
summary in the Appendix in Evans [12], and in many books on measure and
integration, for example, Ambrosia et al. [3]. A major advantage of using
LebesguemeasuretogeneralizethenotionofintegralisthatspacesLp,p≥1of
measurablefunctionsarecomplete.
Consider a real or complex vector space X over the scalar field R or C
(respectively).Anorm||·||:X→[0,∞)isafunctionthatisrequiredtohave
theseproperties:
1.||x||=0ifandonlyifx=0;
2.||αx||=|α|||x||,forallx∈X,andscalarα;and
3.||x+y||≤||x||+||y||forallx,yinX(thetriangleinequality).
A norm defines a metric ρ, expressing the distance between elements x,y ∈ X:
ρ(x,y)=||x−y||.
LetX be a vector space with norm || · ||. A sequence
sequenceifforeveryϵ>0,thereisN=Nϵ>0suchthat
is a Cauchy
Thatis,||xj−xk||→0asj,k→∞.
XiscompleteifeveryCauchysequenceinXconvergestoalimitinX:||xk−
x||→0forsomex∈X.ABanachspaceisacompletenormedvectorspace.Thus,
RnandCnareBanachspaces,andsoisC([a,b])withthenorm||f||=maxa≤x≤b
|f(x)|.
Aninnerproduct(·,·):X×X→Risafunctionsatisfying
1.
forallx,y∈X;
2.(x,x)≥0forallx∈X;
3.(x,x)=0ifandonlyifx=0;and
4.(αx,y)=α(x,y),forallx,y∈X,andscalarα.
Aninnerproductdefinesanormby
.ABanachspaceX that has an
inner product defining its norm is a Hilbert space. Thus, a Hilbert space is a
completeinnerproductspace.
Spaces of infinite sequences of numbers are convenient examples of infinite-
dimensionalvectorspaces.Thespaceℓp(“littleellp”),withp≥1,consistsofall
sequences
satisfying
withnorm||x||p=
.The
spaceℓp is a Banach space. The space ℓ2 is a Hilbert space with inner product
,theinfinite-dimensionalversionoftheEuclideaninnerproduct.
Whenp = ∞, ℓ∞ is the Banach space of all bounded sequences with norm
||x||∞=supk|xk|.
Thespacecofconvergentsequencesxofrealorcomplexnumberswiththeℓ∞
norm||x||∞=supk|xk| is a closed subspace of ℓ∞ and is thus a Banach space.
The subspace c0 consisting of sequences
with limk → ∞ xk = 0 is also a
Banachspace.
The Weierstrass M-test states that if {fk} is a sequence of real- or complexvalued functions defined on a set U ⊂ Rn, and there is a sequence of positive
numbersMksuchthatforallk≥1andallx∈U,
thentheseries
convergesuniformlyonU.
APPENDIXCSystemsofOrdinaryDifferentialEquations
LetF : Rn → Rn be continuous. We say that a C1 curve C = {x = x(t), t ∈ I}
(whereI is an interval and x:I→RnisC1) is an integralcurve of the vector
fieldFif
Integral curves are sometimes called trajectories. Since the ODE system is
autonomous, integral curves are translation invariant: for any t0 ∈ R,x(t + t0)
tracesthesamecurveCinRnastvaries.
ConsiderthenonlinearautonomoussystemoftwoODE:
Equilibriaarepoints(u0,v0)∈R2satisfyingF(u0,v0)=0.Behaviorofthesystem
(C.1)nearanequilibrium(u0,v0)isrelatedtothelinearizedsystem
where
istheJacobian.Let’sassumetheeigenvaluesλ1,λ2ofdF(u0,v0)aredistinct(λ1≠
λ2)andhaveeigenvectorsv1,v2,respectively.Theeigenvaluesandeigenvectors
maybecomplex.Thegeneralrealsolutionofthelinearsystem(C.2)is
whereC1,C2 are arbitrary constants. If the eigenvalues are complex, then these
constantsarecomplex,anditisunderstoodthatsolutionsaretherealpartofthis
formula.TheequilibriumisclassifiedasstableifReλj<0,j=1,2,unstableifRe
λj > 0, j = 1, 2. Stable and unstable spirals correspond to complex conjugate
eigenvalues; the integral curves of F(u, v) spiral into or out of (u0, v0). If the
eigenvalues are real and of the same sign, then the equilibrium is a stable or
unstablenode.Ifλ1,λ2arerealandhaveoppositesigns,thentheequilibriumisa
saddle,andthereareintegralcurvesthataretangenttotheeigenvectorsv1,v2at
(u0,v0).ThesecurvesarethestableandunstablemanifoldsMS,MUof(u0,v0).
References
[1]R.AbeyaratneandJ.K.Knowles.Implicationsofviscosityandstraingradienteffectsforthekineticsof
propagatingphaseboundariesinsolids.SIAMJ.Appl.Math.51:1205–1221,1991.
[2]D.J.Acheson.ElementaryFluidDynamics.OxfordUniversityPress,NewYork,1990.
[3]L.Ambrosio,G.DaPrato,andA.Mennucci.IntroductiontoMeasureTheoryandIntegration,volume10of
EdizionidellaNormale.Springer,NewYork,2011.
[4]S.S.Antman.Theequationsforlargevibrationsofstrings.Am.Math.Monthly87(5):359–370,1980.
[5] D. G. Aronson and H. F. Weinberger. Multidimensional nonlinear diffusion arising in population
genetics.Adv.Math.30:33–76,1978.
[6]A.Bressan.HyperbolicSystemsofConservationLaws:TheOne-DimensionalCauchyProblem,volume20of
MathematicsandItsApplications.OxfordUniversityPress,Oxford,2000.
[7] A. Chorin and J. L. Marsden. A Mathematical Introduction to Fluid Mechanics, volume 3 of Pure and
AppliedMathematics.Springer,NewYork,1990.
[8] S.-N. Chow and J. K. Hale. Methods of Bifurcation Theory, volume 251 of Comprehensive Studies in
Mathematics.Springer,NewYork,1982.
[9] J. D. Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math.
9(3):225–236,1951.
[10] J. D. Conway. Functions of One Complex Variable, 2nd edition, volume 11 of Graduate Texts in
Mathematics.Springer,NewYork,1978.
[11] M. G. Crandall and P. H. Rabinowitz. Bifurcation from simple eigenvalues. J.Funct.Anal. 8(2):321–
340,1971.
[12] L. C. Evans. Partial Differential Equations, volume 19 of Graduate Series in Mathematics. American
MathematicalSociety,Providence,2010.
[13]P.C.FifeandJ.B.McLeod.Theapproachofsolutionsofnonlineardiffusionequationstotravelling
frontsolutions.Arch.RationalMech.Anal.65(4):335–361,1977.
[14]R.A.Fisher.Thewaveofadvanceofanadvantageousgene.Ann.Eugen.7:355–369,1936.
[15]Y.ForterreandO.Pouliquen.Flowsofdensegranularmaterials.Annu.Rev.FluidMech.40:1–24,2008.
[16]P.R.Garabedian.PartialDifferentialEquations.AmericanMathematicalSociety,Chelsea,1998.
[17] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. Korteweg-deVries equation and
generalizations.VI.Methodsforexactsolution.Comm.PureAppl.Math.27(1):97–133,1974.
[18]J.Glimm.Solutionsinthelargefornonlinearhyperbolicsystemsofequations.Comm.PureAppl.Math.
18(4):697–715,1965.
[19]M.G.GolubitskyandD.G.Schaeffer.Singularitiesandgroupsinbifurcationtheory,volume1.Springer,
NewYork,1984.
[20]J.M.N.T.GrayandA.R.Thornton.Atheoryforparticlesizesegregationinshallowgranularfreesurfaceflows.Proc.R.Soc.Lond.A,461(2057):1447–1473,2005.
[21] E. Hopf. The partial differential equation ut + uux = μuxx. Comm. Pure Appl. Math. 3(3):201–230,
1950.
[22]K.HutterandK.R.Rajagopal.Onflowsofgranularmaterials.ContinuumMech.Thermodyn.6:81–139,
1994.
[23]E.Isaacson,D.Marchesin,andB.Plohr.Transitionalwavesforconservationlaws.SIAMJ.Math.Anal.
21:837–866,1990.
[24]D.Jacobs,B.McKinney,andM.Shearer.TravellingwavesolutionsofthemodifiedKorteweg-deVriesBurgersequation.J.Diff.Equ.116(2):448–467,1995.
[25] R. D. James. The propagation of phase boundaries in elastic bars. Arch. Rational Mech. Anal.
73(2):125–158,1980.
[26]A.Jeffrey.AppliedPartialDifferentialEquations,AnIntroduction.AcademicPress,NewYork,2003.
[27]F.John.Formationofsingularitiesinone-dimensionalnonlinearwavepropagation.Comm.PureAppl.
Math.27:377–405,1974.
[28]T.Kato.PerturbationTheoryforLinearOperators,volume132ofDieGrundlehrendermathematischen
Wissenschaften.Springer,NewYork,1966.
[29] B. L. Keyfitz and H. C. Kranzer. A system of non-strictly hyperbolic conservation laws arising in
elasticitytheory.Arch.RationalMech.Anal.72:219–241,1980.
[30]E.Kreyszig.IntroductoryFunctionalAnalysiswithApplications.Wiley,NewYork,2007.
[31] S. N. Kruzkov. First order quasilinear equations in several independent variables. Math. USSR SB,
10(2):217–243,1970.
[32]P.D.Lax.HyperbolicsystemsofconservationlawsII.Comm.PureAppl.Math.10:537–566,1957.
[33] P. D. Lax. Development of singularities of solutions of nonlinear hyperbolic partial differential
equations.J.Math.Phys.5:611–613,1964.
[34] P. LeFloch. Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock
Waves.Birkhauser,Zurich,2002.
[35]A.Majda.CompressibleFlowandSystemsofConservationLawsinSeveralSpaceVariables,volume53of
AppliedMathematicalSciences.Springer,NewYork,1983.
[36]R.M.Miura.TheKorteweg–deVriesequation:Asurveyofresults.SIAMRev.18:412–459,1976.
[37]O.A.Oleinik.UniquenessandstabilityofthegeneralizedsolutionoftheCauchyproblemforaquasilinearequation.UspekhiMat.Nauk14(2):165–170,1959.
[38]W.Rudin.PrinciplesofMathematicalAnalysis.McGraw-Hill,NewYork,1976.
[39]D.Serre.SystemsofConservationLaws,I,II.CambridgeUniversityPress,Cambridge,2000.
[40]M.Shearer.NonuniquenessofadmissiblesolutionsofRiemanninitialvalueproblemsforasystemof
conservationlawsofmixedtype.Arch.RationalMech.Anal.93(1):45–59,1986.
[41] M. Shearer, D. G. Schaeffer, D. Marchesin, and P. Paes-Leme. Solution of the Riemann problem for a
prototype 2 × 2 system of non-strictly hyperbolic conservation laws. Arch. Rational Mech. Anal.
97(4):299–320,1987.
[42] M. Slemrod. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch.
RationalMech.Anal.81(4):301–315,1983.
[43]J.Smoller.ShockWavesandReaction-DiffusionEquations.Springer,NewYork,1994.
[44]E.Stade.FourierAnalysis,volume69ofPureandAppliedMathematics.Wiley,NewYork,2005.
[45]W.Strauss.PartialDifferentialEquations:AnIntroduction,secondedition.Wiley,NewYork,2008.
[46]G.B.Whitham.LinearandNonlinearWaves.Wiley,NewYork,1974.
[47]C.C.Wu.NewtheoryofMHDshockwaves.InM.Shearer,ed.,ViscousProfilesandNumericalMethods
forShockWaves.SIAM,Philadelphia,1991.
Index
absorbingboundarycondition,88–90
amplitude,43,77,114–115,179
avalanchemodel,36–38,209–210,223
backwardcharacteristics,51
balancelaw(seealsoconservationlaw),11,22–26,45,191,196
BanachFixed-PointTheorem,256
Banachspace,102,218,260–261
beamequation,17,94–96
bendingmodulus,94
Benjamin-Bona-Mahoney(BBM)equation,17
Besselfunctions,127
Bessel’sinequality,109–111,156
bifurcation,234–235
bilinearfunctional,168,170,172
bilinearoperator,235
bistableequation,181–186
boundaryconditions,3–4,7
absorbing,88–90
clamped-beam,95
Dirichlet,3,75,82–85,89,101,124,125,127,129,134,146,149,158
homogeneous,4,86,100,126,134–136
inhomogeneous,4,90–91
Neumann,4,72,75,85–87,89,101,102,125
radiating,88–89
Robin,88,101,102
stress-free,53,225
Brownianmotion,65
Burgersequation,5–6,175–176,206–208
Cole-Hopftransformation,5,175,207
inviscid,4–5,7–8,25–26,38–40,189–195
cantileveredbeam,95
Cauchy,Augustine-Louis,17
Cauchyinequality,153
Cauchy-KovalevskayaTheorem,11,17,20–21,81
Cauchyproblem,32–40,204,207–211,215,218,232
approximatesolutions,241–242
energymethodanduniqueness,56–59,73–74
Fouriertransform,116
fundamentalsolution,66–72
heatequation,68–74
methodofcharacteristics,32–38
scalarconservationlaws,196
semi–infinitedomain,53
waveequation,49–51
Cauchy-Schwartzinequality,155,162–164,169,
Cauchysequence,102,111,156,158,166,171,260
centeredrarefactionwaves,190–191,193,194,196,198,226,227,236
chainrule,14,18,30,184,254
changeofvariables,15,30,66,71,116,190,197
characteristics,65,78,85,108,189,237,241,249
initialvalueproblems,29–40
d’Alembert’ssolution,48–56
Laxentropycondition,194–195
systemsofconservationlaws,219–222
undercompressiveshocks,204–211
variablecoefficients,217–219
clamped-beamboundaryconditions,95
classification,12–16
closure,51,253
codimensionone,165
Cole-Hopftransformation,5,175,207
combustion,38,65
compactsupport,51,71,78,120–121,137,144,158,202,232–233,253
completeness,102–106,110–112,156,159,162,166,171,255–256,260
conservationlaw(seealsobalancelaw),4,11,21,24,38,43,143,245,248
scalarconservationlaws,189–211
systemsoffirst-orderhyperbolicPDE,215–242
constitutivelaw,22–25,45,245,248
contactdiscontinuity,250
continuousdependenceondata,11,12,21,50
continuousfunctions,50,81,102–104,135,217–218,259
continuummechanics,21,215
contractionmappingprinciple,218,256
convection-diffusion,175
convergence,66,69,71,81,82,166,171,176
convergenceinthenorm,102
distributions,140
Fouriersineseries,84–85
Gibbsphenomenon,113–114
ℓ2,156–157
L2,110–112
Laplace’sequation,127–130
pointwise,103–108
Poisson’sequation,120–122
powerseries,19–20
testfunctions,137–140
theorems,259–261
uniform,87,108–110
convexfunction,199,201,203,257
convolutionproduct,71,120,121,129,138
d’Alembertsolution,43,48–56,58–61,78,91
dam-breakproblem,239–241
Darcy’slaw,198
density,5,6,21–26,94,116,119,221,223
fluidmechanics,245–247
trafficflow,197–198
dependentvariable,2
differentialoperator,13,14,48,100–101,112,135,157,167,172
Green’sfunctions,144–145
symmetricoperator,167,168
diffusion,1–3,23,65,179,207
diffusionequation,23
Diracdeltafunction,71,106,136,202
Dirichletboundaryconditions,3
boundaryvalueproblem,125
energyprinciple,75
Green’sfunctions,134
heatequation,82–86
Laplace’sequation,127
maximum/minimumprinciple,124
methodofimages,149
Poisson’sequation,146
Poisson’sformula,129
symmetry,101
Dirichletkernel,106,129
dispersionrelation,16–17,95,177
displacement,1–3,24,44,48,49,51,55,133,220,223
distributionalderivative,141
distributions,71,87,136–144,202,232
convergence,140
regular,139,142,144
singular,120,139
divergenceform,167
divergencetheorem,22–24,146,210,246,256
domainofdependence,51–52,58
dominatedconvergencetheorem,138,260
dualspace,138
Duhamel’sprinciple,57–59,77–78
eigenfunction,66,84–86,99–101,112,128
eigenvalueproblem,15,84,112,168,205,230,248,263
absorbingboundaryconditions,89
bifurcation,234
characteristicspeeds,219
eigenfunctionsforODE,99–102
elasticstringequations,223,225–227
Fisher’sequation,180–182
heatequation,84–91
linearsystems,215–216
p-system,221
waveequation,92–93
elasticmodulus,46,94
elasticrod,43–46
elasticstringequations,43,46–48,57,223–226
elasticity,11,17,24,38,43,133,204,220,245
ellipticPDE,13–17,45,103,119,158,161,232
energyfunction,23,56,75,87,247
energymethod,43,56–57,65–66,67,73–75,125,177,180
entropyflux,201,202
entropyinequality,202–203
equationofstate,5,248,249
equivalenceclass,104,154
essentialsupremum,154,158,259
essentiallybounded,154
Euclideannorm,102,157
Eulerequations,6,247–250
Eulerformulas,82,84,91,105
Eulerianvariables,30,44
evolutionequation,2,6
existence,11–12,18–20,50,57,66,73,103,165,218,226
bistableequation,183–185
elliptictheory,168–176
methodofcharacteristics,33–38
Poisson’sequation,161–162
extension,55,60,72–73,82,87,92,113
Fick’slaw,23,179
Fisher’sequation,4–5,179–180
flux,22–26,73,88,125,179,196,256
entropyflux,201
convexfluxfunction,189,196,197,201–204
non-convexfluxfunction,198–199
Fouriercoefficients,82–94,109–113,114,126,129
Fouriermode,16–17,177
Fourierseries,81–114,129,155
convergence,84,90,102–114
convergenceinL2,110–112
Gibbsphenomenon,113–114
pointwiseconvergence,105–108
uniformconvergence,108–110
Fouriertransform,13,95,114–117
Fourier’slawofheatconduction(transfer,flow),23,72,179
frequency,16,17,21,91,95,114–115,177
Fubini’stheorem,257
fundamentalsolution,65,72,116,207
Duhamel’sprinciple,77–78
Green’sfunctions,133–147
heatequation,66–68
Poisson’sequation,119–122
fundamentaltheoremofcalculus,24–25,256
gammafunction,253,255
genuinenonlinearity,219,226,231,249
Gibbs,Josiah,113
GibbsPhenomenon,113–114
Green,George,133
Green’sfunction,65,127,130,133–138,144–149,161
fundamentalsolution,133–147
methodofimages,147–149
Green’sidentity,121,125
Green’stheoremintheplane,256
groupvelocity,17
Hadamard,JacquesS.,11,21
Hamiltonian,178,180,186
Hamilton-Jacobiequations,29
harmonicfunction,120–125,129,146–148
maximumprinciple,123
meanvalueproperty,122
heatequation,2–3,5,13,15,17,23,65–78,81–91
Cauchyproblem,68–72
Duhamel’sprinciple,77
energymethod,65,73–74
fundamentalsolution,65,66–68,72,77–78,116,207
maximumprinciple,65,75–77
separationofvariables,66,82–90
heteroclinicorbit,183–186,205
Hilbertspace,103,104,156,159,164–170,260
Hölder’sinequality,154–155
Holmgren,Erik,20–21
homoclinicorbit,178,182–183
homogeneity,4,94
homogeneousboundarycondition,4,72,86,125,126
eigenfunctions,100–101
Green’sfunctions,134–136
Hooke’slaw,45–46,221,224–225
Huygens’principle,43,59,61
hyperbolicPDE,13–15,43,65,78,215,248
ill-posedness,11,17,21
implicitfunctiontheorem,38,234–235,255
incompressibility,6,119,246
independentvariable,2
inhomogeneity,4
inhomogeneousboundarycondition,90
initialcondition,2–3,7,9,11,21,50,53,58,69,78,102,142,189,190,195
Cole-Hopftransformation,207–211
dambreakproblem,240–241
dispersionrelation,16–18
linearsystems,215–217
methodofcharacteristics,29–39
separationofvariablesforheatequation,83–86
separationofvariablesforwaveequation,91–95
systemsofhyperbolicconservationlaws,231–232
integralaverage,122–124,254
integralkernel,129,133,135
inversefunctiontheorem,33–36,255
inviscidBurgersequation,4,7–8,25–26,38–40,189–195,197
Jacobianmatrix,34,180,181,205,219,255,263
kineticenergy,43,56,177,247
Korteweg-deVries(KdV)equation,5,17,176–186,204
Kovalevskaya,SofiaVasilyevna,17,20,21,81
L∞space,154
ℓ2space,155–156
L2space,100,104,164
Lpspace,154–155
Lagrangianvariables,30,44,221–222
Laplace’sequation,3,13,17,21,66,119–130
cylindricaldomains,127–130
fundamentalsolution,119–122
Hadamardill-posedness,21
polarcoordinates,127
rectangulardomain,125–127
separationofvariables,125–130
sphericaldomains,127–130
Laplace,Pierre-Simon,119
Laplacian,2,119,121,127,146–149,179
Laxentropycondition,183,194–195,196,201–206,211,236–238,241,250
Lax-Milgramtheorem,170–172
Lebesgue-integrablefunctions,104,157
Lebesguemeasure,104,259,260
Leibnizintegralrule,257
Lighthill-Whitham-Richardsmodel,25
linearPDE,2,4
lineartransportequation,2,4–7,16,48,177,216
linearlydegenerate,219,221,249
locallyintegrablefunction,138–139,141,144
logisticequation,5,179
Lyapunov-Schmidtreduction,235
maximumprinciple,12,65,75–77,122–125
mean-valueproperty,122–124,129
measurablefunction,104,154,259–260
measurezero,104,259
methodofcharacteristics,4,65,78,85,108,189,237,241,249
d’Alembert’ssolution,48–56
initialvalueproblems,29–40
Laxentropycondition,194–195
systemsofconservationlaws,219–222
undercompressiveshocks,204–211
variablecoefficients,217–219
methodofimages,147–149
methodofsphericalmeans,59–61
minimumprinciple,77,124
mollifier,137–138
momentum,6,21–24,44–46,177,222–248
monotoneconvergencetheorem,157,259
multi–indexnotation,157–158
Navier-Stokesequations,ix,6,222,245–247
Neumannboundarycondition,4,72,75,85–87,88–89,101,102,125
noncharacteristiccurve,34–36
nondimensionalization,94,246–247
nonlinearheatequation93–94
nonlinearPDE,16
nonlinearsmalldisturbanceequation,16
nonlineartransportequation196,248
openball,121,253
orderofaPDE,2
orthogonal,6,14,15,81,86,88,100,101,111–112,165,220,257
orthonormal,14,108,110–112,156,257
oscillations,17,93,113,180
parabolicboundary,76–77
parabolicmanifold,205
parabolicPDE,13–16,65,71,75,78,103,159,179
Parseval’sidentity,111–113,156
partialdifferentialequation,1–2
periodicextension,82,87,92,113
phasespeed(phasevelocity),17
physicalconfiguration,43–44
planewaves,179
Poincaréinequality,162–164,169
pointwiseconvergence,102–117,129
Poissonkernel,129–130
Poisson’sequation,119–121,161–166
maximumprinciple,122–125
uniqueness,124
Poisson’sformula,129–130
populationmodels,5,65,179–180
porousmediumequation,5,78,93
separationofvariables,93–94
potentialenergy,43,56
powerseries,17–18,19,21,81
principalpart,13–14,16
principalsymbol,13–14,16,173
p-system,215,220–222,223,226
propagationofsingularities,229–230
rarefactionwaves,228
Riemannproblem,236–239
shockformation,230–232
weaksolutionsandshockwaves,232–234
quarter-planesolution,53–54,72
quasilinearfirst-orderequation,32,35
quasilinearPDE,28,157
quasilinearwaveequation,16,220,223,248
radiatingboundarycondition,88–89
randomchoicemethod,242
randomwalk,65
Rankine-Hugoniotcondition,143,202,210,241
Eulerequations,249–250
inviscidBurgersequation,191–198
Riemannproblem,236–237
systems,233–234
rarefactionwave,190–200,219,226–228,236–242,249
realanalytic,19–21
referenceconfiguration,43,44,222–224
regionofinfluence,51–52,71
regularity,12,50,57,66,78,103,110,153,162
Reynoldsnumber,246–247
Riemanninvariant,219–223,230–231,241
Riemann-LebesgueLemma,106–109
Riemannproblem,190–195,200,206,236–241
Rieszrepresentationtheorem,164–166,168–172
Robinboundaryconditions,88,101,102
scaleinvariance,66,189–190
semilinearwaveequation,16
separationofvariables,21,66,81–96,100,125–130,177,210
beamequation,94–95
Fourierseries,81
heatequation,82–90
nonlinearheatequation,93
waveequation,91–92
series,47,66,155,157
convergence,99–117
Fourierseries,81–88,90–93
Laplace’sequation,126–130
powerseries,18–21
Taylorseries,254–255
shallowwaterequations,6,222–223,239–242
shockwave,4,8,38,143,176,186,219,249
admissibility,236–237
bifurcationtheory,234–236
formation,230–232
inviscidBurgersequation,191–195
Laxentropycondition,201–211
p-system,238–239
weaksolutions,232–234
shootingargument,183
singularity,4,35,78,119–121,130,146–147,156,158
smoothness,12,50,57,66,78,103,110,153,162
Sobolevspace,158–159,161–172
solitarywave,5,175–179,183
solutionofaPDE,3
stability,11,211
stablemanifold,182–184
Stokesequations,247
Stokes’theorem,256
strain,45–47,156,220–224
strainsoftening,45,221
stress,21,44–46,53,220–221,224–225
stress-freeboundarycondition,53,225
stricthyperbolicity,215,221,223,225–226,235
strictlyconvexflux,202–203
strongmaximumprinciple,77,123–124
Sturm-Liouvilleproblem,99–101,112–113
supnorm(uniformnorm),102–103,218
superposition,176
supportofafunction,51–52,54,71,78,120–121,137,144,158,202,232–233,253
symmetricdifferentialoperator,101,167,168
systemsofPDE,204,215–241
characteristics,215–219
conservationlaws,219–220
elasticstringequations,223–226
genuinenonlinearity,219,226,231,249
hyperbolicity,219–221,223,225,248–249
linearsystems,215
p-system,215,220–222,223,226,228–232,237–239
Rankine-Hugoniotcondition,233–236
rarefactionwave,219,226–230,236–242,249
Riemanninvariant,219–223,230–231,241
shallowwaterequations,222–223,239
shockformation,230–234,236–237
stricthyperbolicity,219
TaylorSeries(Taylor’stheorem),19,254
testfunctions,136–144,157,167,202,232–233
tracetheorems,159
trafficflowmodel,24–26,191,197–199
translationinvariance,60,68
travelingwave,2,5–8,16–17,91,175–186,204–206,216
triangleinequality,50,70,107,112,154–155,260
Tricomiequation,16
ultrahyperbolicity,15
undercompressiveshock,204–206
uniformconvergence,50,69,85,102–117,137,261
uniformnorm(supnorm),102–103
uniqueness,3,6–7,11–12,18,30,65,124,142,230
Cauchyproblem,49–50
elliptictheory,symmetriccase,168–172
energymethod,56–57,73–74
fixedpoint,218–219
Holmgrentheorem,20
implicitfunctiontheorem,255–256
ODEtheory,33–35
Poisson’sequation,161–166
rarefactionwaves,190–193
shockwaves,236–237
unitnormal,121,253,254
unstablemanifold,182–185
vibratingstring,46–52,91
viscosity,5,6,207,247
waveequation,2–3,13,43–61,71,72,108,158,224,248
balancelaws,23–24
characteristics,48–49
d’Alembert’ssolution,48–19
domainofdependence,51–52,58
Duhamel’sprinciple,57–59,77–78
energy,56–57
methodofsphericalmeans,59–61
nonlinear,17
quasilinear,16
regionofinfluence,51–52,71
semilinear,16
separationofvariables,91–93
system,216–217
uniqueness,56–57
vibratingstring,46–52,91
wavefronttracking,241
wavenumber,13,16–17,21,95,177
wavespeed,2,4,6,7,17,29,43,48,52,71,242,254
Burgersequation,175–177
weakconvergence,87
weakderivative,144,157,158,162
weakformulation,164,232
weakmaximumprinciple,77,124–125
weaksolution,3,40,50,103,172,206,210
inviscidBurgersequation,189–193
Poisson’sequation,161–162
second-orderlinearellipticequations,167–168
systems,231–237
WeierstrassM-test,87,261
well-posedness,ix,11–12,21,49,50,153,191
Whitham,G.B.,17,25,177,208
Young’sinequality,153–154,172