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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. 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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. 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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