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From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
0. Parameters and Notations in Navier-Stokes and Euler equations
The Time t  0
The Space D  R n (region, set) n  2, 3 in 2 or 3 dimensional space filled with a fluid
The Point x  D (position, spatial coordinate, element), x  x(t )  ( x1 ,..., x n )  D  R n
The Velocity vector (field, function), u  u ( x, t )  (u1 ,..., u n )  R n , u ( x, t ) 
dx (t )
dt
The Particle trajectory mapping (this particle of fluid traverses a well defined trajectory)
X (  , t ) :   R n  X ( , t )  R n and X  X (  , t )  ( X 1 ,..., X n ) (location at time t of
a fluid particle initially placed at the point   ( 1 ,...,  n ) at time t  0

The Jacobian of this transformation J ( , t )  det( X ( , t )) like  x u ( x, t )

d
X ( , t )  u ( X ( , t ), t )) and X ( , t  0)   (nonlinear ODE defines the mapping)
dt

J ( , t )  (div x u ) X J ( , t ) for a smooth velocity field (function) u  R n

d
f ( x, t )dx  
( f t  div x ( fu))dx and   R n an
The Transport Formula

X
(

,
t
)
X
(

,
t
)
dt
open, bounded domain with a smooth boundary for any smooth f ( x, t ) , smooth u  R n
The incompressible flow means div u  0 or J ( , t )  1
The Fluid density (field, function)-  ( x, t )
The Pressure (scalar field, function)- p  p( x, t )  R unknown
The External force f  f ( x, t )  ( f1 ,..., f n )  R n
The Dynamic (absolute, material parameter) viscosity  and kinematic viscosity (vorticity
diffusion coefficient)   /  0  0 (ratio of absolute viscosity to density)
Viscosity is the tendency of a fluid to resist shearing motions
n
D


d
The Convective (material) derivative operator
   ui
  u 
Dt t i 1 xi dt
n
Du u
u
The Material derivative

  ui
 ut  u  u  ut (u  )u  R n
Dt t i 1 xi
u
u
u
The Gadient vector
 ( 1 ,..., n )  R n
xi
xi
xi


,...,
) , u  R n  R n , matrix
The Gradient operator   (
x1
x n
2
The Laplace operator    2 , u  R n
i 1 xi
n
The Divergence of a vector field- div u    u 
n
ui
 x
i 1
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Curl curl u    u
i
Page 1
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
1. Gradient, Divergence, Laplace operator, Material derivative and etc
We will see more clearly about them in 2 or 3-dimensional case


Position vector x  x (t )  ( x, y, z )  ( x(t ), y(t ), z (t ))  R 3 of a fluid particle
Velocity vector (derivative by time) of the position vector
  

d 
u  u ( x )  u ( x, y, z )  (u, v, w)  x (t )  ( x (t ), y (t ), z (t ))  R 3
dt
  
Gradient operator   ( , , )
x y z

du dv dw
)  R3
Gradient vector  t u  ( , ,
dt dt dt
(0)
 du dv dw 


 dx dx dx 

du dv dw   u i 
Gradient Matrix  x u  (u , v, w)  

 R3  R3
 dy dy dy   x 
 du dv dw   j  33


 dz dz dz 

 3 u

du dv dw
Divergence div u    u   i  trace(u ) 


R
dx dy dz
i 1 xi
Laplace operator, material derivative (acceleration),
Laplacian (differential operator)    2 
2

2
i 1 xi
3

 2u  3  2u 3  2 v 3  2 w 
   2 ,  2 ,  2   R 3

2
i 1 xi
 i 1 xi i 1 xi i 1 xi 

2
u   u  (u, v, w)  (trace((u)), trace((v)), trace((w)))
3
D





Material derivative
   ui
 t  u    t  u  v  w
Dt t i 1 xi
x
y
z


Laplace operator u   2 u 
3
(0)
Acceleration vector (material derivative), derivative of time of the position vector




u
du d 2 
d 
3
a (t ) 

x (t )  u x(t ), y(t ), z (t ), t   R noting u x 
x
dt dt 2
dt








u t u x u y u z 
a (t ) 



 u t  uu x  vu y  wu z (chain rule)
t t x t y t z t


   Du




a (t )  u t  u  u 
where u    u
v w
Dt
x
y
z
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 2
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
2. Field, Vector integral, Curl, Divergence, rotation, Theorems
We will see more clearly about them in 2 or 3-dimensional case
Vector field, gradient field, potential function, line integral over (along) a plane or space
smooth non-closed curve C , parametric equation



Vector field (function) F ( x, y, z )  P( x, y, z ) i  Q( x, y, z ) j  R( x, y, z ) k



Gradient field (function) f ( x, y, z )  f x ( x, y, z ) i  f y ( x, y, z ) j  f z ( x, y, z ) k
Conservative vector field (function) if there exists a function f such that F  f
Potential function f (scalar function) for F
b
Fundamental Theorem of Calculus
 F ( x)dx  F (b)  F (a) in R
a


Plane curve C , Parametric vector equation r (t )  ( x(t ), y (t ))  x(t ) i  y (t ) j
(0)
b
Line Integral along smooth curve C ,
 f ( x, y)ds   f ( x(t ), y(t ))
CR 2
x 2  y 2 dt
a
Line Integral of Vector field along space smooth (non-closed) curve C
b
 F  dr   F  T ds   F (r (t ))  r (t )dt 
CR
3
C
a
Fundamental Theorem of Line integral
 Pdx  Qdy  Rdz
CR 3
 f  dr  f (r (b))  f (r (a))
CR 3
Space, 3-D, divergence, curl, rotation, irrotational, conservative vector field
Fi
P Q R


R
x y z
i 1
i



i
j
k
 R Q P R Q P  


 
Curl curl F    F  

,

,

 R3

y

z

z

x

x

y

x

y

z


P Q R
Irrotational (conservative) Field F at P if curl F  0
fluid is free from rotations at P , Curl associated with rotations (whirlpool, eddy)
Incompressible field F if div F  0 , always true div (curlF )  0
Divergence div F    F 
3
 x
 trace(F ) 
(0)
2 piecewise smooth curves (paths), independent of path, initial, terminal point, closed or
simple curve, open, connected set (region), conservation of energy, Green’s Theorem,
orientation, plane case, region, integrand (divergence of the vector field)
Independent of path
 F  dr  0 or  F  dr   F  dr
CD  R 2
C1
C2
P Q
Conservative vector field in a plane
or  f such that f  F

y dx
Ts.Batmunkh, Department of Mathematics, University of Wyoming
(0)
Page 3
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
 Q P 

dA in the plane R 2

x y 
C  DR 2
DR 2 

Green’s Theorem (I vector form)  F  dr   (curl F )  kdA
 Pdx  Qdy  
Green’s Theorem
C
D

Green’s Theorem (II vector form)  F  n ds   div F ( x, y )d A
C
D
Space, 3-D, Surface (flux) integrals of vector field , parametric, oriented surface
Stokes theorem (higher version of Green’s theorem), Divergence theorem
Green’s theorem (double integral over a plane region D  R 2 to a line integral around its
plane boundary curve
Stokes theorem (triple) surface integral over a surface S  R 3 to a line integral around its
boundary curve of S (oriented piecewise smooth bounded by a simple, closed piecewise
smooth boundary curve C  S  R 3 with positive orientation)
Surface integral (Flux) of F over the surface S is

 F  dS   F  ndS in R
SR 3
Stokes theorem
S

F

dr

(
curl
F
)

dS

(
curl
F
)

k
dA



C  SR 3
Divergence theorem

SR 3
3
SR 3
(0)
 F  n dS   F  dS   div F ( x, y, z)dV
S E
S
ER 3
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 4
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
3. Fluid Mechanics, Equations of fluid motion
Continuum
Fluid Mechanics
Inviscid flow   0
(Euler equation)
Compressible flow (gas,
density varies , entropy)
Viscous flow  0
(Navier-Stokes equation)
Incompressible flow
(liquid, density constant)
Laminar flow (straight
filament of smoke)
Turbulent flow
(smoke)
Internal flow (bounded
by solid surfaces)
External flow
(free, open channel)
Newtonian flow
External force,
homogenous equation
Entropy condition
Fluid (liquid or gas) is a substance that deforms continuously under the application of a
shear (tangential) stress no matter how small the shear stress may be.
Fluid as continuum (microscopic effects of molecules, an infinitely divisible substance)
Fluid particle (the small mass of fluid of fixed identity of volume)
Fluids with zero viscosity do not exist, however, we do models
Absolute (dynamic) viscosity  and kinematic viscosity  (ratio of absolute viscosity to
density)
Newtonian fluid (water, air, and gasoline under normal condition)
Non-Newtonian fluid (toothpaste, paint)
Ideal flow (governed by Euler equations of an inviscid, incompressible flow)
Euler equation (inviscid, incompressible, flow)
Navier-Stokes equation (viscous, incompressible flow), non-turbulent, Newtonian flow
Turbulent, Laminar flow (internal, external, viscous, incompressible flow)
Euler equation (Burger’s, Shock wave) in gas dynamics (inviscid, compressible flow)
General picture of the equations of Fluid Motion
Entropy flux  (u )
Absolute (dynamic)
viscosity 
Kinematic
viscosity   /  0
Stress tensor
S ( x, t )
Reynolds number
Re  U D / 
Navier-Stokes
Euler
 (u )  const
 (u )  const
 (u )  const
    0
 (u )  const
    0
viscous flow
inviscid flow
 0
 0
 0
viscous flow
~
S ( x, t )   pI  S (T )
viscous flow
Turbulent flow
inviscid flow
viscous flow
inviscid flow
inviscid flow
(non-turbulent)
Laminar flow
(non-turbulent)
Laminar flow
(non-turbulent)
Laminar flow
 0
S ( x, t )   pI
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 5
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
Fluid mass density
 ( x, t )
Divergence
Force
F
Fluid
div u
 ( x, t )  const
 ( x, t )  const
(gas) compressible
Incompressible
compressible
div u  0
F  0 external
 ( x, t )  const
Incompressible
 ( x, t )  const
div u  0
div u  0
div u  0
Incompressible
Incompressible
both
Incompressible
Both
Newtonian
Newtonian
yes
yes
F  0 internal
non-homogenous
homogenous
Non-Newtonian
Newtonian
Blow Up
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Incompressible
Page 6
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
4. Classical Mechanics. Equation of Motion
1. MECHANICS OF A PARTICLE
 Fundamental physical concepts: Space, time t , mass m , force F .
r  r (t ) be the radius vector of a particle from some given origin O .
dr
(rate change of distance with time)
dt
Linear Momentum: p  mv (product of the particle mass and velocity)
dp
 p  0  p -constant, is conserved
Conservation law of linear momentum: F 
dt
Vector Velocity: v 


(1)
(2)
The mechanics of the particle is contained in Newton’s Second Law of Motion.
The vector sum some forces exerted on the particle is the total force F .
Inertial or Galilean system
dp d
dv
 (mv)  m
 ma
dt dt
dt
2
2
Vector acceleration: a  v  d r / dt
Force: F 
The angular momentum of the particle about point
O (Cross product in 3-dimension)
Angular Momentum: L  r  p (cross product of radius and momentum)
dL
(cross product of radius and force)
dt
dL
 0  L -constant, is conserved
Conservation law of angular momentum: N 
dt
 Moment of force (Torque)  twisting or turning the force
Moment of Force or Torque: N  r  F 
Vector Field
(5)
(7)
(11)
A Vector Field is a vector valued function
F : D  R n  R n that assigns to each point x  D  R n a vector F (x)


Complex
analysis
Analytic
function
By algebraically there exists inverse function
Vector field, scalar field, force field, velocity field, gradient field,
gravitational field, force field, electric field, conservative vector field
 Flows are generated by vector fields and vice versa.
f :   C  C , f ( z )  f ( x  iy )  u ( x, y )  iv ( x, y )
f is differentiable at every point z   , then analytic function in a domain
u v
u
v
Cauchy-Riemann’s equations:
and


x y
y
x
  v  0
Harmonic (conjugate) function: u  u xx  u yy
Cauchy Morera theorem
 f ( z)dz(t )  F ( z
2
)  F ( z1 )  0 for closed curve C
C
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 7
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
F  F (e ) upon the particle in going from point 1 to point 2 is:
The Work done by the external force
(12)
2
Work:
Change in Kinetic energy :
W12   F  ds
1
(14)
W12  K 2  K1 (with constant mass)
K
V
i
, E  u  iv  K  iV
s
s
Conservative system (force):  F  ds  0 (closed path)
Conservation law of Energy of a Particle: F  u  iv 
dE d ( K  iV )

 0  K  V -constant, is conserved
ds
ds
Potential energy: F  V (r ) , F  ds  dV if conservative
Conservative, Closed system, closed circuit, contour integral  external force is conservative
If the force field (system) is conservative if the work W12 is the same for any physically possible
Conservation law of Energy: F 


path between points 1 and 2 (initial and end).
Time, Space, Mass, Velocity, Acceleration, Momentum, Impulse, Force, Energy
Matter
Anything that occupies space, has mass, and possesses inertia.
Time, Space
Time plays the role of a fourth dimension. Space has a volume.
Inertia
The tendency of an object to resist changes in its state of motion
Mass
Conservation
law of mass
Velocity
The quantity of matter contained in an object, including inertial
properties with density 
A measure of a substance's mass per init of volume. Some objects are
heavier than other objects, even though they are the same size.
the amount of mass remains constant--mass is neither created nor
destroyed
Rate of change of position with time (first derivative, direction)
Acceleration
Rate of change of velocity with time (second derivative, direction)
Force
a push or pull experienced by a mass m when it is accelerated
Newton’s first
law
Newton’s
second law
Law of inertia: a body at rest or a body in motion continues to move
at a constant velocity unless acted upon by an external force.
Net external force F acting on a body gives it an acceleration
a which is in the direction of the force and has magnitude inversely
proportional to the mass m of the body.
Weak law of action and reaction: For every external action force,
there is a corresponding reaction force which is equal in magnitude and
opposite in direction.
Momentum is a fundamental quantity in mechanics that is conserved in
the absence of external forces F  0 from Newton’s first law.
Momentum can be defined as mass in motion and refers to the quantity
of motion that an object has.
Instantaneous change in momentum
Density
Newton’s
third law
Momentum
Impulse
Momentum,
Force
From Newton’s second law, a force F produces a change in
momentum.
Ts.Batmunkh, Department of Mathematics, University of Wyoming
m   dV
  m /V
m constant
v  dx / dt
a  dv / dt


F  ma

F  0


F  ma


FA   FB


p  mv ,

J  Fdt  dp
F  dp / dt
Page 8
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
Conservation
law
of
Momentum
Energy
Kinetic
energy
Potential
energy
Conservation
law of Energy
Work=Energy
Work energy
theorem
Work, energy,
force
Power
Connection
Momentum of a system is constant if there are no external forces
F  0 acting on the system from Newton’s first law (of inertia).
Energy is an abstract scalar quantity of extreme usefulness in physics
and measured in units of mass times velocity squared.
Energy of Motion: is directly proportional to the square of its speed.
Kinetic energy is a scalar quantity (no direction).
Stored Energy of position: An object can store energy as the result of
its position.
Total mechanical energy, sum of kinetic energy and potential energy a
constant. Energy is the quantity that can be converted from one form
to another but cannot be created or destroyed.
Work is (a scalar quantity) the product of applied force causing a
displacement over a distance, has units of energy.
Total work equals the change in kinetic energy with constant force
Work is an integral with varying force
W  K  E and
d ( p1  p2 )
0
dt
E  mc 2
K  12 mv 2
U  mgh
E  K U
Constant
 
W  F s
W  K 2  K1
x2
W   Fdx
dE
F
dx
x1
Power is (a scalar quantity) Rate of change of work (or energy) with
time like a velocity. And the product of applied force and a velocity.
 
P  F v



W  K 2  K1 , W  U , F  dE / dx , F  U
Force
Internal
Contact force
External
Long distance
(Interaction)
Fundamental
Orbital motions
(Big)
Gravitational
Theory
of
everything TOE
Grand Unified
Theory GUT
horizontal -x
Vertical-y
Frictional, ,
applied
Normal, weight, tensional ,
Air, fluid resistance, spring
Gravitational
between
atoms
Electronic,
magnetic
Electroweak
interaction
Between nucleus of an
atom
Strong interaction
(sun energy)
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Between Neutrinos
(Subatomic)
Weak interaction
(new star, supernova)
Electromagnetic-weak
interaction
Page 9
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
5. Euler equation and conservation laws of hyperbolic systems
ut  f (u) x  0
Velocity u  u ( x, t ) , Mass density    ( x, t ) , Momentum (mass flux)  u
Pressure p  p( x, t ) , Energy (kinetic) E 
1 2
u , Flux function f , g : R m  R m
2


u ( x, t ) 
f (u ( x, t ))  0 and u : R  R  R m
t
x
Hyperbolic system m m Jacobian matrix f (u ) diagonalizable: eigenvalues are all real
In 1-Dim
Instationary flow phenomena of gases are described by Euler equation
In fluid dynamics, Euler equation governs the motion of a compressible (incompressible),
inviscid fluid and corresponds to Navier-Stokes equation with zero viscosity.
Euler equation directly represents conservation law of mass, momentum, and energy.
In gas dynamics (compressible flow) velocities approach the speed of light.
1-D Euler equation in gas dynamics
 u

 
     ( u )

     2
    
u    u  p    u     ( u )u  p   0 , or ut  f (u) x  0 (0)

t
x
t
u ( E  p)
 E 
 E    (u ( E  p )) 


From here
Conservation Laws
1. Conservation Law of Mass  t  ( u ) x  0 (continuity equation)
2. Conservation Law of Momentum ( u ) t  ( u 2  p) x  0
(0)
3. Conservation Law of Energy Et  (u ( E  p)) x  0
In 2-D
2-D Euler equation in gas dynamics u t  f (u ) x  g (u ) y  0
 u

 v

 
 2

 uv

 u 
     u  p   
0


 y  v 2  p 
t  v  x  uv




 
E 
u ( E  p )
v( E  p )
(0)
In 3-D
3-D Euler equation in gas dynamics u t  f (u ) x  g (u ) y  h(u ) z  0
 u

 v

 w

 
 2

 uv



 u 
uw
 u  p 








    v 2  p     vw
0
 v    uv
 y 
 z  2
t   x 


w

uw

vw

w

p






 






 E 
 w( E  p )
u ( E  p )
v ( E  p ) 
Ts.Batmunkh, Department of Mathematics, University of Wyoming
(0)
Page 10
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
6. Equation of fluid motion, Navier-Stokes equation and Euler equation
Assuming constant entropy, neglecting effects due to thermodynamics
The full Navier-Stokes equations consist (5=3+1+1) of the three momentum equations,
the continuity equation and the equation of state in 3-Dim.
Full Navier-Stokes Equation (with constant entropy)
Du
 p  P  F three momentum equations
Dt
The continuity equation (mass conservation law)
 t  div ( u )  0
p  r ( p)
The equation of state

Unknowns are the velocity u  (u , v, w) , the density  and the pressure p

Position x  ( x, y, z ) , given r ( p ), dr / d  0 and given force F
grad p  p  ( p x , p y , p z )

3
( ui )
div ( u)    ( u)  
( u) x  ( v) y  ( w) z
xi
i 1

(0)
Material derivative operator (accelerator) is:
n
D


d




   ui
  u   u  v  w
Dt t i 1 xi dt
t
x
y
z

 (div u ) x 
 u 

 


 
P  (    ) grad (div u )   u  (    ) (div u ) y     v 

 
 w 
 
 (div u ) z 
Condition of incompressibility
Incompressible flow where    0 is a known density constant

 t  0 and the continuity equation became div u  0 , thus volumes are preserved
(0)
After choosing suitable units, we can assume   1
Changing Absolute (dynamic) viscosity  by kinematic viscosity    /  0 we get
Navier-Stokes Equation (viscous, incompressible flow, with constant entropy)

Du
 u t  u  u  p  u  F three momentum equations
Dt
(0)

div u  0
the continuity (mass) equation, (the condition of incompressibility)
u ( x,0)  u 0 ( x)  R n the initial condition (the equation of state)
Compressible and inviscid flow gives us      0 and P  0
Fundamental Euler equation in gas dynamics augmented by an energy equation

Incompressible and inviscid flow gives us      0 and P  0 and div u  0
Euler equation (kinematic viscosity    /  0  0 as a limit of Navier-Stokes equation)
Incompressible case is as a limit of compressible flow
(0)
Euler equation in gas dynamics (inviscid, incompressible flow)
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 11
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation

Du
 u t  u  u  p  F three momentum equations
Dt

div u  0
the continuity (mass) equation, (the condition of incompressibility)
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 12
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
7. Derivation of the Navier-Stokes equation, Conservation Laws
Euler equation, Navier-Stokes equation, blow up of the solution, singularity
Using 3 equations of balance of momentum, 1 equation of continuity (conservation law
of mass), 1 equation of state and adding 1 equation of energy (conservation law of
energy)
Using Transport theorem, Divergence theorem
Conservation law of Mass (Continuity equation)
 

 div ( u ) dx  0 Integral form of conservation law of mass

t

 D 
Differential form (the continuity equation)
 t  div ( u )  0

(0)
By Newton’s second law, using Transport, Gauss’s divergence theorem
stress tensor matrix S ( x, t )
Conservation law of Momentum (balance of momentum)
D

 ( u )  u div (u)  p  b dx  0 Integral form
Dt

 D 
D
( u )  p  b
Differential form (momentum equation)
Dt

(0)
In macroscopic scale we cannot see internal (potential) energy
Conservation law of Energy (Kinetic energy)


 u

 ( u )    u  u   u  p  u  b dV  0 Integral form
 t


 D 
 u

( u )    u  u   u  p  u  b  0
Differential form (energy equation)
 t


Ts.Batmunkh, Department of Mathematics, University of Wyoming
(0)
Page 13
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
8. Navier-Stokes equation of viscous, incompressible fluid flow.
Euler equation, Navier-Stokes equation, blow up of the solution, singularity
Turbulence is most significant macroscopic problem in physics
The Euler and Navier-Stokes equations describe the motion of a fluid in R n , (n  2 or 3)
These equations are to be solved for an unknown velocity vector u( x, t )  R n and
pressure p  p( x, t )  R , defined for position x  R n and time t  0 .
Navier-Stokes equations of incompressible fluid mechanics (dynamics) are a formulation
of Newton’s laws of motion for a continuous distribution of matter in the fluid state
Nonlinear equations of Evolution:
u
(0)
 Au where A is the nonlinear operator
t
Navier-Stokes equation (inviscous, incompressible fluid):
Consider a viscid, incompressible (water) ideal (homogeneous f  0 ) fluid.
Du
  (u t  u  u )  p  u  f
Dt
div u    u  0
(the condition of incompressibility)

(0)
u ( x,0)  u 0 ( x)  R n (the initial condition, velocity field)
Goal is to find smooth functions ( p, u)  R  R n satisfying equations
Existence and smoothness of ( p, u ) solutions of Navier-Stokes equations on R 3 are open
problems?
In gas dynamics (compressible flow) velocities approach the speed of light.
In the summation form
n
Du i u i
u
p

 u j i  
  u i  f i ( x, t ) , 1  i, j  n
Dt
t
x j
xi
j 1
n
div u    u  
i 1
ui
0
xi
(the condition of incompressibility)
(0)
u( x,0)  u t 0 u0 ( x)  u0  R n (the initial condition, velocity field)
In 2-D
2-D Navier Stokes u t  f (u ) x  g (u ) y  F
 u

 v

 F1 
 
 2

 uv

F 
 u 
     u  p   
2



F 
2




 F3 
t  v  x uv
y v  p




 
 
E 
 F4 
u ( E  p )
v( E  p )
Ts.Batmunkh, Department of Mathematics, University of Wyoming
(0)
Page 14
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
In 3-D
3-D Navier Stokes equation (5 equations) ut  f (u) x  g (u ) y  h(u ) z  Ff
 u

 v

 w

 F1 
 
 2

 uv



F 
 u 
uw
 u  p 




 2



 
  2
 



 v  
uv

v  p 
vw
 F   F3 
 y 
 z  2
t   x 

 
 uw

 vw

 w  p 
 F4 
 w
u ( E  p )
v ( E  p ) 
 w( E  p)
 F5 
 E 
 






(0)
Breakdown of u  u ( x, t ) solution of Navier-Stokes equations on R 3 is open problem?
Existence, smoothness and breakdown of solutions of Euler equation on R 3 are also open
problems?
Beale-Kato-Majda theorem for vorticity blow up in finite time on R 3
J.Leray proved that a weak solution ( p, u ) of NSE always exists on R 3 .
Uniqueness of weak solutions of Navier-Stokes equations is not known?
Uniqueness of weak solutions of Euler equation is strikingly false.
Many fascinating problems and conjectures about the behavior of Euler and NSE.
2-Dim problems do not give hints for 3-D problems.
Fluids are important and hard to understand.
Standard methods from PDE appear inadequate to settle the problem.
Instead, we probably need some deep, new ideas.(4. C.Fefferman)
1755, Euler equation of incompressible fluid flows
Euler equation is the nonviscous limit of Navier-Stokes equation
1821, modifying Euler’s equations for viscous flow by Navier
1822-1827, Navier-Stokes equation for fluid flows by Navier, 1831 by Poisson, 1845 by
Stokes for incompressible flow
1906, Priori estimates by S.Bernstein
1934, Leray-Schauder theory
After 1920, Weak solutions (not classical) by B.Levi and L.Tonelli
Existence and regularity of weak solutions are important
1925, Local existence and uniqueness (for a small time interval) of a classical solution for
Euler equation introduced by L.Lichtenstein and more contributions (1966) by V.Arnold
by (1975) R.Temam
In 2-Dim, in 1933 Global existence (for all time) of a classical solution by W.Wolibner,
completed in 1967 by T.Kato
In 3-Dim, the existence of global classical solutions is open?
1933, Existence of Weak global solutions of NSE obtained by J.Leray, and in 1950 by
E.Hopf.
Lax pair
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 15
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
Solution
Local classical
Global (smooth) classical
Existence, smoothness
Blow up
Uniqueness
Regularity
Weak Local
Weak Global
Existence, smoothness
Uniqueness
Regularity
Blow up
singularity
Euler, 2-Dim
1925, 1966, 1975
1933, 1967
Euler, 3 Dim
1925, 1966, 1975
Open
NSE, 2D
NSE, 3D
1933, 1950
Open
Yes weak
existence
Open
Open
Open
Yes weak existence
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 16
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
9. Vorticity equations of Navier-Stokes equations in 1, 2, 3-Dim
For every smooth velocity field, we use Taylor series expansion at a fixed point ( x0 , t 0 )
Gradient matrix u (3x3) has a symmetric part D and an antisymmetric part 
Detormation or rate-of-strain matrix D 
1
(u  u T ) (symmetric part)
2
1
(u  u T ) and u  D   (antisymmetric part)
2
Incompressible flow ( div u  0 gives the trace trD   d ii  0 )
Rotation matrix  
i
 w
v u
w v
u 
Vorticity (curl)     u  curl u  
 , 
,  
 y z z x x y 
1
1
  ( x  x0 )  D( x  x0 ) , h    h
2
2
Infinitesimal translation X ( , t )    u( x0 , t 0 )(t  t 0 ) ,   (0,0,  )
Taylor series u ( x, t 0 )  u ( x0 , t 0 ) 
0   0


By a rotation of axes   ( x  x0 )   0 0 ( x  x0 ) , X 3 ( , t )  x03


0 0 0
1
Particle trajectories X  ( X , X 3 ) as X ( , t )  x0  R(  t )( x   x0 )
2
cos   sin  
Rotation matrix in the x1  x2 plane R( )  

sin  cos  
(0)
RDR T  diag ( 1 ,  2 ,  3 ) and D  diag ( 1 ,  2 , 1   2 ) and traceD  0
0
0
exp(  1 (t  t 0 )


(  x )
X ( , t )  x0  
0
exp(  2 (t  t 0 )
0
0


0
0
exp( ( 1   2 )(t  t 0 )
Every imcompressible velocity field is the sum of infinitesimal translation,
rotation, and deformation velocities
Solutions of Euler and Navier-Stokes equations in 3-Dim
Let D(t ) real, symmetric matrix, trD(t )  0 . The vorticity satisfies the
d
 D(t ) and  ( x,0)   0  R 3 and the antisymmetric
dt
1
matrix  by means of the formula h    h then exact solutions are
2
1
1
u ( x, t )   (t )  x  D(t ) x and p( x, t )   ( Dt (t )  D 2 (t )   2 (t )) x  x
2
2
ODE equation
Ts.Batmunkh, Department of Mathematics, University of Wyoming
(0)
Page 17
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
Proof of Solutions of Euler and Navier-Stokes equations in 3-Dim
n
Du i u i
u
p

 u j i  
  u i  f i ( x, t ) , 1  i, j  n
Dt
t
x j
xi
j 1
Computing the  xk derivative of NSE we get componentwise
(u xi k ) t  u j (u xi k ) x j  u xjk u xi j   p xi xk   (u xi k ) x j x j , note V  (u xi k ) and P  ( p xi xk )
DV
 V 2   P  V
Dt
1
1
Symmetric part D  (V  V T ) Antisymmetric part   (V  V T )
2
2
2
2
2
Quadratic V  (  D )  (D  D) where D and V satisfy this. We get
DD
 D 2   2   P  D and
Symmetric part equation
Dt
D
 D  D   ..
Antisymmetric part equation
Dt
Matrix equation
Vorticity equation for the dynamics of  by h 
(0)
1
  h for antisymmetric part
2
D
 D      u   is derived directly from NSE since
Dt
u  D   , w  w  0 and w  curl u    w
(0)
1
 (t )  x  D(t ) x , and curl u   (t )
2
does not depend on the spatial variable x so that   u    0 .
d
 D .
Vorticity equation reduces to Scalar ODE
dt
Thus, given D(t ) , we can solve this equation for  (t ) .
Symmetric part equation determines the pressure p . Because  determines 
dD
P
 D 2   2 , right hand side is a known symmetric matrix.
dt
Hessian of a scalar function is a symmetric P (t ) matrix.
1
Explicit function p( x.t )  P(t ) x  x
2
By the above construction, u and p satisfy Navier-Stokes equation in 3-D.
Postulating the solutions u ( x, t ) 
Ts.Batmunkh, Department of Mathematics, University of Wyoming
(0)
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From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
Leray formulation (Blow up)
Du
Du
x y
 p  p 
 C N  n
tr (u ( y, t )) 2 dy
N
R
Dt
Dt
x y
(0)
Vorticity equations of Navier-Stokes equations in 2-Dim
Vorticity Stream Formulation
Du
 (u t  u  u )  p  u
Dt
div u    u  0
(the condition of incompressibility)
u ( x,0)  u 0 ( x)  R n (the initial condition, velocity field)
Then D / Dt  D since u  D   and w  w  0 and w  curl u    w
D
 D      u  
Dt
(0)
2-D Vorticity equation


For 2-D flow x  ( x, y,0) , the velocity field u  (u , v,0) , and the vorticity
  (0,0, v x  u y ) is orthogonal to u , and the vorticity stretching term   u  0
Scalar vorticity in 2-D,   v x  u y . The vorticity equation reduces to the
D
 0   .
Scalar vorticity equation
(0)
Dt
Vorticity transport formula for inviscid flow  ( X ( , t ), t )   X ( , t )0 ( )
In 2-D, it implies Conservation of vorticity J ( , t )  det( X ( , t ))  1 along
particle trajectories X ( , t ) that is  ( X ( , t ), t )  1  0 ( )  0 ( ) .
Solution of 2-D Vorticity equation
Because the incompressibility (conservative vector field) divu  0 , there exists
a (unique up to an additive constant)
Stream function  ( x, t ) such that u  ( y , x )   . Computing the curl we get
Poisson equation for  ( x, t ) and    . By a Convolution with Newtonian
potential with,  the solution  ( x, t ) 
1
2
 ln x  y ( x, y) dy . By differentiating,
R2
1  y
x 
  2 ,  2  . It is
u ( x, t )   K 2 ( x  y) ( x, y )dy with the kernel K 2 ( x) 
2  x
x 
R2

(0)
The analog of the Biot-Savart law for the magnetic field induced by a current on a wire
The pressure p can be obtained from Poisson equation  p  tr(u ) 2 
2
u
i , j 1
i
xj
u xji .
The solution has blow up in finite time because of having the kernel in 2, 3-D?.
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 19
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
10. Vorticity model equations of Navier-Stokes equations in 1-Dim
(Constantin-Lax-Majda 1-D Model equation)
In 1-D there is no vorticity. So we define 1-D model equation.
Du
D
 (u t  u  u )  p and
 D ( )
Dt
Dt
3-D Euler equation
div u    u  0
(the condition of incompressibility)
u ( x,0)  u 0 ( x)  R (the initial condition, velocity field) . It reduces to
n
Constantin-Lax-Majda 1-D Model equation.
(0)

  t  H ( )
t
 ( x,0)  0 ( x) and u ( x, t ) 
x
 w( y, t )dy

The convective (material) derivative replaced by  / t .
In 3 Dim D
Convolution operator
In 2 Dim D ( w) w  0
Conservation of vorticity,
In 1 Dim H
There is only one Hilbert operator
In 1 Dim, there is blow up in finite time.
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 20
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
REFERENCES
[1] Constantin,P., Lax,P.D., and Majda,A., A simple One-dimensional Model for the Threedimensional Vorticity Equation. Com. Pure Appl. Math. 38 (1985), 715-724.
[2] Wegert,E., and Vasudeva Murthy,A.S., Blow-Up in a Modified Constantin-Lax-Majda Model
for the Vorticity Equation. Journal for Analysis and its Applications. 18 (1999), No.2, 183-191
[3] Chandrasekhar,S., Hydrodynamic and Hydromagnetic stability. New-York 1961
4. Charles L.Fefferman., Existence and smoothness of the Navier-Stokes equation. Princeton,
2000
5. H.Brezis and Felix Browder, Partial Differential Equations in the 20th century. Advances in
Mathematics 135, 76-144 (1998)
6. A.Majda and A.Bertozzi, Vorticity and Imcompressible Flow. Cambridge, 2002
7. A.Chorin and J.Marsden, A Mathematical Introduction to Fluid Mechanics. Springer, 1990
8. Herbert Goldstein, Classical Mechanics. Addison-Wesley (1980)
9. Sears.F.W, Zemansky.M.W University Physics. Young Freedman (2000
10. R.Fox, Introduction to Fluid Mechanics. John Wiley and Sons (1985)
11. H.Ockendon, Viscous Flow. Cambridge (1995)
12. Randall.LeVeque, Numerical Methods for Conservation Laws. Birkhaeuser (1992)
13. Heinz-Otto Kreis, Initial boundary value problems and Navier-Stokes equations. Academic
press (1989)
14. O.Ladyzhenskaya, The mathematical theory of viscous incompressible flow. (1969)
15. Roger Temam, Navier-Stokes equations. North-Holland (1979)
16. Charles Doering, Applied analysis of the Navier-Stokes equations. Cambridge (1995)
17. Peter Constantin, Navier-Stokes equations Chicago, 1989
18. H.Schey Div grad curl and all that. WW Norton (1997)
19. James Stewart, Calculus, concepts and contents. BrooksCole (2001)
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 21
From Physics to Mathematics
Navier-Stokes, Euler equations and Constantin-Lax-Majda 1-D model for the 3-D vorticity equation
From Physics to Mathematics
(From the Equation of motion, Equation of Fluid motion, Navier-Stokes, and Euler
equations in 1, 2, 3 dimensions
to Constantin-Lax-Majda 1-D model for the 3-D vorticity equation )
0. Parameters and Notations in Navier-Stokes and Euler equations
1. Gradient, Divergence, Laplace operator, Material derivative and etc
2. Field, Vector integral, Curl, Divergence, rotation, Theorems
3. Fluid Mechanics, Equations of fluid motion
4. Classical Mechanics. Equation of Motion
5. Euler equation and conservation laws of hyperbolic systems
6. Equation of fluid motion, Navier-Stokes equation and Euler equation
7. Derivation of the Navier-Stokes equation, Conservation Laws
8. Navier-Stokes equation of viscous, incompressible fluid flow
9. Vorticity equations of Navier-Stokes equations in 1, 2, 3-Dim
10. Constantin-Lax-Majda Vorticity Model equation in 1-Dim
Ts.Batmunkh, Department of Mathematics, University of Wyoming, 2005
Ts.Batmunkh, Department of Mathematics, University of Wyoming
Page 22