0. Parameters and Notations in Navier-Stokes and Euler Equations

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 n The Point x  D (position, spatial coordinate, element), x  x(t)  (x1 ,..., xn )  D  R dx(t) The Velocity vector (field, function), u  u(x,t)  (u ,...,u )  R n , u(x,t)  1 n dt The Particle trajectory mapping (this particle of fluid traverses a well defined trajectory) n n X (,t) :  R  X (,t)  R and X  X ( ,t)  (X 1 ,..., X n ) (location at time t of a fluid particle initially placed at the point   ( ,..., ) at time t  0 1 n   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 u) J (,t) for a smooth velocity field (function) u  R n  x X

d n The Transport Formula f (x,t)dx  ( f t  divx ( fu))dx and   R an dt X (,t) X (,t) open, bounded domain with a smooth boundary for any smooth f (x,t) , smooth u  R n The incompressible flow means divu  0 or J (,t)  1 The Fluid density (field, function)- (x,t) The Pressure (scalar field, function)- p  p(x,t)  R unknown n The External force f  f (x,t)  ( f1 ,..., f n )  R 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 D  n  d The Convective (material) derivative operator   ui   u  Dt t i1 xi dt n Du u u n The Material derivative   ui  ut  u u  ut (u )u  R Dt t i1 xi

u u u n The Gadient vector  ( 1 ,..., n )  R xi xi xi   The Gradient operator   ( ,..., ) , u  R n  R n , matrix x1 xn n  2   n The Laplace operator  2 , u  R i1 xi n u The Divergence of a vector field- divu   u   i Curl curl u    u i1 xi

Ts.Batmunkh, Department of Mathematics, University of Wyoming 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 Gradient vector  u  ( , , )  R 3 t dt dt dt (0)  du dv dw     dx dx dx   du dv dw u  3 3    i Gradient Matrix  x u  (u,v,w)      R  R  dy dy dy  x   j 33  du dv dw     dz dz dz    3 u  du dv dw Divergence divu   u   i  trace(u)     R i1 xi dx dy dz Laplace operator, material derivative (acceleration), 3  2    2  Laplacian (differential operator)  2 i1 xi    3  2u  3  2u 3  2v 3  2 w  u   2u    , ,   R 3 Laplace operator  2  2  2  2  i1 xi  i1 xi i1 xi i1 xi    u   2u  (u,v,w)  (trace((u)),trace((v)),trace((w))) D  3     Material derivative   ui   t  u    t  u  v  w (0) Dt t i1 xi x y z Acceleration vector (material derivative), derivative of time of the position vector    du d 2  d   u a(t)   x(t)  ux(t), y(t), z(t),t R 3 noting u  dt 2 dt x x  dt     u t u x u y u z     a(t)      u  uu  vu  wu (chain rule) t t x t y t z t t x y z      Du     a(t)  u  u u  where u   u  v  w t Dt x y z

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)) x˙ 2  y˙ 2 dt CR2 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   Pdx  Qdy  Rdz CR3 C a CR3 Fundamental Theorem of Line integral f  dr  f (r(b))  f (r(a)) CR3

Space, 3-D, divergence, curl, rotation, irrotational, conservative vector field 3 F P Q R Divergence div F    F  i  trace(F)     R  x x y z i1 i    i j k

 R Q P R Q P     3 Curl curl F    F    ,  ,     R  y z z x x y  x y z (0) P Q R Irrotational (conservative) Field F at P if curlF  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

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) F  dr  F  dr (0) Independent of path  F  dr  0 or   2 CDR C1 C2 P Q Conservative vector field in a plane  or  f such that f  F y dx

 Q P  Green’s Theorem  Pdx  Qdy     dA in the plane R 2 2 2  x y  CDR DR  Green’s Theorem (I vector form)  F  dr  (curl F)  kdA C D  Green’s Theorem (II vector form)  F  n ds  div F(x, y)dA 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)  F dS  F ndS 3 Surface integral (Flux) of F over the surface S is   in R 3 SR S Stokes theorem  F  dr  (curl F)  dS  (curl F)  k dA (0) CSR3 SR3 SR3  Divergence theorem  F  n dS   F  dS  div F(x, y, z)dV S E S ER3

3. Fluid Mechanics, Equations of fluid motion

Continuum Fluid Mechanics

Inviscid flow (Euler Viscous flow equation) (Navier-Stokes equation)

Laminar flow (straight Turbulent flow filament of smoke) (smoke)

Compressible flow (gas, Incompressible flow Internal flow (bounded External flow density varies , entropy) (liquid, density constant) by solid surfaces) (free, open channel)

Newtonian flow Entropy condition External force, homogenous equation

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 Navier-Stokes Euler

Entropy flux  (u)  (u)  const  (u)  const  (u)  const  (u)  const Absolute (dynamic)     0     0  viscosity viscous flow inviscid flow Kinematic viscosity   0   0   0   0    / 0 viscous flow inviscid flow viscous flow inviscid flow ~ Stress tensor S(x,t) S(x, t)   pI  S (T ) S(x,t)   pI viscous flow inviscid flow Reynolds number Turbulent flow (non-turbulent) (non-turbulent) (non-turbulent) Re  UD /  Laminar flow Laminar flow Laminar flow

Fluid mass density (x,t)  const (x,t)  const (x,t)  const (x,t)  const (x,t) (gas) compressible Incompressible Incompressible Incompressible Divergence divu divu  0 divu  0 divu  0 divu  0 compressible Incompressible Incompressible Incompressible Force F F  0 external F  0 internal both Both non-homogenous homogenous Fluid Non-Newtonian Newtonian Newtonian Newtonian Blow Up yes yes

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 Vector Velocity: v  (rate change of distance with time) (1) dt (2) Linear Momentum: p  mv (product of the particle mass and velocity) dp Conservation law of linear momentum: F   p˙  0  p -constant, is conserved dt  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 Force: F   (mv)  m  ma (5) dt dt dt Vector acceleration: a  v˙  d 2 r / dt 2

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) (7) dL Moment of Force or Torque: N  r  F  (cross product of radius and force) (11) dt dL Conservation law of angular momentum: N   0  L -constant, is conserved dt  Moment of force (Torque)  twisting or turning the force

Vector Field 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)  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. Complex f :   C  C , f (z)  f (x  iy)  u(x, y)  iv(x, y) analysis f is differentiable at every point z   , then analytic function in a domain Analytic u v u v Cauchy-Riemann’s equations:  and   function x y y x

Harmonic (conjugate) function: u  uxx  uyy  v  0 f (z)dz(t)  F(z )  F(z )  0 Cauchy Morera theorem  2 1 for closed curve C C

The Work done by the external force F  F (e) upon the particle in going from point 1 to point 2 is: 2 (12) Work: W12  F  ds 1 (14) Change in Kinetic energy : W12  K 2  K1 (with constant mass)

K V Conservation law of Energy of a Particle: F  u  iv   i , E  u  iv  K  iV s s Conservative system (force):  F  ds  0 (closed path) dE d(K  iV ) Conservation law of Energy: F    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 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 The quantity of matter contained in an object, including inertial m  dV properties with density   Density A measure of a substance's mass per init of volume. Some objects are   m /V heavier than other objects, even though they are the same size. Conservation the amount of mass remains constant--mass is neither created nor m constant law of mass destroyed Velocity Rate of change of position with time (first derivative, direction) v  dx / dt Acceleration Rate of change of velocity with time (second derivative, direction) a  dv / dt   Force a push or pull experienced by a mass m when it is accelerated F  ma  Newton’s first Law of inertia: a body at rest or a body in motion continues to move F  0 law at a constant velocity unless acted upon by an external force.    Newton’s Net external force F acting on a body gives it an acceleration a F  ma second law which is in the direction of the force and has magnitude inversely m proportional to the mass of the body.   Newton’s Weak law of action and reaction: For every external action force, F  F third law there is a corresponding reaction force which is equal in magnitude and A B opposite in direction.   Momentum Momentum is a fundamental quantity in mechanics that is conserved in p  mv , 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.  Impulse Instantaneous change in momentum J  Fdt  dp Momentum, From Newton’s second law, a force F produces a change in F  dp / dt Force momentum. Conservation Momentum of a system is constant if there are no external forces d( p1  p2 ) law of F  0 acting on the system from Newton’s first law (of inertia).  0 Momentum dt

Energy Energy is an abstract scalar quantity of extreme usefulness in physics E  mc 2 and measured in units of mass times velocity squared. Kinetic Energy of Motion: is directly proportional to the square of its speed. K  1 mv 2 energy Kinetic energy is a scalar quantity (no direction). 2 Potential Stored Energy of position: An object can store energy as the result of U  mgh energy its position. Conservation Total mechanical energy, sum of kinetic energy and potential energy a E  K U law of Energy constant. Energy is the quantity that can be converted from one form Constant to another but cannot be created or destroyed.   Work=Energy Work is (a scalar quantity) the product of applied force causing a W  F  s displacement over a distance, has units of energy. Work energy Total work equals the change in kinetic energy with constant force W  K 2  K1 theorem Work, energy, dE x2 Work is an integral with varying force and  F W  Fdx force W  K  E x dx 1   Power Power is (a scalar quantity) Rate of change of work (or energy) with P  F  v time like a velocity. And the product of applied force and a velocity. Connection W  K 2  K1 , W  U , F  dE / dx , F  U

Force horizontal -x Vertical-y Internal Contact force Frictional, , Normal, weight, tensional , applied Air, fluid resistance, spring External Long distance Gravitational (Interaction) Orbital motions between Between nucleus of an Between Neutrinos Fundamental (Big) atoms atom (Subatomic) Gravitational Electronic, Strong interaction Weak interaction magnetic (sun energy) (new star, supernova) Theory of Grand Unified Electroweak Electromagnetic-weak everything TOE Theory GUT interaction interaction

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 1 Pressure p  p(x,t) , Energy (kinetic) E  u 2 , Flux function f , g : R m  R m 2   In 1-Dim 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 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       (0) u  u  p  u    (u)u  p  0 , or u  f (u)  0 t   x   t     t x   E  u(E  p) E    (u(E  p))  From here Conservation Laws

1. Conservation Law of Mass t  (u) x  0 (continuity equation) 2 (0) 2. Conservation Law of Momentum (u)t  (u  p) x  0

3. Conservation Law of Energy Et  (u(E  p)) x  0 In 2-D

2-D Euler equation in gas dynamics ut  f (u) x  g(u) y  0   u  v     2     u  u  p  uv (0)          0 t v x uv  y v 2  p        E  u(E  p) v(E  p) In 3-D

3-D Euler equation in gas dynamics ut  f (u) x  g(u) y  h(u) z  0   u  v  w     2      u u  p  uv  uw         (0) v   uv   v 2  p   vw   0 t x   y   z    2  w uw  vw  w  p          E  u(E  p) v(E  p) w(E  p)

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 (0)

t  div(u)  0 The continuity equation (mass conservation law) 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  ( px , p y , pz ) 3  (ui ) div(u)    (u)   (u) x  (v) y  (w) z i1 xi Material derivative operator (accelerator) is: D  n  d       ui   u    u  v  w Dt t i1 xi dt t x y z  (divu)  u      x    P  (  )grad(divu)   u  (  )(divu) y   v     w (divu) z    Condition of incompressibility

Incompressible flow where    0 is a known density constant (0)  t  0 and the continuity equation became divu  0 , thus volumes are preserved 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  ut  u u  p u  F three momentum equations Dt  (0) divu  0 the continuity (mass) equation, (the condition of incompressibility) n u(x,0)  u0 (x)  R 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 divu  0

Euler equation (kinematic viscosity    /  0  0 as a limit of Navier-Stokes equation) Incompressible case is as a limit of compressible flow Euler equation in gas dynamics (inviscid, incompressible flow) Du  ut  u u  p  F three momentum equations (0) Dt  divu  0 the continuity (mass) equation, (the condition of incompressibility)

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 (0) D  t 

t  div(u)  0 Differential form (the continuity equation)

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 D  Dt  (0) D (u)  p  b Differential form (momentum equation) Dt

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 D   t   (0)  u  (u)   u u  u p  u b  0 Differential form (energy equation)  t 

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  2or 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 timet  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  u u)  p u  f Dt t (0) divu   u  0 (the condition of incompressibility) n u(x,0)  u0 (x)  R (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 Dui ui ui p   u j    ui  fi (x,t) , 1  i, j  n Dt t j1 x j xi n u (0) divu   u   i  0 (the condition of incompressibility) i1 xi n u(x,0)  u t0 u0 (x)  u0  R (the initial condition, velocity field)

In 2-D

2-D Navier Stokes ut  f (u) x  g(u) y  F

  u  v  F1     2       u  u  p  uv F (0)       F  2    2   t v x uv  y v  p  F3          E  u(E  p) v(E  p) F4 

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      u u  p    uw F2           (0) v   uv   v 2  p   vw   F  F  t x   y   z 3    2    w uw  vw  w  p  F4  E  u(E  p) v(E  p)           w(E  p) F5  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

Solution Euler, 2-Dim Euler, 3 Dim NSE, 2D NSE, 3D Local classical 1925, 1966, 1975 1925, 1966, 1975 Global (smooth) classical 1933, 1967 Open Existence, smoothness Blow up Uniqueness Regularity Weak Local Weak Global Yes weak existence 1933, 1950 Yes weak existence Existence, smoothness Open Open Uniqueness Regularity Blow up Open Open singularity

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 ,t0 ) Gradient matrix u (3x3) has a symmetric part D and an antisymmetric part  1 Detormation or rate-of-strain matrix D  (u  uT ) (symmetric part) 2 1 Rotation matrix   (u  u T ) and u  D   (antisymmetric part) 2

Incompressible flow ( divu  0 gives the trace trD   dii  0 ) i  w v u w v u  Vorticity (curl)     u  curl u    ,  ,    y z z x x y  1 1 Taylor series u(x,t )  u(x ,t )    (x  x )  D(x  x ) , h    h 0 0 0 2 0 0 2

Infinitesimal translation X (,t)    u(x0 ,t0 )(t  t0 ) ,   (0,0,) 0  0 (0)   3   (x  x )   0 0 (x  x ) X (,t)  x By a rotation of axes 0   0 , 3 0 0 0 0 1 Particle trajectories X  (X , X ) as X (,t)  x  R(  t)(x  x ) 3 0 2 0 cos  sin Rotation matrix in the x1  x2 plane R()    sin cos  T RDR  diag( 1 , 2 , 3 ) and D  diag( 1 , 2 , 1   2 ) and traceD  0

exp( 1 (t  t0 ) 0 0  X (,t)  x   0 exp( (t  t ) 0 (  x ) 0  2 0  0  0 0 exp(( 1   2 )(t  t0 ) 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 ODE equation  D(t) and (x,0)    R 3 and the antisymmetric dt 0 (0) 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)   (D (t)  D 2 (t)   2 (t))x  x 2 2 t

Proof of Solutions of Euler and Navier-Stokes equations in 3-Dim n Dui ui ui p   u j    ui  fi (x,t) , 1  i, j  n Dt t j1 x j xi  Computing the xk derivative of NSE we get componentwise i j i j i i (u )  u (u )  u u   p  (u ) , note V  (u i ) and P  ( p ) xk t xk x j xk x j xi xk xk x j x j xk xi xk DV Matrix equation V 2  P V Dt 1 1 (0) Symmetric part D  (V V T ) Antisymmetric part   (V V T ) 2 2 Quadratic V 2  ( 2  D 2 )  (D  D) where D and V satisfy this. We get DD Symmetric part equation  D 2   2  P D and Dt D Antisymmetric part equation  D  D  .. Dt

1 Vorticity equation for the dynamics of  by h    h for antisymmetric part 2 D (0)  D    u  is derived directly from NSE since Dt u  D   , w w  0 and w  curl u    w

1 Postulating the solutionsu(x,t)  (t)  x  D(t)x , and curl u  (t) 2 does not depend on the spatial variable x so that   u   0 . d Vorticity equation reduces to Scalar ODE  D . dt Thus, given D(t) , we can solve this equation for (t) . Symmetric part equation determines the pressure p . Because  determines  (0) 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.

Leray formulation (Blow up) (0)

Du Du x  y 2  p p   C N n tr(u(y,t)) dy Dt Dt R x  y N

Vorticity equations of Navier-Stokes equations in 2-Dim Vorticity Stream Formulation Du  (u  u u)  p u Dt t divu   u  0 (the condition of incompressibility) n u(x,0)  u0 (x)  R (the initial condition, velocity field) (0) Then D / Dt  D since u  D   and w w  0 and w  curl u    w D  D    u  Dt 2-D Vorticity equation   For 2-D flow x  (x, y,0) , the velocity field u  (u,v,0) , and the vorticity

  (0,0,vx  u y ) is orthogonal to u , and the vorticity stretching term  u  0

Scalar vorticity in 2-D,   vx  u y . The vorticity equation reduces to the D Scalar vorticity equation  0  . (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 1 potential with,  the solution  (x,t)   ln x  y (x, y) dy . By differentiating, 2 2 R (0)   1  y x  u(x,t)  K 2 (x  y)(x, y)dy K (x)   ,  with the kernel 2 2 2 . It is R2 2    x x  The analog of the Biot-Savart law for the magnetic field induced by a current on a wire 2 2 i j The pressure p can be obtained from Poisson equation  p  tr(u)  u u .  x j xi i, j1 The solution has blow up in finite time because of having the kernel in 2, 3-D?.

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 3-D Euler equation  (u  u u)  p and  D() Dt t Dt

divu   u  0 (the condition of incompressibility) n u(x,0)  u0 (x)  R (the initial condition, velocity field) . It reduces to Constantin-Lax-Majda 1-D Model equation. (0)     H () t t x (x,0)   (x) u(x,t) w(y,t)dy 0 and    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.

REFERENCES

[1] Constantin,P., Lax,P.D., and Majda,A., A simple One-dimensional Model for the Three- dimensional 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 20 th 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)

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 21