Navier-Stokes, Euler and other Related Oceanic and Atmospheric Dynamics Models

Edriss S. Titi

Weizmann Institute of Science Texas A&M University

GSSI - Aquila August 18-29, 2014 Rayleigh Bénard Convection / Boussinesq Approximation

• Conservation of Momentum      1    u u  (u )u  p  f k u  gTk t 0 • Incompressibility  u  0

• Heat Transport and Diffusion   T T  (u )T  0 t The Boundary Conditions We partition the boundary of • into:

¡u = f(x; y; z) 2 • : z = 0g;

¡b = f(x; y; z) 2 • : z = ¡hg;

¡s = f(x; y; z) 2 • : (x; y) 2 @M; ¡h · z · 0g:

@v @T on ¡ : = h ¿; w = 0; = ¡®(T ¡ T ); u @z @z ¤ @v @T on ¡ : = 0; w = 0; = 0; b @z @z @v @T on ¡ : v ¢ ~n = 0; £ ~n = 0; = 0; s @~n @~n Sobolev Spaces

  2 ik x s H ()  {  ˆ e L  k kZd such that  2 2 s ˆ  (1 k )  }  k kZd Rayleigh Bénard Convection / Boussinesq Approximation

• Conservation of Momentum      1    u u  (u )u  p  f k u  gTk t 0 • Incompressibility  u  0

• Heat Transport and Diffusion   T T  (u )T  0 t Temperature Estimates

• Maximum Principle

T  C C T L 0 1 0 L

• Gradient Estimates

1 d 2 2  T  T  (u )T Tdx 2 2  2dt L L • Estimate of the Nonlinear Term   (u )T Tdx  c u T T  L6 L3 L2 • Interpolation/Calculus Inequality

1 1   c  2  2  1 L3 L2 L2   H 

  1 3 2 (u )T Tdx  c u T 2 T  L6 L2 L2 • Young’s Inequality

1 p 1 q 1 1 ab  a  b ,  1 p q p q  c  4 2  2 (u )T Tdx  u T 2  T 2  3 6 L L  L 2 

1 d 2  2 c 4 2 T  T  u T 2 dt L2 2 L2  3 L6 L2

By Gronwall’s inequality

t c 4 u( ) 6 d 2 2  3  L T(t)  T(0) e 0 L2 L2 Question:

t u( ) 4 d  K ? Is  L6 0

To answer this question we have to deal with the Navier-Stokes equations. The Navier-Stokes Equations

     1  u u  (u )u  p  f t   0 u  0

Plus Boundary conditions, say periodic in the box  [0, L]3 • We will assume that 0 1

• Denote by   (x)dx     • Observe that if u0  f  0 then u  0.

• Poincaré Inequality

For every   H 1 with   0 we have

  cL  L2 L2 Navier-Stokes Equations Estimates

• Formal Energy estimate

1 d  2  2       u 2  u 2  (u )u u  pu  f , u 2 dt L L    • Observe that since   u  0 we have     (u )u udx  pudx  0

1 d  2  2    u 2  u 2  f , u 2 dt L L By the Cauchy-Schwarz and Poincaré inequalities

1 d  2  2  2  2   u 2  u 2  f u 2  cL f u 2 2 dt L L L2 L L2 L

By the Young’s inequality

2 1 d  2  2 cL  2   2 u 2  u 2  f  u 2 2 dt L L  L2 2 L 2 1 d  2   2 cL  2 u 2  u 2  f 2 dt L 2 L  L2 By Poincaré inequality

2 d  2  2 cL  2 u 2  u 2  f dt L L  L2

By Gronwall’s inequality

4  2  2 cL 2t  2 cL cL 2t u(t) 2  e u(0) 2  1 e  f L L 2 2 t 0,  L and   2    u  2 d  K(L, u0 2 , f ,,)  L L L2 0 Theorem (Leray 1932-34)

For every   0 there exists a weak solution (in the sense of distribution) of the Navier-stokes equations, which also satisfies

 2 2 1 u Cw ([0,], L ())  L ([0,], H ())

The question of uniqueness of weak solutions for the three-dimensional Navier-Stokes equations is still open. Strong Solutions of Navier-Stokes  u C([0,], H1())  L2 ([0,], H 2 ())

Enstrophy

 2  2  2 u    u L2 L2 L2 Formal Enstrophy Estimates

1 d  2  2       u  u  (u )u (u)  p(u)   f (u) 2 dt L2 L2

 Observe that  p (u)dx  0

 2 f   2   2 By Cauchy-Schwarz  f (u)  L  u  4 L2 Nonlinear Estimates One needs to find estimates for the nonlinearity:    (u )u (u)  ?

For example by Hőlder inequality       (u )u (u)  u u u L4 L4 L2 Calculus/Interpolation (Ladyzhenskaya) Inequalities

1 1 c  2  2 2  D L2 L2  4  1 3 L c  4  4 3 D L2 L2

 2 Denote by y  e  u 0 L2 The Three-dimensional Case

 2 Recall that y  e  u 0 L2 y  c( u 4  e2 )y One can show that L6 0

Which implies that

t c ( u( ) 4 e2 )d  L6 0 y(t)  y(0) e 0 The Question Is Again Whether:

 u( ) 4 d  K ?  L6 0

* Which is the $1 Million Question!! * Or the -12 Bulls/Cows!! The Question Is Again Whether:

 u( ) 4 d  K ?  L6 0 Foias-Ladyzhenskaya-Prodi-Serrin Conditions A strong solution of the three-dimensional Navier-Stokes equations exists on the interval [0,T] if and only if 2 3 u 2 Lp((0; T); Lq(•)) for + = 1 p q

For 1 · p · 1: The case of p=1 and q=3 has been established by L. Escauriaza and G. Seregin and V. Sverak. One can instead use the following Sobolev inequality   u  c u L6 L2

 Which leads to y  cy 3 &  y( )d  K 0

Theorem (Leray 1932-1934)   There exists *( u0 1 , f ,, L) such that H L2

y(t)   for every t [0,*). Navier-Stokes Equations

• The Three-dimensional Case * Global existence of the weak solutions * Short time existence of the strong solutions * Uniqueness of the strong solutions

• Open Problems: * Uniqueness of the weak solution. * Global existence of the strong solution. Vorticity Formulation           (u )  ( )u   f t   Vorticity Stretching Term ( )u    Two dimensional case ( )u  0         (u )   f t  2 (x,t) Satisfies a maximum principle. The Three-dimensional Case   ( )u / 0   ~ z   ( )u ~ z2  For large initial data  0 the vorticity balance takes the form

z ~ z2  Potential “Blow Up”!! Euler Equations º=0

1,® Theorem (Lechtenstein-1925): Let u02 C , then there exists T*(u0) > 0, where the solution of the 3D Euler 1,® equations exists and unique, and u 2 C([0, T*], C ).

s Theorem (Ebin-Marsden,Kato,Temam,…): Let u02 H , for s > 5/2, then there exists T*(u0), where the solution of the 3D Euler equations exists and unique, and s u 2 C([0, T*],H ).

Question: Does there exists a global weak solution for the 3D Euler equations? Answer: YES (Wiedemann –February 2011) ● Beale-Kato-Majda   (t) dt   If  L then we have existence and 0 uniqueness on the interval [0,]

 (t) ● That is, one has to “control” the L in some way!! • Constantin, Fefferman and Majda:

Provided sufficient condition involving the Lipschitz regularity of the direction of the vorticity:        Weak Solutions for 3D Euler

•Existence of family (non-uniqueness) of weak solutions to the Cauchy problem of the 3D Euler has been recently proved by Wiedemann (2011).

•DeLellis – Szekelyhidi showed the existence of a non-trivial family of weak solutions to the 3D Euler equations with compact support in space and time. The present proof is not a traditional PDE proof.

It shares many ideas of the proof of the [Nash (1974), Kuiper (1955)] theorem of invariant imbeddings of surfaces. It uses the formalism of differential inclusions, convex integration and accumulation of oscillations. Earlier results were established by Shnirelman and Sheffer. Ill-posedness of 3D Euler

Theorem (Bardos-Titi 2010):

2 (i) Let u1; u3 L (IR) then the shear °ow 2 loc

u(x; t) = (u1(x2); 0; u3(x1 tu1(x2)) ¡ is a weak solution of the Euler equations, in the sense of distribution, in • = IR3.

2 (ii) Let u1; u3 Lper(IR) then the shear °ow above is a weak solution of the Euler equations2, in the sense of distribtions. Furthermore, in this case the energy of this solution is constant.

(iii) There exist shear °ow solutions of the above form which, for t = 0, belong to C0;®, for some ® (0; 1), and for t = 0, they do not belong to C0;¯ for any ¯ > ®2. 2 6 This shear flow was used earlier by DiPerna and Majda to show that weak limit of oscillatory classical solutions of Euler equations does not converge to a solution of Euler equations. Ruling Out Certain Weak Solutions of Euler

The work of DeLellis – Szekelyhidi implies that there are non-unique weak solutions of Euler even with very smooth initial data

(for example u0 = 0).

Question: Is there a natural criterion for ruling out certain weak solutions of Euler? Ruling Out Criterion

In the absence of physical boundaries

A weak solution of Euler which cannot be achieved as a limit of Navier-Stokes, as the viscosity tends to zero, should be ruled out. Vanishing Viscosity Solutions

[Bardos, Titi, Wiedemann, 2012]

For initial data of the form (u1(x2); 0; u3(x1)) there are infinitely many weak solutions of Euler equations. The shear flow solution

(u1(x2); 0; u3(x1 tu1(x2))) ¡ is the vanishing viscosity limit solution. Two-Dimensions Euler      (u )  0 t   u  ( k )    k  

● Yudovich proved a weak version of the maximum principle, that is (t)   . L 0 L The idea of for the uniqueness   2, p  D   c  p  p W  Lp L  2 Special Results of Global Existence for the three-dimensional Navier-Stokes

Theorem (Fujita and Kato)

Let u 1 be small enough . Then the 3D 0 H 2 Navier -Stokes equations are globally well - posed for all time with such initial data. The same result holds if the initial data is small in L3 () (Kato, Giga & Miyakawa) z

x

  Revolution Domain around the z -axis [away from z -axis]

• Let us move to Cylindrical coordinates

Theorem (Ladyzhenskaya) Let  0 0 0 u0 (x, y, z)  (r (r, z), (r, z),z (r, z)) be axi-symmetric initial data. Then the three-dimensional Navier-Stokes equations have globally (in time) strong solution corresponding to such initial data. Moreover, such strong solution remains axi-symmetric. Theorem (Leiboviz, Mahalov and E.S.T.)

Consider the three-dimensional Navier-Stokes equations in an infinite Pipe. Let

 0 0 0 u0  (r (r,n z), (r,n z),z (r,n z))

(Helical symmetry). For such initial data we have global existence and uniqueness. Moreover, such a solution remains helically symmetric. Remarks

For axi-symmetric and helical flows the vorticity stretching term is nontrivial, and the velocity field is three-dimensional.

In the inviscid case, i.e.   0 , the question of global regularity of the three-dimensional helical or axi-symmetrical Euler equations is still open. Except the invariant sub-spaces where the vorticity stretching term is trivial. More about vorticity stretching In the two-and-half dimensional Navier-Stokes and Euler equations (u,v,w)(x,y).

(u,v) satisfy the 2D Navier-Stokes/Euler equations, and w is a passive scalar

The vorticity stretching term is non-trivial.

@w @w @u @v @w @u @w @v ( ; ; 0) ( ; ; 0) = @y ¡ @x ¢ @x @y @y @x ¡ @x @y Does 2D Flow Remain 2D?

[Bardos, Lopes-Filho,Nussenzveig-Lopes, Niu, Titi (2012)]

• Let u0 be a function of on (x,y), then the Leray-Hopf weak solution of the 3D Navier-Stokes remains a function of only (x,y). • Similar result holds for the Navier-Stokes equations with axi-symmetric initial data, or helical initial data.

• Let u0 be a function of (x,y), then the weak solution of the 3D Euler might become a function of (x,y,z). • Also, if the initial data is axi-symmetric or helical symmetric, the weak solutions of Euler might break the symmetry. • Theorem [Cannone, Meyer & Planchon] [Bondarevsky] 1996 Let M be given, as large as we want. Then there exists K(M) such that for every initial data of the form

  2  ik x  0 L u  uˆ  e [VERY OSCILLATORY] 0   k k K (M ) satisfying k u 0 k H 1 · M the three-dimensional Navier-Stokes equations have global existence of strong solutions.

Remark Such initial data satisfies u 1 1. 0 H 2 So, this is a particular case of Kato’s Theorem. Raugel and Sell (Thin Domains)

Global Existence (in time) of strong solutions for the 3D Navier-Stokes equations in thin domains, for large (depending on the thinness of the domain), initial data.

I. Kukavica and M. Ziane, made a significant improvement in the size of the data for which the above result holds, and investigated other physical configurations. Stability of Strong Solutions

• [Ponce, Sideris, Racke & Titi, CMP 1994] The set of globally regular solutions of the 3D Navier-Stokes is stable under small perturbations in the initial data and forcing. The Effect of Rotation  u      (u )u  p  u  0 t  u  0   There is 0 (,u0 ) such that if   0 the solution exists on [0,).    That is there exists 0 (u0,  ) such that the solution exists on [0,0 ).  Observe that 0   as    •Babin, Mahalov and Nicolaenko •Embid and Majda •Chemin, Desjardines, Gallagher, Grenier •Masmoudi •Ziane •Liu and Tadmor An Illustrative Example

Inviscid Burgers Equation

ut  uux  0 in R

u(x,0)  u0 (x)

●If u 0 ( x ) is decreasing function on some subinterval of R then the solution of the above equation develops a singularity (Shock) in finite time. The solution is given implicitly by the relation:

u(x,t)  u0 (x tu(x,t)) The Effect of the Rotation

u C z C

ut  uuz  iu  0

u0 (z)  u(z,0) v(z,t)  eitu(z,t) it vt  e vvz  0 eit 1 v(z,t)  v (z  v(z,t)) 0  i eit 1 v' (z  v(z,t))  0 v   i z eit 1 eit 1 1 v' (z  v(z,t))  i 0  i If  1, (i.e.   0 (u0 ))  v remains finite and the z solution remains regular for all t  0. Fast Rotation = Averaging

t v(z,t)  v(z,0)  eitv(z,t)v (z,t)dt  z 0

Riemann-Lebesgue Lemma

2 lim sin(kt)(t)dt  0 k  0 The above complex system is equivalent to 2D Rotating Burgers:

u  u1  iu 2 , z  x  iy

   0 1 ut  u u   u  0 1 0  Bénard Convection Porous Medium      u  u  p  RTk  0 t   u  0    T T  (u )T  0 t Subject to certain physical boundary conditions. • P. Fabrie [1986] Global Existence & Uniqueness • H.V. Ly E.S.T. [1999] (  0) Same result based on Galerkin numerical procedure. This gived leads to Spatial Analyticity, and exponential rate of convergence of the Galerkin procedure. • M. Oliver and E.S.T. (   0 ) Spatial analyticity of the attractor. Large Scale Oceanic Circulations

Rayleigh Bénard Convection / Boussinesq Approximation

• Conservation of Momentum      1    u u  (u )u  p  f k u  gTk t 0 • Incompressibility  u  0

• Heat Transport and Diffusion   T T  (u )T  0 t Bénard Convection/Boussinesq Approximation

  2   1  v   v  (v  )v  w v   p  f k v  0 H  H 2  H H H H H H H t  z  z 0   2   1  w   w  (v  )w  w w  p Tg  0  H 2  H H t  z  z 0 z   v  w  0 H H z   T T  (v  )T  w T   Q t H H z 0

 Here (vH , w)  u. Typical Scales in the Ocean

6 • horizontal distance L ~ 10 m

• horizontal velocity U ~ 10-1 m/s • depth H ~ 103 m • Coriolis parameter f ~ 10- 4 1/s • gravity g ~ 10 m/s 2 3 3 • density 0 ~ 10 kg/m Calculating the typical values

• Typical vertical W  UH/L ~ 10- 4 m/s velocity

7 • Typical pressure P  0 g H ~ 10 Pa

• Typical time scale   L/U ~ 107 s Scale Analysis of Vertical Motion – The Ideal Case

  1  w  (vH H )w  w w  p Tg  0 t z 0 z W UW W 2 P    Tg  0  L H H0 1011 1011 1011 10 10  0 Hydrostatic Balance

1  p Tg  0 0 z Scale Analysis Horizontal Motion – The Ideal Case

  1  vH  (vH H )vH  w vH  H p  f k vH  0 t z 0 U U 2 UW P    UF  0  L H L 0 108 108 108 102 105  0 Rosby number = Ratio of the intensity Of the nonlinear effects to the intensity of the Coriolis force.

U Ro   O(103) F L Geostrophic Balance

• When R0 1 ¿ 1  H p  f k vH  0 0 The Ideal Planetary Geostrophic equations

1  H p  f k  vH  0 0 1  z p  Tg  0 0

H  vH   zw  0

Tt  (vH H )T  wTz  0 Rayleigh Friction and Horizontal- Viscosity 1  H p  f k vH  F(vH ) 0 1  z p Tg  0 0

H vH   z w  0

Tt  (vH H )T  w zT  0Q v zzT  D(T) Friction, Viscosity and Diffusion Schemes

• Conventional eddy viscosity

F(v )  A  v  A  v H v H H h zz H and D(T )   H  HT

• Linear drag

F(vH )  vH

What should be the diffusion operator D? The Viscous PG Equations

1  H p  f k v H  Kv H vH  Kh zzvH 0 1  z p Tg  0 0

H vH   z w  0

Tt  (vH H )T  wTz  0Q  h HT v zzT The Viscous PG Equations

Weak Solutions

2 2 1 T Cw ([0,], L )  L ([0,], H )

Strong Solutions

T C([0,], H 1)  L2 ([0,], H 2 ) Results

• C. Cao and E.S.T. [CPAM (2003)]

* the uniqueness of weak solutions * the global existence of the strong solutions for any initial data in H 1 * existence of the global attractor. * upper bounds for the dimension of the global attractor. Existence of Global Attractor

• Absorbing Ball B in L 2 (energy estimate)

• Absorbing Ball B in H 1 (energy estimate and the uniform Gronwall inequality)

A  S(t) B  H1. s0 ts The Rayleigh Friction Case

1  H p  f k vH  vH 0 1  z p Tg  0 0

H vH   z w  0

Tt  (vH H )T  wTz  0Q  h HT v zzT Physical Boundary Conditions

• no normal flow   v H n  0 on side and w  0 when z  h, 0

• no heat-flux

  T  0 on the side and n

 zT  0 when z  h,0 The no - flow boundary condition   v n |  0 implies that H s T  1  |  0 where e  (n1  fn2, fn1 f2 ) e s  2  f 2 this is in addition to the no - heat flux boundary  condition  T |  0 n s

Therefore, there are two boundary conditions for the temperature which is governed by a second order parabolic PDE. So it is over-determined, and the problem is ill-posed. This is consistent with the numerical instability observed using this system. Rayleigh Friction and Temperature Horizontal Hyper-Diffusion Model

We therefore propose the following artificial Horizontal Hyper-diffusion model 1  H p  f k vH  vH 0 1  z p Tg  0 0

H vH   z w  0

Tt  (vH H )T  wTz  0Q  H q(T)  zzT With the Boundary Conditions

• no normal flow   on side  , & w  0 when z  h, 0 v H n  0 s

• no heat-flux

 q(T )n  0 on the side and

 zT  0 when z  h,0 Proposed Artificial Hyper-Diffusion

 1 - f/  H =   f/ 1 

T q(T)   H H ( H  (H H T)) +  H Tzz

- K h H T

Which is positive definite (dissipative/stabilizing) with the associate boundary conditions. Hyper Horizontal Diffusion Model

Weak Solutions

 2  2 2 u Cw ([0,], L ), u  L ([0,], L )

Strong Solutions

  u  L ([0,], H 1), u  L2 ([0,], H 2 ) Results

• C. Cao, E.S.T. & M. Ziane [Nonlinearity (2004)]. * The global existence and uniqueness of the weak solutions. * The global existence of the strong solutions. * Existence of the global attractor. * Provide upper bounds for the dimension of the global attractor. The Rayleigh Friction Case Salmon Model

1  H p  f k  vH  vH 0 1 w   z p  Tg  0 0

H  vH   zw  0

Tt  (vH H )T  wTz  0Q  hHT v zzT

Global well-posedness Cao-Titi [2010] Recall Scale Analysis of Vertical Motion –The Ideal Case

  1  w  (vH H )w  w w  p Tg  0 t z 0 z W UW W 2 P    Tg  0  L H H0 1011 1011 1011 10 10  0 Recall Scale Analysis for Horizontal Motion – The Ideal Case

  1  vH  (vH H )vH  w vH  H p  f k vH  0 t z 0 U U 2 UW P    UF  0  L H L 0 108 108 108 102 105  0 The Primitive Equations of Large Scale Oceanic and Atmospheric Dynamics

 tvH  (vH H )vH  w zvH  H p  f k vH

 Ah H vH  Av zzvH

 z p  gT  0

H vH   z w  0

Tt  (vH H )T  wTz  Q  Kh HT  KvTzz ● Introduced by Richardson (1922) For Weather Prediction

● J.L. Lions, R. Temam, S. Wang (1992) Gave Some Asymptotic Derivation of the Model. Primitive Equations

Weak Solutions

 2 2 1 u Cw ([0,], L )L ([0,], H ) \begin{reflist} Strong Solutions  u L ([0,], H 1)L2 ([0,], H 2 ) Previous Results

• J.L. Lions, Temam, S. Wang (1992), and Temam, Ziane (2003) The global existence of the weak solutions (No Uniqueness).

• Guillen-Gonzalez, Masmoudi, Rodriquez-Bellido (2001), and Temam, Ziane (2003) The short time existence of the strong solution

• Temam, Ziane (2003) Global Existence of Strong Solution for the 2-D case.

• C. Hu, Temam, Ziane (2003) Global Regularity for Restricted (Large) Initial Data in Thin Domains. Results • C. Cao and E.S.T. [Annals of Mathematics (2007)]. [arXiv March 1, 2005]. * the global existence of the weak solutions (Galerkin method/announced) * the global existence and uniqueness of the strong solutions. * existence of the global attractor (announced). * See also, [Ning Ju (2007)] for the existence of global attractor. The Primitive Equations of Large Scale Oceanic and Atmospheric Dynamics

 tvH  (vH H )vH  w zvH  H p  f k vH

 Ah H vH  Av zzvH

 z p  gT  0

H vH   z w  0

Tt  (vH H )T  wTz  Q  Kh HT  KvTzz A different formulation of the PE

z w(x, y, z)   H v H (x, y,) d h z p(x, y, z)  ps (x, y)  g T(x, y,) d h 0 1 vH (x, y)  vH (x, y,) d , H vH  0 h h ~ vH (x, y, z)  vH (x, y, z)  vH (x, y) The Barotropic Mode – The Averaged Part of the Horizontal Velocity  tvH  (vH H )vH  w zvH  f k vH  H ps z  AhH vH  H gT dz h The Baroclinic Mode –The Fluctuation Part of the Horizontal Velocity ~ ~ ~ ~ ~ tvH  (vH H )vH  (v H H )vH  (vH H )vH    z  ~ ~  H vH dz  zvH  f k vH   h  ~ ~ ~ ~ (vH H )vH  (H vH )v H  ~ ~ z z Ah H vH  Av zzvH  H gT d  H gT d h h The IDEA – Focus on Burgers Equation

ut u  (u)u  0

We have 2 1 2 1 2  u  1 2  u(x,t)   u(x,t)   i   u  u(x,t)  0 t    2 2 i, j  x j  2 2  A maximum principle for u(x,t) and L bound.

Global Regularity for 1D, 2D and 3D Burgers Equation. The Pressure Term!!

• Is the major difference between Burgers and the Navier-Stokes equations.

• What about in our system? The Averaged Equation is “like” the 2D Navier-Stokes.  tvH  (vH H )vH  w zvH  f k vH  H ps z  AhH vH  H gT dz h

Where ps (x, y)!! The Fluctuation Equation is “like” 3D Burgers Equations – Has No Pressure Term!!

~ ~ ~ ~ ~ tvH  (vH H )vH  (v H H )vH  (vH H )vH    z  ~ ~  H vH dz  zvH  f k vH   h  ~ ~ ~ ~ (vH H )vH  (H vH )v H ~ ~ z z  Ah H vH  Av zzvH  H gT d  H gT d h h A-priori Estimates

 v~  K H L6   v  K H H L2  v  K H L6  v  v  v~  K H L6 H H L6 Q.E.D. One of the Main Estimates Used

By Ladyzhenskaya inequality we usually have:

 h(x, y,z) f(x, y,z) g(x, y,z) dxdydz

1/4 3/4 1/4 3/4  C || h || 2 || h || 1 || f || 2 || f || 1 || g || 2 L () H () L () H () L ()

But here instead we have:

0  h u(x, y,z) dz f(x, y,z) g(x, y,z) dxdydz

1/2 1/2 1/2 1/2  C || u || 2 || u || 1 || f || 2 || f || 1 || g || 2 L () H () L () H () L () Using this observation about the pressure that it is effectively a function of two variables:

I. Kukavica and M. Ziane, (Nonlinearity 2007) obtained similar results for the case of Dirichlet boundary conditions, on top and bottom, and periodic boundary conditions in the horizontal direction. Back to the 3D Navier-Stokes Equations      1  u u  (u )u  p  f t 0  u  0 New Criterion for Global Regularity of the 3D Navier-Stokes Equations

Theorem (C. Cao and E.S.T. 2008) : The strong solution of the 3D Navier -Stokes equations exists on the interval [0,] for as long as 2 3 20 21  p  Lr ((0,), Ls ()),   , where r 1 and s  . z r s 7 16

This is different than the result of L. Berselli and G.P. Galdi (2002) and of Y. Zhou (2005) where the assumption is on p. Recently this result was improved by Zhou and Pokorny

• There are other global regularity criteria involving the pressure due to:

Chae Kukavica and Struwe. • Sereigen and Sverak:

Global regularity if the pressure is bounded from below. Most Recent Criterion for Global Regularity

Theorem (Cao and T.) [Arch. Rational Mech. Anal. 2011]

The three-dimensional Navier-Stokes equations has a unique storng solution on the interval [0; T ] if and only if for some j; k 1; 2; 3 2 f g @u j L¯([0; T ]; L®(IR3)); when k = j; and where @xk 2 6 3 2 ® + 3 ® > 3; 1 · ¯ < ; and + · ; 1 ® ¯ 2® or @u j L¯([0; T ]; L®(IR3)); where ® > 2; 1 · ¯ < ; @xj 2 1 3 2 3(® + 2) and + · : ® ¯ 4® The Primitive Equations for Stratified Fluid Flows

1  tvH  (vH H )vH  w zvH  H p  f k  vH 0

 AhH vH  Av zzvH

 z p  gT

H  vH   zw  0

Tt  (vH H )T  wTz  0 Global Regularity of the Primitive Equations with vertical Diffusion Turbulence Mixing

Cao-Titi. [Comm. Math. Physics 2011] 1  tvH  (vH H )vH  w zvH  H p  f k  vH 0

 AhH vH  Av zzvH

 z p  gT

H  vH   zw  0

Tt  (vH H )T  wTz  TTzz Global Regularity of the Primitive Equations with Vertical Diffusion Turbulence Mixing

Cao-Li-Titi [ARMA 2014]

• Improvement of the results with Vertical Diffusion Turbulence Mixing Global Regularity of the Primitive Equations with Horizontal Diffusion Mixing Cao-Li-Titi [JDE 2014]

• Global Regularity of PE with Horizontal Diffusion Turbulence Mixing 1  tvH  (vH H )vH  w zvH  H p  f k  vH 0

 AhH vH  Av zzvH

 z p  gT

H  vH   zw  0

Tt  (vH H )T  wTz  hHT Global Regularity of the Primitive Equations with Horizontal Viscosity & Diffusion Mixing Cao-Li-Titi [2014]

• Global Regularity of PE with Horizontal Viscosity and Horizontal Diffusion 1  tvH  (vH H )vH  w zvH  H p  f k  vH 0

 AhH vH

 z p  gT

H  vH   zw  0

Tt  (vH H )T  wTz  hHT Generalization of the Brezis-Gallouet inequality Gronwall’s Inequality for Systems Blowup of Inviscid Primitive Equations Cao-Ibrahim-Nakanishi-Titi (2012) [Comm. Math Phys. 2013], see also Wong (2012)] ut + uux + vuy + wuz + px = 0;

vt + uvx + vvy + wvz + py = 0;

pz + T = 0;

Tt + uTx + vTy + wTz = ·H ¢H T + ·3Tzz;

ux + vy + wz = 0: In the horizontal channel

• = (x; y; z) : 0 · z · H; (x; y) IR2 f 2 g Reduced Inviscid Primitive Equations

We take v 0 = 0 and T 0 = 0 . This implies that v 0 and T 0 . This yields the reduced ´ ´ 2D system ut + uux + wuz + px = 0;

pz = 0;

ux + wz = 0: in the strip M = (x; z) : 0 · z · H ; x IR : f 2 g Boundary Conditions and the Pressure

w = w = 0 jz=H jz=0 u, p and w are periodic in the x-direction, with period L. u and p are odd functions of x, and w is an even function. 2 H p (x; t) = ¡ u(x; z; t)u (x; z; t) dz x H x Z0 The Reduced System

2 H uxt + (uux)x + wxuz + wuxz (uux)x dz = 0 ¡ H Z0

ux + wz = 0

Since u is an odd function and and w is and even with respect to x we have:

u(0; z; t) = wx(0; z; t) = 0 The Reduced Equation at x=0

Denote by W ( z ; t ) = w ( 0 ; z ; t ) then

2 2 H 2 Wtz (Wz) + WWzz + (Wz) dz = 0 ¡ H 0 R Which blows up in finite time. [Childress- Ierley-Spiegel-Young, 1989] Self-similar Blowup Solution

'(z) W(z; t) = 1 t ; with'(0) = '(H) = 0 ¡

Then

2 2 H 2 '0 ('0) + ''00 + ('0(z)) dz = 0 ¡ H 0 R Which has a non-trivial solutions.