on the global regularity of the 3d navier-stokes equations ...on the global regularity of the 3d...

On the Global Regularity of the 3D Navier-Stokes Equations and

Relevant Geophysical ModelsEdriss S. Titi

University of California – Irvineand

Weizmann Institute of Science

University of Maryland, October 24-26, 2006

Rayleigh Be’nard Convection / Boussinesq Approximation

• Conservation of Momentum

• Incompressibility

• Heat Transport and Diffusion

kgTukfpuuuut

=×+∇+∇⋅+∆−∂∂

0=⋅∇ u

0)( =∇⋅+∆−∂∂ TuTTt

Temperature Estimates

• Maximum Principle

• Gradient Estimates

∞∞ +≤LL

TCCT 010

dxTTuTTdtd

LL∫ ∆⋅∇⋅=∆+∇ )(

• Estimate of the Nonlinear Term

• Interpolation/Calculus Inequality

• Young’s Inequality

236)(LLL

TTucdxTTu ∆∇≤∆⋅∇⋅∫

226)(LLL

TTucdxTTu ∆∇≤∆⋅∇⋅

223 LLLc ϕϕϕ ∇≤ 1H∈∀ϕ

,11 qp bq

ba +≤⋅ 111=+

TucTTdtd

TTucdxTTuL

∇≤∆+∇

∆+∇≤∆⋅∇⋅∫

∫∇≤∇

LLeTtT 0

)(22 )0()(

ττκ

By Gronwall’s inequality

Question:

46 Kdu

L≤∫ ττIs

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

The Navier-Stokes Equations

fpuuuut

=∇+∇⋅+∆−∂∂

0=⋅∇ uPlus Boundary conditions, say periodic in the box

3],0[ L=Ω

• We will assume that

• Denote by

• Observe that if then

• Poncare’ Inequality

For every with we have

∫Ω= dxx )(ϕϕ

00 == fu

22 LLcL ϕϕ ∇≤

1H∈ϕ 0=ϕ

10 =ρ

Sobolev Spaces

such that

ϕϕπ

Navier-Stokes Equations Estimates

• Formal Energy estimate

• Observe that since we have

( )ufupuuuuudtd

22 =⋅∇+⋅∇⋅+∇+ ∫∫υ

0=⋅∇ u

0)( =⋅∇=⋅∇⋅ ∫∫ dxupdxuuu

( )ufuudtd

22 =∇+⇒ υ

222222

LLLLLLufcLufuu

∇≤≤∇+υ

fcLuudtd

ufcLuudtd

≤∇+

∇+≤∇+

By the Cauchy-Schwarz and Poincare’ inequalities

By the Young’s inequality

By Poincare’ inequality22

LLLfcLu

2 1)0()(L

fecLuetu−− −− −+≤ υυ

( ) ),,,,(222 0

2 Τ≤∇∫Τ

υττυLLL

fuLKdu

By Gronwall’s inequality

[ ]Τ∈∀ ,0t

Theorem (Leray 1932-34)

))(],,0([))(],,0([ 122 ΩΤΩΤ∈ HLLCu w ∩

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

The uniqueness of weak solutions in the threedimensional Navier-Stokes equations case is still an open question.

Strong Solutions of Navier-Stokes

))(],,0([))(],,0([ 221 ΩΤΩΤ∈ HLHCu ∩

Enstrophy222

222 LLLuu ∇==×∇ ω

Formal Enstrophy Estimates

)()()()(21 22

22 ufupuuuuudtd

LL∆−⋅∫=∆−∇∫+∆−⋅∇⋅∫+∆+∇ υ

0)( =∆−⋅∇∫ dxup2

fuf ∆+≤∆−⋅∫

244)()(LLL

uuuuuu ∆∇≤∆−⋅∇⋅∫

Observe that

By Cauchy-Schwarz

By Hőlder inequality

LLL −∇

−∇≤

ϕϕϕ

Calculus/Interpolation (Ladyzhenskaya) Inequatlities

Denote by 20 2L

uey ∇+=

)()( &0

2 Τ≤≤ ∫Τ

Kdyycy ττ

)(~)( Τ≤⇒ Kty

The Two-dimensional Case

Global regularity of strong solutions to the two-dimensional Navier-Stokes equations.

Navier-Stokes Equations

• Two-dimensional Case

* Global Existence and Uniqueness of weak and strong solutions

* Finite dimension global attractor

The Three-dimensional Case2

0 2Luey ∇+=

yeucyL

46 +≤

∫≤

L deuc

eyty 0

46 ))((

)0()(ττ

Recall that

One can show that

Which implies that

The Question Is Again Whether:

46 Kdu

L≤∫

One can instead use the following Sobolev inequality

26 LLucu ∇≤

≤≤0

3 )(& Kdycyy ττWhich leads to

Theorem (Leray 1932-1934)),,,(

220* LfuLLυΤ

∞<)(ty ).,0[ *Τ∈tThere exists such that

for every

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

×∇=∇⋅−∇⋅+∆−∂∂ )()( ωωωνω

×∇=∇⋅+∆−∂∂ ωωνω )(

2),( txω

Two dimensional case

Satisfies a maximum principle.

Vorticity Stretching Term u)( ∇⋅ω

0)( ≡∇⋅ uω

u∇⋅ω

The Three-dimensional Case

Potential “Blow Up”!!

⇒2~ zz

For large initial data the vorticity balance takes the form

Euler Equations Three-Dimensional case

such that we have existence and uniqueness on

Beale-Kato-Majda

If then we have existence and

uniqueness on the interval

That is, one has to “control” the

in some way!!

)( 0u∗Τ∃).,0[ ∗Τ

∞<∞

)( dttL

∞Lt )(ω

],0[ Τ

• Constantin and Fefferman:

Provided sufficient condition involving the Lipschitzregularity of the direction of the vorticity:

ωωξ =

Two-Dimensions Euler

×∇=

=∇⋅+∂∂

Yudovich proved a weak version of the

maximum principle, that is .)( 0 ∞∞ ≤ LLt ωω

ppp LLWpcD ψψψ

α ∆⋅≤= ∑≤ 2

Special Results of Global Existence for the three-dimensional Navier-Stokes

Miyakawa) & Giga (Kato, )(in small is data initial theif holdsresult same The data.

initialsuch with timeallfor posed-wellglobally are equations Stokes-Navier

3D Then the . enough small be Let

(Kato) Theorem

axis]- from[away axis- thearoundDomain Revolution

zz−Ω•

)),(),,(),,((),,( 0000 zrzrzrzyxu zr ϕϕϕ θ=

• Let us move to Cylindrical coordinates

Theorem (Ladyzhenskaya) Let

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 solutionremains axi-symmetric.

Theorem (Leiboviz, Mahalov and E.S.T.)

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

)),(),,(),,(( 0000 znrznrznru zr αθϕαθϕαθϕ θ +++=

(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 vorticitystretching term is nontrivial, and the velocity field is three-dimensional.

• In the inviscid case, i.e. , 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.

• 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

∑≥

0 ˆMKk

k euuπ

the three-dimensional Navier-Stokes equations have global existence of strong solutions.

[VERY OSCILLATORY]

Remark Such initial data satisfies

So, this is a particular case of Kato’s Theorem.

.1210 <<

The Effect of Rotation

=⋅∇

=×Ω+∇+∇⋅+∂∂

upuutu

Tadmor. andLiu ... Masmoudi, Granier, Ghalagher, Chemin,

Majda.-Embid .Nicolaenko-Mahalov-Babin

thatObserve ).,0[on exists

solution thesuch that )( exists thereisThat

).,0[on exists solution the ifsuch that ),( is There

••••

∞→Ω∞→Τ

ΩΤ•

Ω>ΩΤΩ•

(x))0,(in 0

EquationBurgersInviscid

0uxuuuu xt

If 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:

)(0 xu

)),((),( 0 txtuxutxu −=

An Illustrative Example

The Effect of the Rotation

)0,()(0

0 zuzuuiuuu

==Ω++

∈∈ CC

),(),( tzuetzv tiΩ=

)),(1(11

)),(1(

)),(1(),(

ezvtzv

Ω−−

−Ω−−

Ω−−

∂∂

Ω−−

Ω−Ω−

.0 allfor regular remainssolution

theand finite remains

))( i.e.( ,1 If 00

≥∂∂

Ω>Ω>>Ω

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −Ω+∇⋅+

iyxziuuu

The above complex system is equivalent to 2D Rotating Burgers:

existence. global have we)(For 0)()cos(

0)(generally More

00 vvFdivtv

uFdivu

Ω>Ω=Ω+•

=+• (Short time existence)

)()(0)(

),(),(by denote and sinLet

00 xuxv)w(x,wFdivw

xtvxwt

)()sin()( satisfies whenever isThat exists.

solution the)()( interval in the For

∗∗

Τ≤ΩΩ

≤Τ−

Τ≤≤Τ− ττ

Bénard Convection Porous Medium

.conditionsboundary physicalcertain Subject to

=∇⋅+∆−∂∂

=⋅∇

=−∇++∂∂

kRTpuut

• P. Fabrie [1986] Global Existence & Uniqueness

• H.V. Ly E.S.T. [1999]

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.

Spatial analyticity of the attractor.

)0( =γ

)0( >γ

Large Scale Oceanic Circulations

Be’nard Convection/BoussinesqApproximation

.),( Here

wTvTTt

vkfpvz

HHHHHHHHH

=∂∂

+∇⋅+∆−∂∂

=∂∂

+⋅∇

=+∂∂

+∂∂

+∇⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∆−∂∂

=×+∇+∂∂

+∇⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∆−∂∂

Typical Scales in the Ocean• horizontal distance

• horizontal velocity

• depth

• Coriolis parameter

• gravity

• density

m/s 10 ~ U -1

1/s 10 ~ f -4

m 10 ~ L 6

m 10 ~ H 3

2m/s 10 ~ g33

0 kg/m 10 ~ ρ

Calculating the typical values

• Typical vertical velocity

• Typical pressure

• Typical time scale

m/s 10 ~ UH/LW -4=

Pa 10 ~ H g P 70ρ=

s 10 ~ L/U 7=Τ

Scale Analysis of Vertical Motion –The Ideal Case

01010101010

1111110

=++++Τ

=+∂∂

+∂∂

+∇⋅+∂∂

−−−

wwvwt HH

Hydrostatic Balance

=+∂∂ Tgpzρ

Scale Analysis – The Ideal Case

01010101010

528880

=++++Τ

=×+∇+∂∂

+∇⋅+∂∂

−−−−−

vkfpvz

wvvvt HHHHHHH

Rossby Number

LFUR =

Geostrophic Balance

• When 1<<R

=×+∇ HH vkfpρ

The Ideal Planetary Geostrophicequations

TQwTTvTwv

zzzHHt

∂+=+∇⋅+=∂+⋅∇

=×+∇

Rayleigh Friction and Horizontal-Diffusion

TDTQTwTvTwv

vFvkfp

zzvzHHt

+∂+=∂+∇⋅+=∂+⋅∇

=×+∇

Friction, Viscosity and Diffusion Schemes

• Conventional eddy viscosity

• Linear drag

What should be the diffusion operator D?

HzzhHHvH vAvAvF ∂+∆=)(

HH vvF ε−=)(

TTD HH∆=κ)(

The Viscous PG Equations

TTQwTTvTwv

vKvKvkfp

zzvHhzHHt

HzzhHHvH H

∂+∆+=+∇⋅+=∂+⋅∇

∂+∆=×+∇

κκρ

The Viscous PG Equations

Weak Solutions

)],,0([)],,0([ 122 HLLCT w ΤΤ∈ ∩

Strong Solutions

)],,0([)],,0([ 221 HLHCT ΤΤ∈ ∩

Results

• Samelson, Temam and Wang (1998)* the existence of the weak solutions,

but no uniqueness,* the short time existence of the strong solutions.

• Samelson, Temam and Wang (2000)* global existence of the strong solution if initial data is bounded, i.e. in .∞L

Results

• Cao and E.S.T. (2003)

* the uniqueness of weak solutions* the global existence of the strong solutions for any initial data in

* existence of the global attractor.* upper bounds for the dimension of the global attractor.

Existence of Global Attractor

• Absorbing Ball B in (energy estimate)

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

.H B S(t) 1

⊂=> >∩∪

The Rayleigh Friction Case

TTQwTTvTwv

zzvHhzHHt

∂+∆+=+∇⋅+=∂+⋅∇

−=×+∇

κκρ

Natural Boundary Conditions

• no normal flow

on side and

• no heat-flux

on the side and

0=⋅nvH 0,when0 hzw −==

0=∂∂ Tn

0, when 0 hzTz −==∂

0| condition

boundaryflux heat -no theoaddition tin is this

)(1 where0|

thatimplies 0|conditionboundary flow-no The

21,2122

=∂∂

−−+

==∂∂

ffnfnnf

εεε

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

TTqQwTTvTwv

zzHzHHt

∂+⋅∇+=+∇⋅+=∂+⋅∇

−=×+∇

We therefore propose the following artificial Horizontal Hyper-diffusion model

With the Boundary Conditions

• no normal flow

on side

• no heat-flux

on the side and

0=⋅nvH0,when0 hzw −==

0)( =⋅ nTq

0, when 0 hzTz −==∂

& ,sΓ

Proposed Artificial Hyper-Diffusion

T K -T + T)) (H ( H q(T)

1f/f/-1

∇∇∇⋅∇∇=

⎟⎟⎠

⎞⎜⎜⎝

Which is positive definite (dissipative/stabilizing) with the associate boundary conditions.

Hyper Horizontal Diffusion Model

Weak Solutions

)],,0([),],,0([ 222 LLuLCu w Τ∈∆Τ∈

Strong Solutions

)],,0([),],,0([ 221 HLuHLu Τ∈∆Τ∈∇ ∞

Results

• Cao, E.S.T., Ziane (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 theglobal attractor.

Recall Scale Analysis of Vertical Motion –The Ideal Case

01010101010

1111110

=++++Τ

=+∂∂

+∂∂

+∇⋅+∂∂

−−−

wwvwt HH

Recall Scale Analysis for Horizontal Motion – The Ideal Case

01010101010

528880

=++++Τ

=×+∇+∂∂

+∇⋅+∂∂

−−−−−

vkfpvz

wvvvt HHHHHHH

The Primitive Equations of Large Scale Oceanic and Atmospheric

Dynamics

zzvHhzHHt

HzzvHHh

HHHzHHHHt

TKTKQwTTvTwv

gTpvAvA

vkfpvwvvv

+∆+=+∇⋅+=∂+⋅∇

=+∂∂+∆=

×+∇+∂+∇⋅+∂

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

∩ )],,0([)],,0([ 122 HLLCu w ΤΤ∈

Strong Solutions

∩ )],,0([)],,0([ 221 HLHLu ΤΤ∈ ∞

Previous Results

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

•• GuillenGuillen--Gonzalez, Masmoudi, RodriquezGonzalez, Masmoudi, Rodriquez--Bellido (2001), and Bellido (2001), and Temam, Temam, Ziane (2003)Ziane (2003)The short time existence of the strong solutionThe 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• Cao and E.S.T. Annals of Mathematics

(2007) (to appear)* the global existence of the weak solutions(Galerkin method)

* the global existence and uniqueness of the strong solutions.

* existence of the global attractor.* upper bound for the dimension of theglobal attractor.

A different formulation of the PE

),(),,(),,(~0,),,(),(

),,(),(),,(

),,(),,(

yxvzyxvzyxv

vdyxvyxv

dyxTgyxpzyxp

dyxvzyxw

HHh HhH

=⋅∇=

⋅∇−=

∫∫

The Barotropic Mode – The Averaged Part of the Horizontal

Velocity

∫−∇+∆=

∇+×+∂+∇⋅+∂z

sHHHzHHHHt

pvkfvwvvv )(

The Baroclinic Mode –The Fluctuation Part of the Horizontal

Velocity

∫∫

−−

∇−∇+∂+∆

=⋅∇+∇⋅

−×+∂⎟⎠⎞⎜

⎝⎛ ⋅∇−

+∇⋅+⋅∇+∇⋅+∂

hHHzzvHHh

HHHHHH

HHHHHHHHHHt

dgTdgTvAvA

vkfvdzv

vvvvvvv

ξξ~~

~)~(~)~(

~)()~(~)~(~

The IDEA – Focus on Burgers Equation

0)( =∇⋅+∆− uuuut ν

0),(21 ),(

22 =∇⋅+⎟⎟⎠

⎞⎜⎜⎝

∂∂

+∆−∂ ∑ txuuxutxutxu

bound. and ),( 2 ∞Ltxu

We have

A maximum principle for

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.

∫−∇+∆=

∇+×+∂+∇⋅+∂z

sHHHzHHHHt

pvkfvwvvv )(

!)!,( yxpsWhere

The Fluctuation Equation is “like”3D Burgers Equations – Has No

Pressure Term!!

∫∫

−−

∇−∇+∂+∆=

⋅∇+∇⋅

−×+∂⎟⎠⎞⎜

⎝⎛ ⋅∇−

+∇⋅+⋅∇+∇⋅+∂

hHHzzvHHh

HHHHHH

HHHHHHHHHHt

dgTdgTvAvA

vkfvdzv

vvvvvvv

ξξ~~

~)~(~)~(

~)()~(~)~(~

A-priori Estimates

Q.E.D.

≤+=⇒

≤⇒

≤∇•

≤•

One of the Main Estimates Used

( )[ ])(L

1/2)(H

1/2)(L

1/2)(H

1/2)(L

21212 ||g||||f||||f||||u||||u|| C

dxdydz )y,g(x, )y,f(x, d )y,u(x,

ΩΩΩΩΩ

∫∫ zzzzh

Back to The 3D Navier-Stokes Equations

fpuuuut

=∇+∇⋅+∆−∂∂

0=⋅∇ u

New Criterion for Global Regularity of the 3D Navier-Stokes Equations

.2 and 3 where)),(),,0(( as long asfor ],0[ interval theon the exists equations

Stokes-Navier 3D theofsolution strong The:2005) E.S.T. and Cao (C. Theorem

>>ΩΤ∈∂

srLLp srz

This is different that the result of Y. Zhou (2005) where the assumption is on .p∇

Inviscid Regularazation of the 3D Euler Equations

2 =∇+∇⋅+∂∂

+∂∂

∆− puuut

0=⋅∇ u

Modified Energy

const. )),(),(( 222=∇+∫ dxtxutxu α

Inviscid Regularization of the Surface Quasi-Geostrophic

( ) θ

θθθα

−⊥ ∆−∇=

=∇⋅++∆−

on the global regularity of the 3d navier-stokes equations ...on the global regularity of the 3d...

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