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
1)(ρ
υ
0=⋅∇ u
0)( =∇⋅+∆−∂∂ TuTTt
κ
Temperature Estimates
• Maximum Principle
• Gradient Estimates
∞∞ +≤LL
TCCT 010
dxTTuTTdtd
LL∫ ∆⋅∇⋅=∆+∇ )(
21 22
22κ
• Estimate of the Nonlinear Term
• Interpolation/Calculus Inequality
• Young’s Inequality
236)(LLL
TTucdxTTu ∆∇≤∆⋅∇⋅∫
2321
226)(LLL
TTucdxTTu ∆∇≤∆⋅∇⋅
⇒
∫
21
21
223 LLLc ϕϕϕ ∇≤ 1H∈∀ϕ
,11 qp bq
ap
ba +≤⋅ 111=+
qp
243
22
2243
2622
226
221
2)(
LLLL
LL
TucTTdtd
TTucdxTTuL
∇≤∆+∇
⇒
∆+∇≤∆⋅∇⋅∫
κκ
κκ
∫∇≤∇
t
L duc
LLeTtT 0
463
22
)(22 )0()(
ττκ
By Gronwall’s inequality
Question:
?)(0
46 Kdu
t
L≤∫ ττIs
To answer this question we have to deal with the Navier-Stokes equations.
The Navier-Stokes Equations
fpuuuut
=∇+∇⋅+∆−∂∂
0
1)(ρ
υ
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
.0=u
10 =ρ
Sobolev Spaces
)1(ˆ
such that
ˆ)(
22
2
∞<+
==Ω
∑
∑
∈
∈
⋅
s
kk
k
Lxki
ks
k
eHd
dZ
Z
ϕ
ϕϕπ
Navier-Stokes Equations Estimates
• Formal Energy estimate
• Observe that since we have
( )ufupuuuuudtd
LL,)(
21 22
22 =⋅∇+⋅∇⋅+∇+ ∫∫υ
0=⋅∇ u
0)( =⋅∇=⋅∇⋅ ∫∫ dxupdxuuu
( )ufuudtd
LL,
21 22
22 =∇+⇒ υ
222222
2222
21
LLLLLLufcLufuu
dtd
∇≤≤∇+υ
2222
22222
222
2222
221
221
LLL
LLLL
fcLuudtd
ufcLuudtd
υυ
υυ
υ
≤∇+
∇+≤∇+
By the Cauchy-Schwarz and Poincare’ inequalities
By the Young’s inequality
By Poincare’ inequality22
22
2222
LLLfcLu
Lcu
dtd
υυ
≤+
( ) 2
2
422
2
2
2
2
2 1)0()(L
tLcL
tLcL
fecLuetu−− −− −+≤ υυ
υ
( ) ),,,,(222 0
0
2 Τ≤∇∫Τ
υττυLLL
fuLKdu
By Gronwall’s inequality
and
[ ]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
0>Τ
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
2
2
2
4)(
LL u
fuf ∆+≤∆−⋅∫
υυ
244)()(LLL
uuuuuu ∆∇≤∆−⋅∇⋅∫
Observe that
By Cauchy-Schwarz
By Hőlder inequality
DcDc
LL
LLL −∇
−∇≤
32
43
24
1
2
21
22
1
2
4
ϕϕ
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
)( 20
46 +≤
∫≤
+t
L deuc
eyty 0
20
46 ))((
)0()(ττ
Recall that
One can show that
Which implies that
The Question Is Again Whether:
?)(0
46 Kdu
L≤∫
Τ
ττ
One can instead use the following Sobolev inequality
26 LLucu ∇≤
∫Τ
≤≤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
fuut
×∇=∇⋅−∇⋅+∆−∂∂ )()( ωωωνω
fut
×∇=∇⋅+∆−∂∂ ωωνω )(
2),( txω
Two dimensional case
Satisfies a maximum principle.
Vorticity Stretching Term u)( ∇⋅ω
0)( ≡∇⋅ uω
0ω
0)( ≡∇⋅ uω
2~)(~
zz
u∇⋅ω
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
∫Τ
∞<∞
0
)( dttL
ω
∞Lt )(ω
0=υ
],0[ Τ
• Constantin and Fefferman:
Provided sufficient condition involving the Lipschitzregularity of the direction of the vorticity:
ωωξ =
Two-Dimensions Euler
ωψ
ψ
ωω
=∆
×∇=
=∇⋅+∂∂
k
ku
ut
)(
0)(
Yudovich proved a weak version of the
maximum principle, that is .)( 0 ∞∞ ≤ LLt ωω
ppp LLWpcD ψψψ
α
α ∆⋅≤= ∑≤ 2
,2
IDEA
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
3
0 21
ΩL
uH
axis]- from[away axis- thearoundDomain Revolution
zz−Ω•
)),(),,(),,((),,( 0000 zrzrzrzyxu zr ϕϕϕ θ=
z
x
• 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.
0=υ
• 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
∑≥
⋅=
)(
20
0 ˆMKk
Lxki
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 <<
Hu
The Effect of Rotation
0
0)(
=⋅∇
=×Ω+∇+∇⋅+∂∂
u
upuutu
Tadmor. andLiu ... Masmoudi, Granier, Ghalagher, Chemin,
Majda.-Embid .Nicolaenko-Mahalov-Babin
as
thatObserve ).,0[on exists
solution thesuch that )( exists thereisThat
).,0[on exists solution the ifsuch that ),( is There
0
0
,00
000
••••
∞→Ω∞→Τ
Τ
ΩΤ•
Τ
Ω>ΩΤΩ•
u
u
(x))0,(in 0
EquationBurgersInviscid
0uxuuuu xt
==+ R
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
zu
zt
==Ω++
∈∈ CC
),(),( tzuetzv tiΩ=
)),(1(11
)),(1(
)),(1(),(
0
'0
'0
0
tzvi
ezvi
e
tzvi
ezvv
z
tzvi
ezvtzv
vvev
titi
ti
tiz
tit
Ω−−
−Ω−−
+
Ω−−
−=
∂∂
Ω−−
−=
=+
Ω−Ω−
Ω−
Ω−
Ω−
.0 allfor regular remainssolution
theand finite remains
))( i.e.( ,1 If 00
≥∂∂
Ω>Ω>>Ω
t
vz
u
00110
,21
=⎟⎟⎠
⎞⎜⎜⎝
⎛ −Ω+∇⋅+
+=+=
uuuu
iyxziuuu
t
The above complex system is equivalent to 2D Rotating Burgers:
existence. global have we)(For 0)()cos(
0)(generally More
00 vvFdivtv
uFdivu
t
t
Ω>Ω=Ω+•
=+• (Short time existence)
)()(0)(
Then
),(),(by denote and sinLet
00 xuxv)w(x,wFdivw
xtvxwt
===+
=ΩΩ
=
0τ
ττ
)()sin()( satisfies whenever isThat exists.
solution the)()( interval in the For
00
00
vtvt
wvv
∗∗
∗∗
Τ≤ΩΩ
≤Τ−
Τ≤≤Τ− ττ
Bénard Convection Porous Medium
.conditionsboundary physicalcertain Subject to
0)(
0
0
=∇⋅+∆−∂∂
=⋅∇
=−∇++∂∂
TuTTt
u
kRTpuut
κ
γ
• 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
)(
0
01)(
01)(
0
02
2
02
2
uwv
QTz
wTvTTt
wz
v
Tgpz
wz
wwvwz
wt
vkfpvz
wvvvz
vt
H
HH
HH
HHH
HHHHHHHHH
=
=∂∂
+∇⋅+∆−∂∂
=∂∂
+⋅∇
=+∂∂
+∂∂
+∇⋅+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∆−∂∂
=×+∇+∂∂
+∇⋅+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∆−∂∂
ρκ
ρυ
ρυ
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
0
01)(
1111110
20
=++++
=++++Τ
=+∂∂
+∂∂
+∇⋅+∂∂
−−−
TgHP
HW
LUWW
Tgpz
wz
wwvwt HH
ρ
ρ
Hydrostatic Balance
01
0
=+∂∂ Tgpzρ
Scale Analysis – The Ideal Case
01010101010
0
01)(
528880
20
=++++
=++++Τ
=×+∇+∂∂
+∇⋅+∂∂
−−−−−
UFLP
HUW
LUU
vkfpvz
wvvvt HHHHHHH
ρ
ρ
Rossby Number
LFUR =
Geostrophic Balance
• When 1<<R
01
0
=×+∇ HH vkfpρ
The Ideal Planetary Geostrophicequations
TQwTTvTwv
Tgp
vkfp
zzzHHt
zHH
z
HH
∂+=+∇⋅+=∂+⋅∇
=+∂
=×+∇
κρ
ρ
ρ
0
0
0
)(0
01
01
Rayleigh Friction and Horizontal-Diffusion
)()(0
01
)(1
0
0
0
TDTQTwTvTwv
Tgp
vFvkfp
zzvzHHt
zHH
z
HHH
+∂+=∂+∇⋅+=∂+⋅∇
=+∂
=×+∇
κρ
ρ
ρ
Friction, Viscosity and Diffusion Schemes
• Conventional eddy viscosity
and
• Linear drag
What should be the diffusion operator D?
HzzhHHvH vAvAvF ∂+∆=)(
HH vvF ε−=)(
TTD HH∆=κ)(
The Viscous PG Equations
TTQwTTvTwv
Tgp
vKvKvkfp
zzvHhzHHt
zHH
z
HzzhHHvH H
∂+∆+=+∇⋅+=∂+⋅∇
=+∂
∂+∆=×+∇
κκρ
ρ
ρ
0
0
0
)(0
01
1
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.
1H
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
0s
⊂=> >∩∪
st
A
1H
2L
The Rayleigh Friction Case
TTQwTTvTwv
Tgp
vvkfp
zzvHhzHHt
zHH
z
HHH
∂+∆+=+∇⋅+=∂+⋅∇
=+∂
−=×+∇
κκρ
ρ
ερ
0
0
0
)(0
01
1
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
=∂∂
−−+
==∂∂
=⋅
Γ
Γ
Γ
s
s
s
Tn
ffnfnnf
eeT
nvH
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
Tgp
vvkfp
zzHzHHt
zHH
z
HHH
∂+⋅∇+=+∇⋅+=∂+⋅∇
=+∂
−=×+∇
κρ
ρ
ερ
)()(0
01
1
0
0
0
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
= H
h
zzT
H
HHHH
∇∇∇⋅∇∇=
⎟⎟⎠
⎞⎜⎜⎝
⎛
µλ
εε
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
0
01)(
1111110
20
=++++
=++++Τ
=+∂∂
+∂∂
+∇⋅+∂∂
−−−
TgHP
HW
LUWW
Tgpz
wz
wwvwt HH
ρ
ρ
Recall Scale Analysis for Horizontal Motion – The Ideal Case
01010101010
0
01)(
528880
20
=++++
=++++Τ
=×+∇+∂∂
+∇⋅+∂∂
−−−−−
UFLP
HUW
LUU
vkfpvz
wvvvt HHHHHHH
ρ
ρ
The Primitive Equations of Large Scale Oceanic and Atmospheric
Dynamics
zzvHhzHHt
zHH
z
HzzvHHh
HHHzHHHHt
TKTKQwTTvTwv
gTpvAvA
vkfpvwvvv
+∆+=+∇⋅+=∂+⋅∇
=+∂∂+∆=
×+∇+∂+∇⋅+∂
)(0
0
)(
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,),,(),(
),,(),(),,(
),,(),,(
01
yxvzyxvzyxv
vdyxvyxv
dyxTgyxpzyxp
dyxvzyxw
HHH
HHh HhH
z
hs
z
h HH
−=
=⋅∇=
−=
⋅∇−=
∫∫
∫
−
−
−
ξξ
ξξ
ξξ
The Barotropic Mode – The Averaged Part of the Horizontal
Velocity
∫−∇+∆=
∇+×+∂+∇⋅+∂z
hHHHh
sHHHzHHHHt
dzTvA
pvkfvwvvv )(
The Baroclinic Mode –The Fluctuation Part of the Horizontal
Velocity
∫∫
∫
−−
−
∇−∇+∂+∆
=⋅∇+∇⋅
−×+∂⎟⎠⎞⎜
⎝⎛ ⋅∇−
+∇⋅+⋅∇+∇⋅+∂
z
hH
z
hHHzzvHHh
HHHHHH
HHz
z
h HH
HHHHHHHHHHt
dgTdgTvAvA
vvvv
vkfvdzv
vvvvvvv
ξξ~~
~)~(~)~(
~~
~)()~(~)~(~
The IDEA – Focus on Burgers Equation
0)( =∇⋅+∆− uuuut ν
0),(21 ),(
21),(
21 2
2
,
22 =∇⋅+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∆−∂ ∑ txuuxutxutxu
j
i
jit
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
hHHHh
sHHHzHHHHt
dzTvA
pvkfvwvvv )(
!)!,( yxpsWhere
The Fluctuation Equation is “like”3D Burgers Equations – Has No
Pressure Term!!
∫∫
∫
−−
−
∇−∇+∂+∆=
⋅∇+∇⋅
−×+∂⎟⎠⎞⎜
⎝⎛ ⋅∇−
+∇⋅+⋅∇+∇⋅+∂
z
hH
z
hHHzzvHHh
HHHHHH
HHz
z
h HH
HHHHHHHHHHt
dgTdgTvAvA
vvvv
vkfvdzv
vvvvvvv
ξξ~~
~)~(~)~(
~~
~)()~(~)~(~
A-priori Estimates
Q.E.D.
~
~
66
6
2
6
Kvvv
Kv
Kv
Kv
LHHLH
LH
LHH
LH
≤+=⇒
≤⇒
≤∇•
≤•
One of the Main Estimates Used
( )[ ])(L
1/2)(H
1/2)(L
1/2)(H
1/2)(L
0
21212 ||g||||f||||f||||u||||u|| C
dxdydz )y,g(x, )y,f(x, d )y,u(x,
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Back to The 3D Navier-Stokes Equations
fpuuuut
=∇+∇⋅+∆−∂∂
0
1)(ρ
υ
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∈∂
Τ
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
01)(0
2 =∇+∇⋅+∂∂
+∂∂
∆− puuut
ut ρ
α
0=⋅∇ u
Modified Energy
const. )),(),(( 222=∇+∫ dxtxutxu α
Inviscid Regularization of the Surface Quasi-Geostrophic
( ) θ
θθθα
2/1
2
0
−⊥ ∆−∇=
=∇⋅++∆−
u
utt
