Recent results on 2D Surface Quasi-Geostrophic Equation Alexander - PowerPoint PPT Presentation

Recent results on 2D Surface Quasi-Geostrophic Equation Alexander Kiselev UW-Madison Based on work joint with Fedor Nazarov, Roman Shterenberg and Alexander Volberg

2D Surface Dissipative Quasi-Geostrophic Equation (SQGE) Appears in the studies of strongly rotating fluids (as a boundary condition on the planar boundary). θ t = u · ∇ θ − ( − ∆) α θ, θ ( x, 0) = θ 0 ( x ) , θ scalar, real valued. Here � � iξ 2 θ ( ξ ) , − iξ 1 u ( ξ ) = ( − R 2 ˆ θ ( ξ ) , R 1 ˆ ˆ ˆ ˆ θ ( ξ )) = θ ( ξ ) . | ξ | | ξ | In this talk, I consider the periodic setting ( T 2 ). Introduced by Constantin, Majda and Tabak (1994). Nonlin- earity exhibits many properties of the 3D Euler equation, but is simpler. In the conservative case, equation for ∇ ⊥ θ is very similar to 3D Euler equation for vorticity. Finite time blow up?

Studied recently in particular by Caffarelli, Constantin, A. Cor- doba, D. Cordoba, Dong, Fefferman, T. Hou, Ju, Li, Majda, Resnick, Rodrigo, Tabak, Stefanov, Vasseur, Wu, Yu. Perhaps the simplest equation of fluid dynamics for which exis- tence of global regular solution is not known. The next step towards simplification is 1D dissipative Burgers equation. θ t = θθ x − ( − ∆) α θ, θ ( x, 0) = θ 0 ( x ) .

Plan of the talk: 1. Background results and history. 2. The critical dissipation case: global regularity by the moduli of continuity method. 3. Other applications: Burgers equation. 4. Growth of high order Sobolev norms for the conservative SQGE.

SQGE: Basic properties and some known results. T 2 θ 2 dx is non-increasing. The energy � Resnick, Cordoba&Cordoba showed � | θ | p − 2 θ ( − ∆) α θ dx ≥ 0 , 1 < p < ∞ . � θ � L ∞ is non-increasing. This yields a maximum principle: It makes the value α = 1 / 2 critical.

If α > 1 / 2 , for any θ 0 ∈ H s , s > 2 − 2 α, s ≥ 0 there exists global smooth (for any t > 0) solution. (Resnick, Ju, Wu) If α < 1 / 2 , only local existence is known. For the Burgers equation, blow up can happen if α < 1 / 2 (Kise- lev, Nazarov, Shterenberg; Alibaud, Droniou and Vovelle - whole line case).

The critical case α = 1 / 2 . Has been studied especially actively since this case has physical significance. Constantin, D. Cordoba, Wu: Global smooth (analytic in x for any t > 0) solution exists if θ 0 ∈ H 2 , � θ 0 � L ∞ is small. Other re- sults due to A. Cordoba and D. Cordoba, Dong, Ju, Li, Stefanov. Theorem 1 (KNV) Assume the initial data θ 0 is periodic and θ 0 ∈ H 1 . Then the critical quasi-geostrophic equation has a unique global solution which is real analytic in x for any t > 0 . Caffarelli-Vasseur: a similar result by a completely different method.

Main idea: a nonlocal maximum principle. Let us call ω ( x ) a modulus of continuity if ω ( x ) : [0 , ∞ ) �→ [0 , ∞ ) is increasing, continuous, concave, and ω (0) = 0 . We say f has ω if | f ( x ) − f ( y ) | ≤ ω ( | x − y | ) . We will construct a family of unbounded moduli of continuity ω A ( ξ ) = ω ( Aξ ) , A > 0 , which will be preserved by the evolution: if the initial data θ 0 has ω A , then so does θ ( x, t ) for any t > 0 . We will have ω ′ (0) = 1 , thus giving us global control of �∇ θ ( x, t ) � L ∞ . It is sufficient to show preservation of ω, ω A follows by scaling.

Given control over �∇ θ � L ∞ and local existence of smooth solu- tion, standard techniques allow to show uniform in time estimates for the Sobolev norms of arbitrary order. For example, one can derive a differential inequality ∂ t � θ � H s ≤ C �∇ θ � a ( s ) L ∞ � θ � 3 − a ( s ) H s +1 / 2 − � θ � 2 H s +1 / 2 with a ( s ) > 1 for all s large enough.

Properties of ω ( ξ ) . (i) Increasing, concave. (ii) Near zero, ω ( ξ ) = ξ − ξ 3 / 2 , so ω ′ (0) = 1 and ω ′′ (0) = −∞ . (iii) As ξ → ∞ , ω ( ξ ) ∼ c log log ξ. Corollary. The following estimate holds: �∇ θ ( x, t ) � ∞ ≤ C �∇ θ 0 � ∞ exp exp( C � θ 0 � ∞ ) . Reason: we just need to find A so that θ 0 ( x ) has the modulus of continuity ω A . Then |∇ θ ( x, t ) | is bounded by A for all times.

Breakdown scenario. How can solution lose the moduli of continuity ω ? Claim: If θ 0 has ω and it is lost, there must be a time t 0 and two distinct points x, y where θ ( x, t 0 ) − θ ( y, t 0 ) = ω ( | x − y | ) , θ has ω for t ≤ t 0 and loses it for t > t 0 . Reason: The only alternative to the Claim is that |∇ θ ( x, t 0 ) | = ω ′ (0) at some x (instead of two distinct points). But since ω ′′ (0) = −∞ , this would imply that ω has already been violated at time t 0 .

Fix x, y, t 0 as in Claim with | x − y | ≡ ξ, will show that ∂ t ( θ ( x, t ) − θ ( y, t )) | t = t 0 < 0! This contradicts the assumption that solution has modulus of continuity ω up to t 0 . Have to control the flow term and the dissipation term contri- butions: ∂ ∂t ( θ ( x, t ) − θ ( y, t )) = ( u · ∇ θ )( x ) − ( u · ∇ θ )( y ) − ( − ∆) 1 / 2 θ ( x ) + ( − ∆) 1 / 2 θ ( y ) .

The flow term. Lemma. If the function θ has modulus of continuity ω , then u = ( − R 2 θ, R 1 θ ) has modulus of continuity �� ξ � ∞ � ω ( η ) ω ( η ) Ω( ξ ) = B dη + ξ dη (1) η 2 η 0 ξ with some universal constant B > 0 . Note ( u · ∇ θ )( x ) = d � � dhθ ( x + hu ( x )) . � � h =0 Now θ ( x + hu ( x )) − θ ( y + hu ( y )) ≤ ω ( | x − y | + h | u ( x ) − u ( y ) | ) ≤ ω ( ξ + h Ω( ξ )) . Since θ ( x ) − θ ( y ) = ω ( ξ ) , we conclude that ( u · ∇ θ )( x ) − ( u · ∇ θ )( y ) ≤ Ω( ξ ) ω ′ ( ξ ) .

The dissipation term. Observe that − ( − ∆) 1 / 2 θ ( x ) = d � � dh P h ∗ θ , � � h =0 where P h is the 2D planar Poisson kernel, and θ is periodization to R 2 . After a computation, we get that − ( − ∆) 1 / 2 θ ( x ) + ( − ∆) 1 / 2 θ ( y ) is bounded from above by � ξ 1 ω ( ξ + 2 η ) + ω ( ξ − 2 η ) − 2 ω ( ξ ) 2 dη η 2 π 0 � ∞ +1 ω (2 η + ξ ) − ω (2 η − ξ ) − 2 ω ( ξ ) dη . η 2 ξ π 2 Both terms are negative due to concavity.

To conclude, have to check the inequality �� ξ � ∞ � ω ( η ) ω ( η ) ω ′ ( ξ )+ dη + ξ B dη η 2 η 0 ξ � ξ 1 ω ( ξ + 2 η ) + ω ( ξ − 2 η ) − 2 ω ( ξ ) 2 dη η 2 π 0 � ∞ +1 ω (2 η + ξ ) − ω (2 η − ξ ) − 2 ω ( ξ ) dη < 0 ξ η 2 π 2 Explicit form of ω : Set ω ( ξ ) = ξ − ξ 3 / 2 , when 0 ≤ ξ ≤ δ and γ ω ′ ( ξ ) = ξ (4 + log( ξ/δ )) when ξ > δ. Here 0 < γ < δ are small, can be chosen to ensure the inequality is true. Moreover the contribution of the dissipative term is ≤ − 2Ω( ξ ) ω ′ ( ξ ) .

Consider, for example, ξ ≥ δ case. Then � ξ ω ( η ) dη ≤ δ + ω ( ξ ) log ξ 2 + log ξ � � δ ≤ ω ( ξ ) η δ 0 if δ is small enough. Also � ∞ � ∞ ω ( η ) dη = ω ( ξ ) η 2 (4 + log( η/δ )) ≤ 2 ω ( ξ ) dη + γ η 2 ξ ξ ξ ξ if γ, δ are small enough. Thus the positive term does not exceed 4 + log ξ ω ′ ( ξ ) = Bγω ( ξ ) � � Bω ( ξ ) . δ ξ

Now if ξ ≥ δ and γ is small, then ω (2 ξ ) ≤ 3 2 ω ( ξ ) . Due to concavity, � ∞ 1 ω (2 η + ξ ) − ω (2 η − ξ ) − 2 ω ( ξ ) dη ≤ η 2 π ξ/ 2 � ∞ − 1 ω ( ξ ) η 2 dη = − ω ( ξ ) πξ . 2 π ξ/ 2 We have − ω ( ξ ) vs. Bγω ( ξ ) , πξ ξ just take γ small enough!

Analyticity. Smoothness of solution follows from control of �∇ θ � ∞ and stan- dard estimates on Sobolev norms. These estimates yield uniform in time control of the Sobolev norms of the solution. θ ( k, t ) exp( 1 Set ξ k ( t ) = ˆ 2 | k | t ) . Define | k | 4 | ξ k ( t ) | 2 . � Y ( t ) = k One can show that dY dt ≤ C 1 Y 3 / 2 + ( C 2 Y 1 / 2 t − 1) | k | 5 | ξ k | 2 . � k Thus we have control of Y on a small time interval [0 , τ ] , which implies analyticity. Uniform in time control of � θ � H 2 allows to continue with a fixed time step.

Other applications: Theorem 2 (KNS). Let α ≥ 1 / 2 , θ 0 ∈ L p , 1 < p < ∞ . Then there exists a global solution real analytic in x for t > 0 such that � θ ( x, t ) − θ 0 ( x ) � p → 0 as t → 0 . Uniqueness is not known! A related model: 2D Surface Critical Dissipative Dispersive QGE equation: θ t = u · ∇ θ − ( − ∆) 1 / 2 θ + Fu 2 , For α ≤ 1 / 2 and θ 0 ∈ H 1 , the dispersive Theorem 3 (KN). SQGE has unique global solution which is smooth for t > 0 .

Rough solutions. 1. Approximate θ 0 in L p by smooth θ 0 ,k , construct θ k ( x, t ) . 2. Get a-priori estimates: a. Set M k ( t ) = � θ k ( x, t ) � ∞ . Then uniformly in k k ( t ) ≤ c 1 M 1+ p/ 2 M ′ + c 2 M k ⇒ t 2 /p M k ( t ) ≤ C k b. Using arguments similar to the proof of conservation of ω for regular data, one can show that θ k ( x, t ) have modulus of continuity ω A ( t ) , with certain A ( t ) > 0 , A ( t ) → ∞ as t → 0 . c. Passing to the limit gives a regular for t > 0 solution. d. Uniqueness is not known.