The quasi-geostrophic equation C regularity for the critical case - PowerPoint PPT Presentation

The quasi-geostrophic equation C α regularity for the critical case Luis Caffarelli and Alexis Vasseur caffarel@math.utexas.edu, vasseur@math.utexas.edu Department of Mathematics University of Texas at Austin The quasi-geostrophic equation – p. 1/17

Outline of the talk • The equation • Discussion on the model • The result • The method • Brief sketch of the proof The quasi-geostrophic equation – p. 2/17

The equation We consider the temperature function θ : R 2 → R solution to: ∂ ∂tθ + u · ∇ θ = − Λ θ, (1) u = R ⊥ θ. • � Λ θ = | ξ | ˆ θ : models the Ekman pumping, • R ⊥ θ = ( R 2 θ, − R 1 θ ) : perp of the Riesz transform where: R i θ = ξ i � ˆ θ. | ξ | Note that: div u = 0 (incompressible flow). The quasi-geostrophic equation – p. 3/17

Energy inequality We consider solutions with finite initial energy (no smallness assumption): � θ 2 0 dx < + ∞ , which verify the energy inequality: � � ∂ θ 2 dx + 2 | Λ 1 / 2 θ | 2 dx ≤ 0 . (2) ∂t • Solution "a la Leray". • Weak solutions have been constructed in Resnick 1995 The quasi-geostrophic equation – p. 4/17

Discussion on the model • Toy model for the study of possible blow-up in 3D fluid dynamics. Constantin, Wu 1999; Cordoba, Cordoba 2004... • If the dissipation term is replaced by − Λ α θ , α > 1 (sub-critical case): existence of global smooth solutions. Constantin, Wu 1999. • For the critical case, α = 1 (case studied here): global solutions were known to exist ONLY for small initial data in L ∞ . Constantin, Cordoba, Wu 2001. The quasi-geostrophic equation – p. 5/17

The result Theorem 1 (critical case: α = 1 ) Let θ 0 ∈ L 2 ( R 2 ) . Then for every t 0 > 0 , θ lies in C β (( t 0 , ∞ ) × R 2 ) for a β > 0 . • Condition on initial data is finite energy. • No condition of smallness. • The results holds true in R N for every N > 1 provided that each component of u is a linear combination of Riesz transform verifying div u = 0 . The quasi-geostrophic equation – p. 6/17

The method • De Giorgi (57): C α regularity for elliptic equation with rough coefficients. • reminescent of results of Nash in the parabolic case. Other application of the method in fluid dynamics: • V.: A new proof of partial regularity for incompressible Navier-Stokes equations (To appear in NoDEA), • Mellet, V.: L p bounds for quantities advected by a compressible fluid (preprint) motivated by the study of the velocity field near vacuum for compressible Navier-Stokes equations. The quasi-geostrophic equation – p. 7/17

First step: L ∞ bound Lemma 1 Any solution to (1)(2) with θ 0 ∈ L 2 ( R 2 ) verifies for every ε > 0 : θ ∈ L ∞ (( ε, + ∞ ) × R 2 ) . It relies on: • De Giorgi techniques, • "Maximum principle" due to Cordoba and Cordoba (2004). The quasi-geostrophic equation – p. 8/17

Maximum principle • (Cordoba and Cordoba 2004) For any convex function φ we have (despite the non-locality of the diffusion term !): − φ ′ ( θ )Λ θ ≤ − Λ( φ ( θ )) . • Then we can derive an equation on the De Giorgi truncations: θ k = ( θ − C k ) + . We have: � � ∂ | Λ 1 / 2 θ k | 2 dx ≤ 0 . θ 2 k dx + 2 ∂t The quasi-geostrophic equation – p. 9/17

2 nd step: local energy inequality • C α regularity is a LOCAL property. • We need a local version (in x and t ) of the energy equality on the truncation θ k . • The "maximum principle" of Cordoba and Cordoba is not sufficient. • We need to extend θ with a new variable z > 0 to keep memory of the non locality of the diffusion term. The quasi-geostrophic equation – p. 10/17

Extension for z > 0 • θ ( t, x, z ) defined by: x ∈ R 2 , z > 0 , ∆ x,z θ = 0 θ ( t, x, 0) = θ ( t, x ) . • We have: Λ θ ( t, x ) = ∂ z θ ( t, x, 0) . • We get a nice local estimate: �� � � 0 � � 1 |∇ θ k | 2 dz dx dt θ 2 sup k dx + − 1 ≤ t ≤ 0 B 1 − 1 B 1 0 �� 0 � � � 0 � � 2 2 θ p ≤ C p k dx dt + θ k dx dt . − 2 B 2 − 2 B 2 0 The quasi-geostrophic equation – p. 11/17

Remarks • The "maximum principle" of Cordoba and Cordoba follows naturally in the extension framework. • The local energy inequality allows to use De Giorgi techniques. • However, the degeneracy of the estimates makes the proof more tedious (some of the terms hold only on the trace θ of θ at z = 0 ). The quasi-geostrophic equation – p. 12/17

Oscillation lemma Q 1 = ( − 1 , 0) × ( − 1 , 1) 2 , Q 1 / 2 = ( − 1 / 2 , 0) × ( − 1 / 2 , 1 / 2) 2 . � � � � � � u � L ∞ ( − 1 , 0; BMO (( − 1 , 1) 2 )) ≤ C, � Q 1 u dx dt � ≤ C. Lemma 2 There exists 0 < λ < 1 s.t. for any θ solution to (1)(2) for ( t, x ) ∈ Q 1 , with � � � � � � � sup θ − inf Q 1 θ � ≤ 1 . Q 1 Then: � � � � � � � sup θ − inf Q 1 / 2 θ � ≤ λ. � � Q 1 / 2 The quasi-geostrophic equation – p. 13/17

Scaling • The final point is to use the invariant scaling of the equation: θ ε ( t, x ) = θ ( εt, εx ) . • Applying the oscillation lemma recursively on θ ε n with ε n = 2 − n gives: | u ( t, x ) − u ( s, y ) | ≤ λ n , sup ( t,x ) , ( s,y ) ∈ Q n Q n dyadic cubes centered on a fixed t 0 , x 0 . • Finally θ ∈ C α for α = ln 2 (1 /λ ) . The quasi-geostrophic equation – p. 14/17

scaling of the velocity u from the Riesz transform: u = R ⊥ θ Thus L ∞ bound on θ gives that for any ε > 0 : u ∈ L ∞ (( ε, ∞ ) , BMO ( R 2 )) . • BMO estimate is invariant by the scaling. • But it does not control the mean values ! • Additional difficulties to control the mean drift from scale to scale ! The quasi-geostrophic equation – p. 15/17

conclusion • CLAIM: De Giorgi method is a powerful tool for the study of possible blow-up in PDEs. • The L ∞ part (the easiest one !) works also for SYSTEM. The quasi-geostrophic equation – p. 16/17

Proof of the L ∞ bound • Fix t 0 > 0 and set T k = t 0 (1 − 2 − k ) . • Consider C k = M (1 − 2 − k ) , M > 0 chosen later. � ∞ � � � | Λ 1 / 2 θ k | 2 dx dt + • U k = θ 2 sup t ≥ T k k dx . T k • We can compare the energy level U k from the previous one U k − 1 in a non linear way: U k ≤ C M (1 + 1 /t 0 )2 Ck U γ k , for a γ > 1 . • So for M big enough (depending on t 0 ): k →∞ U k = 0 , lim so θ ≤ M. The quasi-geostrophic equation – p. 17/17