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Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. (English) Zbl 1204.35063

Motivated by the quasi-geostrophic model, the authors study the equation:

t θ+v·θ=-Λθ,x N ,divv=0,

where Λθ=(-Δ) 1/2 θ and they prove the following theorems:

(1) Let θ(t,x) be a function in L (0,T;L 2 ( N ))L 2 (0,T;H 1/2 ( N )). For λ>0, set θ λ :=(θ-λ) + . If θ (and -θ) satisfies for every λ>0 the level set energy inequalities

N θ λ 2 (t 2 ,x)dx+2 t 1 t 2 N |Λ 1/2 θ λ | 2 dxdt N θ λ 2 (t 1 ,x)dx,0<t 1 <t 2 ,

then

sup x N |θ(T,x)|C * θ 0 L 2 T N/2 ,

where C * >0 is a constant.

(2) Let Q r =[-r,0]×[-r,r] N , for r>0. Assume that θ(t,x) is bounded in [-1,0]× N and v| Q 1 L (-1,0;BMO); then θ is C α in Q 1/2 .

From these two theorems, the regularity of solutions to the quasi-geostrophic equation follows.

(3) Let θ be a solution to an equation

t θ+u·θ=-Λθ,x N ,divu=0,

with u j =R ¯ j [θ], R ¯ j a singular integral operator. Assume also that θ verifies the level set energy inequalities. Then, for every t 0 >0, there exists α such that θ is bounded in C α ([t 0 ,[× N ).


MSC:
35B65Smoothness and regularity of solutions of PDE
35B45A priori estimates for solutions of PDE
42B37Harmonic analysis and PDE
86A99Miscellaneous topics in geophysics
76D03Existence, uniqueness, and regularity theory