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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:

${\partial }_{t}\theta +v·\nabla \theta =-{\Lambda }\theta ,\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{N},\phantom{\rule{2.em}{0ex}}divv=0,$

where ${\Lambda }\theta ={\left(-{\Delta }\right)}^{1/2}\theta$ and they prove the following theorems:

(1) Let $\theta \left(t,x\right)$ be a function in ${L}^{\infty }\left(0,T;{L}^{2}\left({ℝ}^{N}\right)\right)\cap {L}^{2}\left(0,T;{H}^{1/2}\left({ℝ}^{N}\right)\right)$. For $\lambda >0$, set ${\theta }_{\lambda }:={\left(\theta -\lambda \right)}_{+}$. If $\theta$ (and $-\theta$) satisfies for every $\lambda >0$ the level set energy inequalities

${\int }_{{ℝ}^{N}}{\theta }_{\lambda }^{2}\left({t}_{2},x\right)\phantom{\rule{0.166667em}{0ex}}dx+2{\int }_{{t}_{1}}^{{t}_{2}}{\int }_{{ℝ}^{N}}{|{{\Lambda }}^{1/2}{\theta }_{\lambda }|}^{2}\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}dt\le {\int }_{{ℝ}^{N}}{\theta }_{\lambda }^{2}\left({t}_{1},x\right)\phantom{\rule{0.166667em}{0ex}}dx,\phantom{\rule{1.em}{0ex}}0<{t}_{1}<{t}_{2},$

then

$\underset{x\in {ℝ}^{N}}{sup}|\theta \left(T,x\right)|\le {C}^{*}\frac{\parallel {\theta }_{0}{\parallel }_{{L}^{2}}}{{T}^{N/2}},$

where ${C}^{*}>0$ is a constant.

(2) Let ${Q}_{r}=\left[-r,0\right]×{\left[-r,r\right]}^{N}$, for $r>0$. Assume that $\theta \left(t,x\right)$ is bounded in $\left[-1,0\right]×{ℝ}^{N}$ and ${v|}_{{Q}_{1}}\in {L}^{\infty }\left(-1,0;\text{BMO}\right)$; then $\theta$ is ${C}^{\alpha }$ in ${Q}_{1/2}$.

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

(3) Let $\theta$ be a solution to an equation

${\partial }_{t}\theta +u·\nabla \theta =-{\Lambda }\theta ,\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{N},\phantom{\rule{2.em}{0ex}}divu=0,$

with ${u}_{j}={\overline{R}}_{j}\left[\theta \right]$, ${\overline{R}}_{j}$ a singular integral operator. Assume also that $\theta$ verifies the level set energy inequalities. Then, for every ${t}_{0}>0$, there exists $\alpha$ such that $\theta$ is bounded in ${C}^{\alpha }\left(\left[{t}_{0},\infty \left[×{ℝ}^{N}\right)$.

##### MSC:
 35B65 Smoothness and regularity of solutions of PDE 35B45 A priori estimates for solutions of PDE 42B37 Harmonic analysis and PDE 86A99 Miscellaneous topics in geophysics 76D03 Existence, uniqueness, and regularity theory