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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:
$\partial_{t}\theta+v\cdot \nabla \theta= -\Lambda \theta, \quad x\in \mathbb R^N,\qquad \operatorname{div} v=0,$ where $$\Lambda \theta = (-\Delta)^{1/2}\theta$$ and they prove the following theorems:
(1) Let $$\theta(t,x)$$ be a function in $$L^\infty(0,T;L^{2}(\mathbb R^N))\cap L^2(0,T;H^{1/2}(\mathbb R^N))$$. For $$\lambda >0$$, set $$\theta_\lambda:=(\theta-\lambda)_+$$. If $$\theta$$ (and $$-\theta$$) satisfies for every $$\lambda>0$$ the level set energy inequalities
$\int_{\mathbb R^N} \theta_\lambda^2(t_2,x)\,dx+2 \int_{t_1}^{t_2}\int_{\mathbb R^N} |\Lambda^{1/2}\theta_\lambda|^2\,dx\,dt\leq \int_{\mathbb R^N} \theta_\lambda^2(t_1,x)\,dx, \quad 0<t_1<t_2,$ then
$\sup_{x\in\mathbb R^N}|\theta(T,x)|\leq C^* \frac{\|\theta_0\|_{L^2}}{T^{N/2}},$ where $$C^*>0$$ is a constant.
(2) Let $$Q_r=[-r,0]\times[-r,r]^N$$, for $$r>0$$. Assume that $$\theta(t,x)$$ is bounded in $$[-1,0]\times \mathbb R^N$$ and $$v|_{Q_1}\in L^\infty(-1,0;\text{BMO})$$; 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
$\partial_{t}\theta+u\cdot\nabla \theta= -\Lambda \theta, \quad x\in\mathbb R^N,\qquad \operatorname{div} u=0,$ with $$u_j=\overline{R}_j[\theta]$$, $$\overline{R}_j$$ 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 $$C^\alpha([t_0,\infty[\times\mathbb R^N)$$.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35B45 A priori estimates in context of PDEs 42B37 Harmonic analysis and PDEs 86A99 Geophysics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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