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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
\[ \int_{\mathbb R^N} \theta_\lambda^2(t_2,x)\,dx+2 \int_{t_1}^{t_2}\int_{\mathbb R^N} |\Lambda^{1/2}\theta_\lambda|^2\,dx\,dt\leq \int_{\mathbb R^N} \theta_\lambda^2(t_1,x)\,dx, \quad 0<t_1<t_2, \]
\[ \sup_{x\in\mathbb R^N}|\theta(T,x)|\leq C^* \frac{\|\theta_0\|_{L^2}}{T^{N/2}}, \]
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 \(C^\alpha([t_0,\infty[\times\mathbb R^N)\).

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
Full Text: DOI Link
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