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Regularity criteria for the 2D Boussinesq equations with supercritical dissipation. (English) Zbl 1355.35152
The purpose of this paper is to study the two-dimensional Boussinesc equation with fractional dissipation $\begin{cases} \partial_t v+(v\cdot \nabla)v+\nu\Lambda^{\alpha} v=-\nabla p+\theta e_2,\;x\in \mathbb{R}^2. \;t>0, \\\partial_t\theta+ (v\cdot \nabla)\theta+\kappa\Lambda^{\beta}\theta =0, \\ \nabla\cdot v=0, \\v(x,0)=v_0(x),\;\theta(x,0)=\theta_0(x), \end{cases}$ where $$\nu\geq 0$$, $$\kappa\geq 0$$, $$\alpha,\beta\in(0,2)$$ and $$\Lambda=(-\Delta)^{\tfrac 12}$$ denotes the Zygmund operator. The first main theorem states that for $$0<\alpha<0$$ and $$0\leq\beta<\alpha$$, and an $$L^1$$ assumption for $$\theta$$, the local solution of the system (1) can be extended to a certain interval. There are also two similar theorems. The proofs consist of several steps and use Hölder, Gronvall, Minkowski, Bernstein, Young inequalities, Fourier localization operator, Riesz transform and integration by parts.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B65 Smoothness and regularity of solutions to PDEs 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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