Regularity criteria for the 2D Boussinesq equations with supercritical dissipation (Q345989)

The purpose of this paper is to study the two-dimensional Boussinesc equation with fractional dissipation NEWLINE\[NEWLINE\begin{cases} \partial_t v+(v\cdot \nabla)v+\nu\Lambda^{\alpha} v=-\nabla p+\theta e_2,\;x\in \mathbb{R}^2. \;t>0, \\\partial_t\theta+ (v\cdot \nabla)\theta+\kappa\Lambda^{\beta}\theta =0, \\ \nabla\cdot v=0, \\v(x,0)=v_0(x),\;\theta(x,0)=\theta_0(x), \end{cases}NEWLINE\]NEWLINE 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.