Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity

Abstract

We address the persistence of regularity for the 2D $\alpha$-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for $(u_0,{\rho }_0)\in W^{s,q}({\mathbb{R}}^2)\times W^{s,q}({\mathbb{R}}^2)$, where $s>1$ and $q\in (2,\infty )$. We prove that the solution $(u(t),\rho (t))$ exists and belongs to $W^{s,q}({\mathbb{R}}^2)\times W^{s,q}({\mathbb{R}}^2)$ for all positive time $t$ for $q>2$, where $\alpha \in (1,2)$ is arbitrary.

Pure and Applied Functional Analysis, to appear