Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions

Abstract

We consider a Stokes-Magneto system in ${\mathbb{R}}^d$ ($d\ge 2$) with fractional diffusions ${\Lambda }^{2\alpha }\mathbf{u}$ and ${\Lambda }^{2\beta }\mathbf{b}$ for the velocity $\mathbf{u}$ and the magnetic field $\mathbf{b}$, respectively. Here $\alpha ,\beta$ are positive constants and ${\Lambda }^s=(-\Delta {)}^{s/2}$ is the fractional Laplacian of order $s$. We establish global existence of weak solutions of the Stokes-Magneto system for any initial data in $L_2$ when $\alpha$, $\beta$ satisfy $1/2<\alpha <(d+1)/2$, $\beta >0$, and $\min \{\alpha +\beta ,2\alpha +\beta -1\}>d/2$. It is also shown that weak solutions are unique if $\beta \ge 1$ and $\min \{\alpha +\beta ,2\alpha +\beta -1\}\ge d/2+1$, in addition.

40 pages