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

We consider a Stokes-Magneto system in $\mathbb{R}^d$ ($d\geq 2$) with fractional diffusions $\Lambda^{2\alpha}\boldsymbol{u}$ and $\Lambda^{2\beta}\boldsymbol{b}$ for the velocity $\boldsymbol{u}$ and the magnetic field $\boldsymbol{b}$, 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 $\beta \geq 1$ and $\min \{\alpha+\beta,2\alpha+\beta-1\}\geq d/2+1$, in addition.