Global solutions to Stokes-Magneto equations with fractional dissipations

Abstract

In this paper, we investigate a Stokes-Magneto system with fractional diffusions. We first deal with the non-resistive case in ${\mathbb{T}}^d$ and establish the local and global well-posedness with initial magnetic field ${\mathbf{b}}_0\in H^s({\mathbb{T}}^d)$. We also show the existence of a unique mild solution of the resistive case with initial data ${\mathbf{b}}_0$ in the critical $L^p({\mathbb{R}}^d)$ space. Moreover, we show that $\Vert \mathbf{b}(t){\Vert }_{L^p}$ converges to zero as $t\to \infty$ when the initial data is sufficiently small.

arXiv comment18 pages

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