Global solutions to Stokes-Magneto equations with fractional dissipations

TL;DR

This paper studies the Stokes-Magneto system with fractional diffusions, establishing well-posedness and long-term behavior of solutions in various function spaces.

Contribution

It provides new results on local and global well-posedness, existence of mild solutions, and decay properties for the fractional Stokes-Magneto equations.

Findings

Global well-posedness for non-resistive case

Existence of unique mild solutions in critical spaces

Magnetic field decay to zero over time for small initial data

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 $\|\mathbf{b}(t)\|_{L^{p}}$ converges to zero as $t\rightarrow\infty$ when the initial data is sufficiently small.