Propagation of regularity for the MHD system in optimal Sobolev space

TL;DR

This paper investigates how regularity of solutions to the incompressible viscous non-resistive MHD system propagates in critical Sobolev spaces, establishing conditions under which solutions maintain their regularity over time.

Contribution

It demonstrates that solutions with initial regularity above a critical threshold remain regular for a short time, extending understanding of regularity propagation in MHD systems.

Findings

Regularity propagates for solutions with initial data in Sobolev spaces above critical level.

Propagation of regularity holds under certain initial conditions and known uniqueness.

Results apply to solutions in critical Sobolev spaces for the MHD system.

Abstract

We study the problem of propagation of regularity of solutions to the incompressible viscous non-resistive magneto-hydrodynamics system. According to scaling, the Sobolev space $H^{\frac n2-1}(\mathbb R^n)\times H^{\frac n2}(\mathbb R^n)$ is critical for the system. We show that if a weak solution $(u(t),b(t))$ is in $H^{s}(\mathbb R^n)\times H^{s+1}(\mathbb R^n)$ with $s>\frac n2-1$ at a certain time $t_0$, then it will stay in the space for a short time, provided the initial velocity $u(0)\in H^s(\mathbb R^n)$. In the case that the uniqueness of weak solution in $H^{s}(\mathbb R^n)\times H^{s+1}(\mathbb R^n)$ is known, the assumption of $u(0)\in H^s(\mathbb R^n)$ is not necessary.