Propagation of regularity for the MHD system in optimal Sobolev space

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.