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 $H^{\frac{n}{2}-1}({\mathbb{R}}^n)\times H^{\frac{n}{2}}({\mathbb{R}}^n)$ 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 $s>\frac{n}{2}-1$ 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.

References in corpus (1)

• Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces