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

Abstract

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in ${\mathbb{R}}^d$, $d=2,3$, with initial data $B_0\in H^s({\mathbb{R}}^d)$ and $u_0\in H^{s-1+\epsilon }({\mathbb{R}}^d)$ for $s>d/2$ and any $0<\epsilon <1$. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking $\epsilon =0$ is explained by the failure of solutions of the heat equation with initial data $u_0\in H^{s-1}$ to satisfy $u\in L^1(0,T;H^{s+1})$; we provide an explicit example of this phenomenon.

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