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 R-d, where d = 2, 3, with initial data B-0 is an element of H-s(R-d) and u(0) is an element of Hs-1+epsilon(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) is an element of H(s-1)t o satisfy u is an element of L-1 (0, T; Hs+1); we provide an explicit example of this phenomenon.