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

2016/02/08 by Charles L. Fefferman, David S. McCormick, James C. Robinson +1 · 1 citation
Mathematics · #math.AP

paper · pdf · doi:10.1007/s00205-016-1042-7

arxiv created 2016/02/08 · arxiv updated 2016/09/21

Abstract

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in ℝ^d, d=2,3, with initial data B₀∈ H^s(ℝ^d) and u₀∈ H^s-1+ε(ℝ^d) for s>d/2 and any 0<ε<1. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ε=0 is explained by the failure of solutions of the heat equation with initial data u₀∈ H^s-1 to satisfy u∈ L¹(0,T;H^s+1); we provide an explicit example of this phenomenon.

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