Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rᵈ , where d = 2, 3, with initial data B₀∈ Hˢ(Rᵈ) and u₀∈ Hs-1+ε(Rᵈ) 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∈ Hˢ⁻¹ to satisfy u∈ L¹(0,T; Hˢ⁺¹) ; we provide an explicit example of this phenomenon.