Local existence for the non-resistive MHD equations in Besov spaces

Abstract

In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of Rn, n=2,3, for divergence-free initial data in certain Besov spaces, namely u0∈B2,1n/2−1 and B0∈B2,1n/2. The a priori estimates include the term ∫0t‖u(s)‖Hn/22ds on the right-hand side, which thus requires an auxiliary bound in Hn/2−1. In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in H1/2 is required, which we prove using the splitting method of Calderón (1990) [2]. By contrast, our proof that such solutions are unique only applies to the 3D case.