Higher order commutator estimates and local existence for the non-resistive MHD equations and related models

Abstract

This paper establishes the local-in-time existence and uniqueness of strong solutions in H-s for s > n/2 to the viscous, non-resistive magnetohydrodynamics (MUD) equations in R-n, n = 2,3, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato and Ponce (1988) (Comm. Pure Appl. Math. 41(7), 891–907, 1988).