Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

Abstract

This paper examines the uniqueness of weak solutions to the ddimensional magnetohydrodynamic (MHD) equations with the fractional dissipation (−∆) α u and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case α = 1. This paper discovers that there are new phenomena with the case α < 1. The approach for α = 1 can not be directly extended to α < 1. We establish that, for α < 1, any initial data(R d ) guarantees the existence and uniqueness. These regularity requirements appear to be optimal.