Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum

Abstract

In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic magnetohydrodynamic flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in L2, and that the initial density and the gradients of initial velocity/magnetic field are bounded in L∞ and L2, respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy.