Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values

Abstract

This paper is concerned with the weak solution theory for the MHD system with large $L^3$-initial data. Due to the fact that the natural boundary condition on the magnetic field $H$ is the slip boundary condition, the Leray-Schauder fixed-point theorem, which have used to investigate the weak solution theory of the Navier-Stokes system, becomes invalid. To address such difficulty, we will invoke the Leray's approximation technique and the perturbation theory to seek a global weak solution to the Cauchy problem for MHD equations with large $L^3$-initial data. Our strategy provides a simple alternative (self-contained) proof of weak $L^3$-solution theory of incompressible Navier-Stokes system. Moreover, this weak solution is unique under some restrictions.