Global Existence of Strong Solutions to Incompressible MHD

Abstract

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for incompressible MHD equations in a bounded smooth domain of three spatial dimensions with initial density being allowed to have vacuum, in particular, the initial density can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $|{\sqrt{\rho }}_0{u}_0{|}_{L^2(\Omega )}^2+|H_0{|}_{L^2(\Omega )}^2$ and $|\nabla u_0{|}_{L^2(\Omega )}^2+|\nabla H_0{|}_{L^2(\Omega )}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.

arXiv comment10 pages. Communications on Pure and Applied Analysis, 2014