Global Existence of Strong Solutions to Incompressible MHD

TL;DR

This paper proves the global existence and uniqueness of strong solutions for the incompressible MHD equations in three-dimensional bounded domains, allowing initial vacuum states and small initial data conditions.

Contribution

It establishes the first global strong solution results for incompressible MHD with vacuum initial density in bounded domains.

Findings

Global existence and uniqueness of strong solutions

Initial vacuum density allowed in the model

Solutions exist under small initial data conditions

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 $|\sqrt\rho_0u_0|_{L^2(\Omega)}^2+|H_0|_{L^2(\Omega)}^2$ 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.