Global existence of strong solutions to incompressible MHD

Huajun Gong,Jinkai Li

doi:10.3934/cpaa.2014.13.1553

Abstract

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it 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.

Highlights

Magnetohydrodynamics (MHD for short) is the study of the interaction between magnetic fields and moving conducting fluids, which can be described by the following system ρt + div(ρu) = 0, ρ(ut + (u · ∇)u) − μ∆u + ∇p = (∇ × H) × H, Ht − λ∆H = ∇ × (u × H), divu = divH = 0, (1.1) (1.2) (1.3) (1.4)
The aim of this paper is to prove the global existence and uniqueness of strong solutions to system (1.1)–(1.6) with the initial data being allowed to have vacuum
Recall the boundary condition (1.6) and the identity ∆H = ∇divH − ∇ × (∇ × H), it follows from integrating by parts that

Summary

Introduction

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. Incompressible MHD; global existence and uniqueness; strong solutions.

Objectives

Results

Conclusion