Global existence of strong solutions to incompressible MHD

Huajun Gong; Jinkai Li

doi:10.3934/cpaa.2014.13.1553

Article Contents

2014, Volume 13, Issue 4: 1553-1561. Doi: 10.3934/cpaa.2014.13.1553

This issue Previous Article General decay estimates for a Cauchy viscoelastic wave problem Next Article Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data

Global existence of strong solutions to incompressible MHD

Huajun Gong^1, and
Jinkai Li^2,

1.
The Institute of Mathematical Sciences, University of Science and Technology of China, Anhui, 230026
2.
The Institute of Mathematical Sciences, The Chinese University of Hong Kong

Received: September 2013

Revised: November 2013

Published: July 2015

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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.

Keywords:

Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D05.

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