Communications on Pure and Applied Analysis (CPAA)

Global existence of strong solutions to incompressible MHD

Pages: 1553 - 1561, Volume 13, Issue 4, July 2014      doi:10.3934/cpaa.2014.13.1553

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Huajun Gong - The Institute of Mathematical Sciences, University of Science and Technology of China, Anhui, 230026, China (email)
Jinkai Li - The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong (email)

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:  Incompressible MHD, global existence and uniqueness, strong solutions.
Mathematics Subject Classification:  Primary: 35Q35, 35B65; Secondary: 76D05.

Received: September 2013;      Revised: November 2013;      Available Online: February 2014.