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July  2009, 25(2): 575-583. doi: 10.3934/dcds.2009.25.575

BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity

Zhen Lei 1, and  Yi Zhou 1,

1. 

Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, China

Received  October 2008 Revised  January 2009 Published  June 2009

In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.
Keywords: weak solutions, zero viscosity., magnetohydrodynamics, Beale-Kato-Majda's criterion.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q6.
Citation: Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575
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