BKM's criterion and global weak solutionsfor magnetohydrodynamics with zero viscosity

Zhen Lei; Yi Zhou

doi:10.3934/dcds.2009.25.575

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2009, Volume 25, Issue 2: 575-583. Doi: 10.3934/dcds.2009.25.575

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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

Received: October 2008

Revised: January 2009

Published: June 2009

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Abstract

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 $H^1(\mathbb{R}^2)$.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q60.

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