BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity

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

to appear in DCDS-A, 2009

References in corpus (1)

• The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations

Cited by in corpus (2)

• Two regularity criteria for the 3D MHD equations
• Remarks on the Blowup Criteria for Oldroyd Models