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

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

2009-01-18

arxiv.org/abs/0901.2683v1

Mathematics

Analysis of PDEs

to appear in DCDS-A, 2009

Scientific paper

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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