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metric spaces
BKM's criterion and global weak solutions
for magnetohydrodynamics with zero viscosity
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 $H^1(\mathbb{R}^2)$.