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BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity. (English) Zbl 1171.35452
Summary: 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 \(H^1(\mathbb{R}^2)\).

MSC:
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
76W05 Magnetohydrodynamics and electrohydrodynamics
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