BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity. (English) Zbl 1171.35452

Lei, Zhen; Zhou, Yi

Discrete Contin. Dyn. Syst. 25, No. 2, 575-583 (2009).

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

Keywords:

Beale-Kato-Majda’s criterion; weak solutions; magnetohydrodynamics; zero viscosity