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arXiv · 1302.6633

On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

Chuong V. Tran·Xinwei Yu·Zhichun Zhai
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Abstract

In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- \nu (- \triangle)^{\alpha} u$ and $- \kappa (-\triangle)^{\beta} b$. We show that smooth solutions are global in the following three cases: $\alpha \geqslant 1 / 2, \beta \geqslant 1$; $0 \leqslant \alpha < 1 / 2, 2 \alpha + \beta > 2$; $\alpha \geqslant 2, \beta = 0$. We also show that in the inviscid case $\nu = 0$, if $\beta > 1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.

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Chuong V. Tran, Xinwei Yu, Zhichun Zhai. 2013-02-27. On Global Regularity of 2D Generalized Magnetohydrodynamic Equations. https://arxiv.org/abs/1302.6633

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