On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

Abstract

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

19 pages. Journal of Differential Equations, to appear