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 $-\nu (-△{)}^{\alpha }u$ and $-\kappa (-△{)}^{\beta }b$. We show that smooth solutions are global in the following three cases: $\alpha ⩾1/2,\beta ⩾1$; $0⩽\alpha <1/2,2\alpha +\beta >2$; $\alpha ⩾2,\beta =0$. 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.

19 pages. Journal of Differential Equations, to appear

References in corpus (1)

• Nonlinear maximum principles for dissipative linear nonlocal operators and applications