On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

TL;DR

This paper proves the global regularity of solutions for 2D generalized magnetohydrodynamic equations under various dissipation conditions, extending understanding of when solutions remain smooth over time.

Contribution

It establishes new conditions on dissipation exponents ensuring global regularity of 2D GMHD solutions, including inviscid cases with magnetic field smoothness.

Findings

Global regularity for $eta > 1$ in inviscid case

Smooth solutions are global for $eta eq 0$ with certain $eta$ and $eta$ ranges

Conditions on $ u$, $eta$, and $eta$ ensure solution smoothness

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.