On the global regularity of two-dimensional generalized magnetohydrodynamics system

TL;DR

This paper proves that for a 2D generalized magnetohydrodynamics system with fractional dissipation, solutions remain smooth globally if the dissipation power exceeds 1/3, extending previous results that required higher dissipation.

Contribution

The study demonstrates that the regularity of solutions holds under weaker dissipation conditions than previously established, specifically for the case when the diffusion power is exactly 1.

Findings

Solutions are globally smooth for ta > 1/3 when eta=1.

Extends previous regularity results to lower dissipation powers.

Provides new insights into the critical dissipation threshold for 2D MHD systems.

Abstract

We study the two-dimensional generalized magnetohydrodynamics system with dissipation and diffusion in terms of fractional Laplacians. In particular, we show that in case the diffusion term has the power $\beta = 1$, in contrast to the previous result of $\alpha \geq \frac{1}{2}$, we show that $\alpha > \frac{1}{3}$ suffices in order for the solution pair to remain smooth for all time.