A remark on the two-dimensional magneto-hydrodynamics-alpha system

TL;DR

This paper proves that solutions to the 2D magneto-hydrodynamics-alpha system with fractional Laplacians maintain their initial regularity globally when the dissipative and diffusive powers sum to one, resolving a previously noted open problem.

Contribution

It establishes the global regularity of solutions for the 2D MHD-alpha system with fractional Laplacians under a specific condition on the powers, addressing a key open question.

Findings

Solutions preserve initial regularity when powers sum to one

Global regularity is achieved in the general case

Addresses a previously unresolved problem in the field

Abstract

We study the two-dimensional generalized magnetohydrodynamics-$\alpha$ system with fractional Laplacians in the dissipative and diffusive terms. We show that the solution pair of velocity and magnetic fields preserves their initial regularity in all cases when the powers add up to one. This settles the global regularity issue in the general case which was remarked by the authors in [33] to be a problem.