Global Regularity of 2D almost resistive MHD Equations

TL;DR

This paper proves the global smoothness of solutions to 2D almost resistive MHD equations with very weak dissipation, advancing understanding of the equations' regularity under minimal dissipative conditions.

Contribution

It establishes the global regularity of 2D almost resistive MHD equations with dissipative operators weaker than any fractional Laplacian power, improving previous results.

Findings

Proved global regularity under minimal dissipation conditions

Extended the class of dissipative operators ensuring smooth solutions

Improved upon previous regularity results for 2D generalized MHD equations

Abstract

Whether or not the solution to 2D resistive MHD equations is globally smooth remains open. This paper establishes the global regularity of solutions to the 2D almost resistive MHD equations, which require the dissipative operators $\mathcal{L}$ weaker than any power of the fractional Laplacian. The result is an improvement of the one of Fan et al. (Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., 175 (2014), pp. 127-131) which ask for $\alpha>0, \beta=1$.