Global regularity for the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion

TL;DR

This paper proves the global regularity of solutions to the 2D magnetohydrodynamics equations with only horizontal dissipation and magnetic diffusion, using anisotropic analysis techniques, and also examines a simplified climate model.

Contribution

It establishes the global regularity for the 2D MHD system with horizontal dissipation and diffusion, and analyzes the asymptotic behavior of solutions, extending understanding in anisotropic PDEs.

Findings

Global regularity of classical solutions under small initial data

Asymptotic behavior of solutions analyzed

Existence and uniqueness for a simplified climate model

Abstract

This paper establishes the global regularity of classical solution to the 2D MHD system with only horizontal dissipation and horizontal magnetic diffusion in a strip domain $\mathbb{T}\times\mathbb{R}$ when the initial data is suitable small. To prove this, we combine the Littlewood-Paley decomposition with anisotropic inequalities to establish a crucial commutator estimate. We also analysis the asymptotic behavior of the solution. In addition, the global existence and uniqueness of classical solution is obtained for the 2D simplified tropical climate model with only horizontal dissipations.