Hydrostatic approximation of the 2D MHD system in a thin strip with a small analytic data

TL;DR

This paper proves the global well-posedness of a 2D anisotropic MHD system with small analytic initial data in a strip and justifies the hydrostatic limit as the strip height approaches zero, connecting it to a Prandtl-like system.

Contribution

It establishes the global existence of solutions for the 2D MHD system with analytic data and rigorously derives the hydrostatic limit in a thin strip setting.

Findings

Global well-posedness for small analytic initial data

Justification of the hydrostatic limit as strip height tends to zero

Connection to a Prandtl-like system in the limit

Abstract

In this paper, we study the global well-posedness of the two dimensional rescaled anisotropic MHD system and also to the hydrostatic MHD with no slip boundary condition in a strip, for small initial data, which are analytic in the horizontal variable. We also justify the limit of the MHD system (when $\epsilon\to 0$ where $\epsilon$ denotes the height of the initial strip) is given by a couple of a Prandtl like system for the velocity and a magnetic field equation when the initial data are small and analytic.