Long time well-posedness of compressible magnetohydrodynamics boundary layer equations in Sobolev space

TL;DR

This paper proves long-term well-posedness for 2D compressible MHD boundary layer equations with small initial perturbations, showing solutions exist for a time scale inversely proportional to a power of the perturbation size.

Contribution

It establishes the long-time existence of solutions in Sobolev space for small perturbations around steady states, extending previous local well-posedness results.

Findings

Solutions exist for time > ε^(-4/3)

Small perturbations lead to long-term stability

Provides rigorous mathematical foundation for boundary layer behavior

Abstract

In this paper we consider the long time well-posedness of solutions to two dimensional compressible magnetohydrodynamics (MHD) boundary layer equations. When the initial data is a small perturbation of a steady solution with size of $\varepsilon$ and the far-field state is also a small perturbation around such a steady solution in Sobolev space, then the lifespan of solutions is proved to be greater than $\varepsilon^{-\frac43}$.