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

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 $\epsilon$ 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 ${\epsilon }^{-\frac{4}{3}}$.