Long time well-posedness of compressible magnetohydrodynamic boundary layer equations in Sobolev spaces

In this paper we consider the long time well-posedness of solutions to two-dimensional compressible magnetohydrodynamic (MHD) boundary layer equations. When the initial data is a small perturbation of a steady solution with size of $ \varepsilon $, then the lifespan of solutions in Sobolev spaces is proved to be greater than $ \varepsilon^{-\frac43} $. And such a result can be extended to the case that both initial data and far-field state are small perturbations around the steady states. Moreover, it holds true for both isentropic and non-isentropic magnetohydrodynamic boundary layer equations.