Global regularity of solutions to the 2D steady compressible Prandtl equations

In this paper, we study the global C-infinity regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary y = 0 for any x > 0.