Global $C^∞$ regularity of the steady Prandtl equation with favorable pressure gradient

In the case of favorable pressure gradient , Oleinik obtained the global-in- x solutions to the steady Prandtl equations with low regularity (see Oleinik and Samokhin [9], P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in- x higher regularity established by Guo and Iyer [5]. In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).