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

Abstract

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).