Global $C^{\infty }$ 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).