# Global $C_{∞}$ regularity of the steady Prandtl equation with favorable pressure gradient

### Yue Wang

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China### Zhifei Zhang

School of Mathematical Sciences, Peking University, 100871, Beijing, China

## 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).

## Cite this article

Yue Wang, Zhifei Zhang, Global $C_{∞}$ regularity of the steady Prandtl equation with favorable pressure gradient. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 6, pp. 1989–2004

DOI 10.1016/J.ANIHPC.2021.02.007