Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points

TL;DR

This paper establishes local well-posedness for the Prandtl equations with non-degenerate critical points in Gevrey spaces, extending previous results to a broader class of regularity indices without requiring monotonicity.

Contribution

It proves well-posedness in Gevrey class $G^\sigma$ for $\sigma ext{ in } [3/2, 2]$, solving an open problem for the case $\sigma=7/4$.

Findings

Well-posedness in Gevrey spaces for $\sigma ext{ in } [3/2, 2]$

No monotonicity condition needed for tangential velocity

Extension of previous results to a wider regularity range

Abstract

In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index $\sigma\in[3/2, 2],$ we obtain the local in time well-posedness in the space of Gevrey class $G^\sigma$ in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. G\'{e}rard-Varet and N. Masmoudi [{\it Ann. Sci. \'{E}c. Norm. Sup\'{e}r}. (4) 48 (2015), no. 6, 1273-1325], in which the case $\sigma=7/4$ is solved.