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

Abstract

We study the well-posedness of the Prandtl system without monotonicity and analyticity assumption. Precisely, 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 by D. Gérard-Varet and N. Masmoudi [Ann. Sci. École Norm. Sup. (4) 48 (2015), no. 6, 1273–1325], who solved the case $\sigma =7/4$.