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

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.