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

We study the well-posedness of the Prandtl system without monotonicity and analyticity assumption. Precisely, for any index sigma is an element of invertef right perpendicular3/2, 2inverted left perpendicular, 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 an open question raised by D. Gerard-Varet and N. Masmoudi [Ann. Sci. Ecole Norm. Sup. (4) 48 (2015), 1273-1325], who solved the case sigma = 7/4.