Well-posedness in Gevrey function space for the three-dimensional Prandtl equations

In the paper, we study the three-dimensional Prandtl equations without any monotonicity condition on the velocity field. We prove that when one tangential component of the velocity field has a single curve of non-degenerate critical points with respect to the normal variable, the system is locally well-posed in the Gevrey function space with Gevrey index in $]1, 2].$ The proof is based on some new observation of cancellation mechanism in the three space dimensional system in addition to those in the two-dimensional setting obtained in [1,7,19,22].