We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with some of the old ideas to overcome the difficulty of the loss of derivatives in the system.

This shows that the three dimensional instabilities in the system leading to ill-posedness are not worse than the two dimensional ones. This is a joint work with Wei-xi Li and Nader Masmoudi.