Well-Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption

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.