We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for three-dimensional Prandtl system without any structural assumption. The proof combines a novel way new cancellation some old ideas to overcome difficulty loss derivatives system. This shows that instabilities leading ill-posedness are not worse than two-dimensional ones. © 2021 Wiley Periodicals LLC.

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has damping term due to effect of Hartmann boundary layer. A global-in-time well-posedness is obtained Gevrey function space with optimal index 2. The proof based on cancellation mechanism through some auxiliary functions study equation and an observation about structure loss one order tangential derivatives twice ...