Well-posedness of the Prandtl equation without any structural assumption

TL;DR

This paper proves local well-posedness for the Prandtl equation with Gevrey 2 regularity in x and H^1 in y, without structural assumptions on initial data, extending the understanding of its mathematical properties.

Contribution

It establishes the first well-posedness result for the Prandtl equation without any structural assumptions on initial data, using optimal regularity conditions.

Findings

Proves local well-posedness for initial data with Gevrey 2 regularity in x and H^1 in y.

No structural assumptions such as monotonicity are needed.

Result is optimal considering known ill-posedness results.

Abstract

We show the local in time well-posedness of the Prandtl equation for data with Gevrey $2$ regularity in $x$ and $H^1$ regularity in $y$. The main novelty of our result is that we do not make any assumption on the structure of the initial data: no monotonicity or hypothesis on the critical points. Moreover, our general result is optimal in terms of regularity, in view of the ill-posedness result of [9].