Well-posedness for the Prandtl system without analyticity or monotonicity

TL;DR

This paper proves local well-posedness of the Prandtl system for Gevrey class 7/4 data in the horizontal variable, removing the need for analyticity or monotonicity assumptions, using new energy estimates.

Contribution

It establishes well-posedness for the Prandtl system with Gevrey class 7/4 data, extending beyond previous analyticity requirements.

Findings

Well-posedness for Gevrey class 7/4 data in the Prandtl system.

Introduction of new energy estimates based on non-quadratic functionals.

Improvement over classical results requiring analyticity in the horizontal variable.

Abstract

It has been thought for a while that the Prandtl system is only well-posed under the Oleinik monotonicity assumption or under an analyticity assumption. We show that the Prandtl system is actually locally well-posed for data that belong to the Gevrey class 7/4 in the horizontal variable x. Our result improves the classical local well-posedness result for data that are analytic in x (that is Gevrey class 1). The proof uses new estimates, based on non-quadratic energy functionals.