Gevrey well-posedness of the hyperbolic Prandtl equations

TL;DR

This paper proves the well-posedness of 2D and 3D degenerate hyperbolic Prandtl equations in the Gevrey class with index up to 2, using an elementary energy estimate without classical cancellation techniques.

Contribution

It establishes Gevrey well-posedness for hyperbolic Prandtl equations without relying on structural assumptions or classical cancellation methods.

Findings

Gevrey well-posedness with index ≤ 2 for hyperbolic Prandtl equations

Elementary $L^2$ energy estimates suffice for the proof

No structural assumptions or cancellation mechanisms needed

Abstract

We study the 2D and 3D Prandtl equations of degenerate hyperbolic type, and establish without any structural assumption the Gevrey well-posedness with Gevrey index $\leq 2$. Compared with the classical parabolic Prandtl equations, the loss of the derivatives, caused by the hyperbolic feature coupled with the degeneracy, can't be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart. Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature, we give in this text a straightforward proof, basing on an elementary $L^2$ energy estimate. In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.