Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow

TL;DR

This paper establishes the well-posedness of the linearized Prandtl equation around non-monotonic shear flows within a specific Gevrey class, nearly matching the known ill-posedness results and advancing understanding of boundary layer equations.

Contribution

It proves the well-posedness of the linearized Prandtl equation around non-monotonic shear flows in Gevrey class 2−θ, nearly optimal compared to previous ill-posedness results.

Findings

Well-posedness established in Gevrey class 2−θ for non-monotonic shear flows

Nearly optimal result matching ill-posedness growth rates

Advances theoretical understanding of boundary layer stability

Abstract

In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class $2-\theta$ for any $\theta>0$. This result is almost optimal by the ill-posedness result proved by G\'{e}rard-Varet and Dormy, who construct a class of solution with the growth like $e^{\sqrt{k}t}$ for the linearized Prandtl equation around a non-monotonic shear flow.