On the $L^{\infty }$ stability of Prandtl expansions in Gevrey class

Abstract

In this paper, we prove the $L^{\infty }\cap L^2$ stability of Prandtl expansions of shear flow type as $(U(y/\sqrt{\nu }),0)$ for the initial perturbation in the Gevrey class, where $U(y)$ is a monotone and concave function and $\nu$ is the viscosity coefficient. To this end, we develop the direct resolvent estimate method for the linearized Orr-Sommerfeld operator instead of the Rayleigh-Airy iteration method introduced by Grenier, Guo and Nguyen.