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

Qi Chen; Di Wu; Zhifei Zhang

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

Analysis of PDEs 2021-01-05 v3

Authors: Qi Chen , Di Wu , Zhifei Zhang

Abstract

In this paper, we prove the $L^\infty\cap L^2$ stability of Prandtl expansions of shear flow type as $\big(U(y/\sqrt{\nu}),0\big)$ 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.

Keywords

Prandtl expansions L^\infty stability Gevrey class Orr-Sommerfeld operator resolvent estimate shear flow

Cite

@article{arxiv.2004.09755,
  title  = {On the $L^\infty$ stability of Prandtl expansions in Gevrey class},
  author = {Qi Chen and Di Wu and Zhifei Zhang},
  journal= {arXiv preprint arXiv:2004.09755},
  year   = {2021}
}

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