On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows

TL;DR

This paper demonstrates the nonlinear instability of Prandtl's boundary layer expansions for Sobolev solutions and shows that stable monotonic shear flows become unstable when viscosity is small but positive.

Contribution

It proves the invalidity of Prandtl's expansion in Sobolev spaces and establishes the nonlinear instability of stable shear flows at small positive viscosity.

Findings

Prandtl's expansion fails in Sobolev regularity.

Monotonic boundary layer profiles are nonlinearly unstable for small viscosity.

The instability persists up to $O( u^{1/4})$ in $L^ abla$ norm.

Abstract

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to $0$. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to $O(\nu^{1/4})$ order terms in $L^\infty$ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when $\nu = 0$, are nonlinearly unstable when $\nu > 0$, provided $\nu$ is small enough, up to $O(\nu^{1/4})$ terms in $L^\infty$ norm.