$L^\infty$ instability of Prandtl layers

TL;DR

This paper demonstrates that for certain shear flows unstable in Euler equations, the classical boundary layer approximation fails in Sobolev spaces, and such flows are nonlinearly unstable in Navier-Stokes as viscosity diminishes.

Contribution

It proves the failure of Prandtl's boundary layer ansatz in Sobolev regularity for unstable shear flows and establishes nonlinear instability of these flows in Navier-Stokes.

Findings

Prandtl's boundary layer approximation fails for Sobolev solutions of unstable shear flows.

Unstable shear flows in Euler are nonlinearly unstable in Navier-Stokes at low viscosity.

The result contrasts with the analytic case where the approximation holds.

Abstract

In $1904$, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to $0$. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$ in the inviscid limit. In this paper we prove that, for a class of smooth solutions of Navier Stokes equations, namely for shear layer profiles which are unstable for Rayleigh equations, this Ansatz is false if we consider solutions with Sobolev regularity, in strong contrast with the analytic case, pioneered by R.E. Caflisch and M. Sammartino \cite{SammartinoCaflisch1,SammartinoCaflisch2}. Meanwhile we address the classical problem of the nonlinear stability of shear layers near a boundary and prove that if a shear flow is spectrally unstable for Euler equations, then it is non linearly unstable for the Navier Stokes equations provided the viscosity is small enough.