# Sublayer of Prandtl boundary layers

## TL;DR

This paper explores the stability relationship between Prandtl boundary layers and their sublayers in the vanishing viscosity limit, revealing that both cannot be simultaneously stable in the supremum norm.

## Contribution

It establishes a link between the stability of Prandtl layers and their viscous sublayers, showing mutual instability in the nonlinear regime.

## Key findings

- Prandtl boundary layer and sublayer cannot both be nonlinearly stable in $L^
ablafty$.
- The instability of one layer implies the instability of the other.
- The work connects boundary layer stability to sublayer behavior in viscous flows.

## Abstract

The aim of this paper is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: $\nu \to 0$. In \cite{Grenier}, one of the authors proved that there exists no asymptotic expansion involving one Prandtl's boundary layer with thickness of order $\sqrt\nu$, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order $\nu^{3/4}$. In this paper, we point out how the stability of the classical Prandtl's layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in $L^\infty$. That is, either the Prandtl's layer or the boundary sublayer is nonlinearly unstable in the sup norm.