Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1139–1151 | Cite as

Sublayer of Prandtl Boundary Layers

  • Emmanuel Grenier
  • Toan T. Nguyen


The aim of this paper is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit \({\nu \to 0}\). In Grenier (Commun Pure Appl Math 53(9):1067–1091, 2000), one of the authors proved that there exists no asymptotic expansion involving one of 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.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Equipe Projet Inria NUMED, INRIA Rhône AlpesUnité de Mathématiques Pures et Appliquées., UMR 5669, CNRS et École Normale Supérieure de LyonLyon Cedex 07France
  2. 2.Department of MathematicsPenn State UniversityState CollegeUSA

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