La limite faible viscosité pour Navier–Stokes 2D dans un domaine rugueux

Translated title of the contribution: The vanishing viscosity limit for 2D Navier–Stokes in a rough domain

David Gérard-Varet, Christophe Lacave, Toan Nguyen, Frédéric Rousset

Research output: Contribution to journalArticle

Abstract

We study the high Reynolds number limit of a viscous fluid in the presence of a rough boundary. We consider the two-dimensional incompressible Navier–Stokes equations with Navier slip boundary condition, in a domain whose boundaries exhibit fast oscillations in the form x21+αη(x1/ε), α>0. Under suitable conditions on the oscillating parameter ε and the viscosity ν, we show that solutions of the Navier–Stokes system converge to solutions of the Euler system in the vanishing limit of both ν and ε. The main issue is that the curvature of the boundary is unbounded as ε→0, which precludes the use of standard methods to obtain the inviscid limit. Our approach is to first construct an accurate boundary layer approximation to the Euler solution in the rough domain, and then to derive stability estimates for this approximation under the Navier–Stokes evolution.

Original languageFrench
Pages (from-to)45-84
Number of pages40
JournalJournal des Mathematiques Pures et Appliquees
Volume119
DOIs
StatePublished - Nov 1 2018

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Vanishing Viscosity
Navier-Stokes
Rough
Viscosity
Inviscid Limit
Incompressible Navier-Stokes
Euler System
Slip Boundary Condition
Navier-Stokes System
Stability Estimates
Approximation
Viscous Fluid
Reynolds number
Euler
Boundary Layer
Navier-Stokes Equations
Boundary layers
Curvature
Boundary conditions
Oscillation

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

Gérard-Varet, David ; Lacave, Christophe ; Nguyen, Toan ; Rousset, Frédéric. / La limite faible viscosité pour Navier–Stokes 2D dans un domaine rugueux. In: Journal des Mathematiques Pures et Appliquees. 2018 ; Vol. 119. pp. 45-84.
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La limite faible viscosité pour Navier–Stokes 2D dans un domaine rugueux. / Gérard-Varet, David; Lacave, Christophe; Nguyen, Toan; Rousset, Frédéric.

In: Journal des Mathematiques Pures et Appliquees, Vol. 119, 01.11.2018, p. 45-84.

Research output: Contribution to journalArticle

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