# The vanishing viscosity limit for 2D Navier-Stokes in a rough domain

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 $x_2 = \varepsilon^{1+\alpha} \eta(x_1/\varepsilon)$, $\alpha > 0$. Under suitable conditions on the oscillating parameter $\varepsilon$ and the viscosity $\nu$, we show that solutions of the Navier-Stokes system converge to solutions of the Euler system in the vanishing limit of both $\nu$ and $\varepsilon$. The main issue is that the curvature of the boundary is unbounded as $\varepsilon \rightarrow 0$, 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.
Type de document :
Pré-publication, Document de travail
IF_PREPUB. 2017
Domaine :

https://hal.archives-ouvertes.fr/hal-01451399
Contributeur : Christophe Lacave <>
Soumis le : mercredi 1 février 2017 - 09:51:11
Dernière modification le : jeudi 20 juillet 2017 - 09:27:22

### Identifiants

• HAL Id : hal-01451399, version 1
• ARXIV : 1701.08808

### Citation

David Gérard-Varet, Christophe Lacave, Toan T. Nguyen, Frédéric Rousset. The vanishing viscosity limit for 2D Navier-Stokes in a rough domain. IF_PREPUB. 2017. <hal-01451399>

### Métriques

Consultations de la notice