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 x2=ε1+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 language | French |
|---|---|
| Pages (from-to) | 45-84 |
| Number of pages | 40 |
| Journal | Journal des Mathematiques Pures et Appliquees |
| Volume | 119 |
| DOIs | |
| State | Published - Nov 1 2018 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
- Applied Mathematics
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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 journal › Article
TY - JOUR
T1 - La limite faible viscosité pour Navier–Stokes 2D dans un domaine rugueux
AU - Gérard-Varet, David
AU - Lacave, Christophe
AU - Nguyen, Toan
AU - Rousset, Frédéric
PY - 2018/11/1
Y1 - 2018/11/1
N2 - 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 x2=ε1+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.
AB - 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 x2=ε1+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.
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UR - http://www.scopus.com/inward/citedby.url?scp=85053130531&partnerID=8YFLogxK
U2 - 10.1016/j.matpur.2017.10.009
DO - 10.1016/j.matpur.2017.10.009
M3 - Article
AN - SCOPUS:85053130531
VL - 119
SP - 45
EP - 84
JO - Journal des Mathematiques Pures et Appliquees
JF - Journal des Mathematiques Pures et Appliquees
SN - 0021-7824
ER -