The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions

Wang, Ya-Guang; Williams, Mark

Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2257-2314.

Nous étudions des solutions avec couches limites des équations de Navier-Stokes compressibles isentropiques avec des conditions de frottement de Navier au bord, lorsque la constante de viscosité figurant dans l’équation sur la quantité de mouvement est proportionnelle à un petit paramètre $ϵ$. Ces conditions aux limites sont caractéristiques pour le problème non visqueux sous-jacent, le système d’ équations d’Euler compressibles.

Les conditions aux limites impliquent que la vitesse au bord est proportionnelle à la composante tangentielle des contraintes. La composante normale de la vitesse est nulle au bord. Nous construisons tout d’abord une solution approchée à un ordre élevé de la solution, décrivant la présence d’une couche limite. La contribution principale de la couche limite apparait dans la composante tangentielle de la vitesse, est de taille $\sqrt{ϵ}$ et d’amplitude $O\left(\sqrt{ϵ}\right)$. Nous prouvons ensuite que cette solution approchée est effectivement asymptotique à la solution exacte, sur un intervalle de temps indépendant de $ϵ$. Un corollaire immédiat est que la solution des équations de Navier-Stokes converge dans $L^{\infty }$, lorsque la viscosité tend vers $0$, vers la solution du système d’Euler compressible avec composante normale de la vitesse nulle au bord.

We study boundary layer solutions of the isentropic, compressible Navier-Stokes equations with Navier-friction boundary conditions when the viscosity constants appearing in the momentum equation are proportional to a small parameter $ϵ$. These boundary conditions are characteristic for the underlying inviscid problem, the compressible Euler equations.

The boundary condition implies that the velocity on the boundary is proportional to the tangential component of the stress. The normal component of velocity is zero on the boundary. We first construct a high-order approximate solution that exhibits a boundary layer. The main contribution to the layer appears in the tangential velocity and is of width $\sqrt{ϵ}$ and amplitude $O\left(\sqrt{ϵ}\right)$. Next we prove that the approximate solution stays close to the exact Navier-Stokes solution on a fixed time interval independent of $ϵ$. As an immediate corollary we show that the Navier-Stokes solution converges in $L^{\infty }$ in the small viscosity limit to the solution of the compressible Euler equations with normal velocity equal to zero on the boundary.

Classification : 76N20, 76N17
Keywords: characteristic boundary layers, compressible Navier-Stokes equations, Navier boundary conditions, inviscid limit
Mot clés : couches limites caractéristiques, équations de Navier-Stokes compressibles, conditions de frottement de Navier au bord, limite non visqueuse

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