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

Ya-Guang Wang[1]; Mark Williams[2]

• [1] Department of Mathematics, Shanghai Jiao Tong University 200240 Shanghai, China
• [2] Department of Mathematics, University of North Carolina at Chapel Hill NC 27599, USA
• Volume: 62, Issue: 6, page 2257-2314
• ISSN: 0373-0956

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## Abstract

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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.

## How to cite

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Wang, Ya-Guang, and Williams, Mark. "The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions." Annales de l’institut Fourier 62.6 (2012): 2257-2314. <http://eudml.org/doc/251143>.

@article{Wang2012,
abstract = {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 $\epsilon$. 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 $\sqrt\{\epsilon \}$ and amplitude $O(\sqrt\{\epsilon \})$. Next we prove that the approximate solution stays close to the exact Navier-Stokes solution on a fixed time interval independent of $\epsilon$. 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.},
affiliation = {Department of Mathematics, Shanghai Jiao Tong University 200240 Shanghai, China; Department of Mathematics, University of North Carolina at Chapel Hill NC 27599, USA},
author = {Wang, Ya-Guang, Williams, Mark},
journal = {Annales de l’institut Fourier},
keywords = {characteristic boundary layers; compressible Navier-Stokes equations; Navier boundary conditions; inviscid limit},
language = {eng},
number = {6},
pages = {2257-2314},
publisher = {Association des Annales de l’institut Fourier},
title = {The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions},
url = {http://eudml.org/doc/251143},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Wang, Ya-Guang
AU - Williams, Mark
TI - The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2257
EP - 2314
AB - 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 $\epsilon$. 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 $\sqrt{\epsilon }$ and amplitude $O(\sqrt{\epsilon })$. Next we prove that the approximate solution stays close to the exact Navier-Stokes solution on a fixed time interval independent of $\epsilon$. 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.
LA - eng
KW - characteristic boundary layers; compressible Navier-Stokes equations; Navier boundary conditions; inviscid limit
UR - http://eudml.org/doc/251143
ER -

## References

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