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, Volume 62 (2012) no. 6, pp. 2257-2314.

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 ϵ and amplitude O(ϵ). 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 in the small viscosity limit to the solution of the compressible Euler equations with normal velocity equal to zero on the boundary.

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 ϵ et d’amplitude O(ϵ). 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 , lorsque la viscosité tend vers 0, vers la solution du système d’Euler compressible avec composante normale de la vitesse nulle au bord.

DOI: 10.5802/aif.2749
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

Wang, Ya-Guang 1; Williams, Mark 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
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Wang, Ya-Guang; 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, Volume 62 (2012) no. 6, pp. 2257-2314. doi : 10.5802/aif.2749. https://aif.centre-mersenne.org/articles/10.5802/aif.2749/

[1] Clopeau, T.; Mikelić, R. A.and Robert On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions., Nonlinearity, Volume 11 (1998), pp. 1625-1636 | MR | Zbl

[2] Feirisl, E. Dynamics of Viscous Compressible Fluids., Oxford Mathematical Monographs, Oxford University Press, 2004 | MR | Zbl

[3] Gérard-Varet, D.; Dormy, E. On the ill-posedness of the Prandtl equation., Journal A.M.S., Volume 23 (2010), pp. 591-609 | MR | Zbl

[4] Grenier, E. On the nonlinear instability of Euler and Prandtl equations., Comm. Pure Appl. Math., Volume 53 (2000), pp. 1067-1091 | MR | Zbl

[5] Guès, O. Problème mixte hyperbolique quasi-linéaire caractéristique., Comm. PDE, Volume 15 (1990), pp. 595-645 | MR | Zbl

[6] Guès, O.; Métivier, G.; Williams, M.; Zumbrun, K. Navier-Stokes regularization of multidimensional Euler shocks., Ann. Sci. École Norm. Sup., Volume 39 (2006), pp. 75-175 | MR | Zbl

[7] Guès, O.; Métivier, G.; Williams, M.; Zumbrun, K. Existence and stability of noncharacteristic boundary layers for the compressible Navier-Stokes and viscous MHD equations., Arch. Rat. Mech. Analy., Volume 197 (2010), pp. 1-87 | MR | Zbl

[8] Hoff, D. Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data., J. Diff. Eqns., Volume 120 (1995), pp. 215-254 | MR | Zbl

[9] Hoff, D. Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech., Volume 7 (2005), pp. 315-338 | MR | Zbl

[10] Iftimie, D.; Sueur, F. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Rat. Mech. Analy., Volume 199 (2011) no. 1, pp. 145-175 | MR | Zbl

[11] Kelliher, J. Navier-stokes equations with Navier boundary conditions for a bounded domain in the plane., SIAM J. Math. Anal., Volume 38 (2006), pp. 201-232 | MR | Zbl

[12] Lopes Filho, M. C.; Mazzucato, A. L.; Nussenzveig Lopes, H. J.; Taylor, M. Vanishing viscosity limits and boundary layers for circularly symmetric 2d flows., Bull. Braz. Math. Soc., Volume 39 (2008), p. 471-453 | MR | Zbl

[13] Lopes Filho, M. C.; Nussenzveig Lopes, H. J.; Planas, G. On the inviscid limit for two-dimensional incompressible flow with Navier friction condition., SIAM J. Math. Anal., Volume 36 (2005), pp. 1130-1141 | MR | Zbl

[14] Matsumura, A.; Nishida, T. The initial value problem for the equations of motion of viscous and heat-conductive gases., J. Math. Kyoto Univ., Volume 20 (1980), pp. 67-104 | MR | Zbl

[15] Métivier, G. Uber Flüssigkeitsbewegungen bei sehr kleiner Reibung., Verh. Int. Math. Kongr., Heidelberg 1904, Teubner, 1905, pp. 484-494 | JFM

[16] Navier, C. L. M. H. Sur les lois de l’équilibrie et du mouvement des corps élastiques., Mem. Acad. R. Sci. Inst. France, Volume 6 (1827), pp. 369

[17] Oleinik, O. A.; Samokhin, V. N. Mathematical Models in Boundary Layers Theory, Chapman & Hall/CRC, 1999 | MR | Zbl

[18] Qian, T.; Wang, X. P.; Sheng, P. Molecular scale contact line hydrodynamics of immiscible flows., Physical Review E, Volume 68 (2003), pp. 1-15 | MR

[19] Sammartino, M.; Caflisch, R. E. Zero viscosity limit or analytic solutions of the Navier-Stokes equations on a half-space, I. Existence for Euler and Prandtl equations., Comm. Math. Phys., Volume 192 (1998), pp. 433-461 | MR | Zbl

[20] Temam, R.; Wang, X. Boundary layers associated with incompressible Navier–Stokes equations: the noncharacteristic boundary case., J. Diff. Eq., Volume 179 (2002), pp. 647-686 | MR | Zbl

[21] Wang, X. P.; Wang, Y. G.; Xin, Z. P. Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit, Comm. Math. Sci., Volume 8 (2010), pp. 965-998 | MR | Zbl

[22] Xin, Z.P.; Yanagisawa, T. Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane., Comm. Pure Appl. Math., Volume 52 (1999), pp. 479-541 | MR | Zbl

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