Abstract
We consider the Navier–Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier–Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary. The convergence is strong in Sobolev’s spaces \(W^{1}_{p}, p>2\), which correspond to the spaces of the data.
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Alekseev, G.V.: The solvability of an inhomogeneous boundary value problem for two-dimensional non-stationary equations of the dynamics of an ideal fluid. In: Dinamika Splošn. Sredy Vyp. 24. Dinamika Zidk. so Svobod. Granicami, vol. 169, pp. 15–35 (1976) (Russian)
Amrouche, C., Rodriguez-Belido, M.A.: On the Regularity for the Laplace Equation and the Stokes System. Monografias de la Real Academia de Ciencias de Zaragoza, vol. 38, pp. 1–20 (2012)
Arnal, D., Juillen, J.C., Reneaux, J., Gasparian, G.: Effect of wall suction on leading edge contamination. Aerosp. Sci. Technol. 8, 505–517 (1997)
Bardos, C., Titi, E.S.: Euler equations for incompressible ideal fluids. Russ. Math. Surv. 62(3), 409–451 (2007)
Beirão da Veiga, H., Crispo, F.: Concerning the W k,p-inviscid limit for 3-D flows under a slip boundary condition. J. Math. Fluid Mech. 13, 117–135 (2011)
Beirão da Veiga, H., Crispo, F.: The 3-D inviscid limit result under slip boundary conditions. A negative answer. J. Math. Fluid Mech. 14, 55–59 (2012)
Black, T.L., Sarnecki, A.J.: The turbulent boundary layer with suction or injection. Aeronautical Research Council Reports and Memoranda, N. 3387 (October, 1958), London (1965)
Boyer, F.: Trace theorems and spatial continuity properties for the solutions of the transport equation. Differ. Integral Equ. 18(8), 891–934 (2005)
Braslow, A.L.: A history of suction-type laminar-flow control with emphasis on flight research. NASA History Division (1999)
Bucur, D., Feireisl, E., Necasova, S.: Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions. Arch. Ration. Mech. Anal. 197, 117–138 (2010)
Caflisch, R., Sammartino, M.: Navier–Stokes equations on an exterior circular domain: construction of the solution and the zero viscosity limit. C. R. Acad. Sci., Ser. 1 Math. 324(8), 861–866 (1997)
Caflisch, R., Sammartino, M.: Existence and singularities for the Prandtl boundary layer equations. Z. Angew. Math. Mech. 80(11–12), 733–744 (2000)
Chemetov, N.V., Antontsev, S.N.: Euler equations with non-homogeneous Navier slip boundary condition. Physica D 237(1), 92–105 (2008)
Clopeau, T., Mikelic, A., Robert, R.: On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions. Nonlinearity 11, 1625–1636 (1998)
Constantin, P.: Euler and Navier–Stokes equations. Publ. Mat. 52(2), 235–265 (2008)
DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)
Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92(1), 102–163 (1970)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations. Springer, Berlin (2001)
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)
Iftimie, D., Sueur, F.: Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199(1), 145–175 (2011)
Jager, W., Mikelic, A.: On the roughness-induced effective boundary conditions for a viscous flow. J. Differ. Equ. 170, 96–122 (2001)
Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28 (1984)
Kelliher, J.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)
Kruzkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sb. 10, 217–243 (1970)
Kufner, A., John, O., Fučík, S.: Function Spaces. Academia Publishing House of the Czechoslovak Academia of Sciences, Prague (1977)
Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Math., vol. 14. AMS, Providence (2001)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, vol. 1. Clarendon Press, Oxford University Press, New York (1996)
Lions, J.L., Magenes, E.: Problèmes aux limites non Homogénes et Applications, vol. 2. Dunod, Paris (1968)
Lopes Filho, M.C., Nussenzveig Lopes, H.J., Planas, G.: On the inviscid limit for 2D incompressible flow with Navier friction condition. SIAM J. Math. Anal. 36(4), 1130–1141 (2005)
Malek, J., Necas, J., Rokyta, M., Ruzicka, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996)
Marshall, L.A.: Boundary-layer transition results from the F-16XL-2 supersonic laminar flow control experiment. NASA/TM-1999-209013, Dryden Flight Research Center Edwards, California 93523-0273, December (1999)
Mucha, P.: On the inviscid limit of the Navier–Stokes equations for flows with large flux. Nonlinearity 16, 1715–1732 (2003)
Necas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (2010)
Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Chapman & Hall/CRC, London (1999)
Otto, F.: Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729–734 (1996)
Priezjev, N.V., Troian, S.M.: Influence of periodic wall roughness on the slip behavior at liquid/solid interfaces: molecular-scale simulations versus continuum predictions. J. Fluid Mech. 554, 25–46 (2006)
Priezjev, N.V., Darhuber, A.A., Troian, S.M.: Slip behavior in liquid films on surfaces of patterned wettability: comparison between continuum and molecular dynamics simulations. Phys. Rev. E 71, 041608 (2005)
Sammartino, M., Caflisch, R.E.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. II. Construction of the Navier–Stokes solution. Commun. Math. Phys. 192, 463–491 (1998)
Schlichting, H., Gersten, K.: Boundary-Layer Theory. Springer, Berlin (2003)
Simon, J.: Compact sets in the space L p(0,T;B). Ann. Mat. Pura Appl. 146(1), 65–96 (1986)
Temam, R., Wang, X.: Boundary layers associated with incompressible Navier–Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 179(2), 647–686 (2002)
Xiao, Y., Xin, Z.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60(7), 1027–1055 (2007)
Acknowledgements
We would like to thank the referees for several corrections and comments which improved the presentation of the article.
N.V. Chemetov thanks for support from FCT and FEDER through the project POCTI/ISFL/209 of Centro de Matemática e Aplicações Fundamentais da Universidade de Lisboa (CMAF/UL). The research work of F. Cipriano was supported by the projects FCT-PTDC/MAT/104173/2008 and EURATOM/IST.
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Communicated by Andrew Szeri.
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Chemetov, N.V., Cipriano, F. The Inviscid Limit for the Navier–Stokes Equations with Slip Condition on Permeable Walls. J Nonlinear Sci 23, 731–750 (2013). https://doi.org/10.1007/s00332-013-9166-5
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DOI: https://doi.org/10.1007/s00332-013-9166-5
Keywords
- Navier–Stokes equations
- Euler equations
- Navier slip boundary conditions
- Vanishing viscosity
- Boundary layer
- Turbulence