Journal of Nonlinear Science

, Volume 23, Issue 5, pp 731–750 | Cite as

The Inviscid Limit for the Navier–Stokes Equations with Slip Condition on Permeable Walls



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.


Navier–Stokes equations Euler equations Navier slip boundary conditions Vanishing viscosity Boundary layer Turbulence 

Mathematics Subject Classification (2000)

35D05 76B03 76B47 76D09 



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.


  1. 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) Google Scholar
  2. 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) Google Scholar
  3. Arnal, D., Juillen, J.C., Reneaux, J., Gasparian, G.: Effect of wall suction on leading edge contamination. Aerosp. Sci. Technol. 8, 505–517 (1997) CrossRefGoogle Scholar
  4. Bardos, C., Titi, E.S.: Euler equations for incompressible ideal fluids. Russ. Math. Surv. 62(3), 409–451 (2007) MathSciNetCrossRefMATHGoogle Scholar
  5. 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) MathSciNetCrossRefMATHGoogle Scholar
  6. 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) MathSciNetCrossRefGoogle Scholar
  7. 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) Google Scholar
  8. Boyer, F.: Trace theorems and spatial continuity properties for the solutions of the transport equation. Differ. Integral Equ. 18(8), 891–934 (2005) MathSciNetMATHGoogle Scholar
  9. Braslow, A.L.: A history of suction-type laminar-flow control with emphasis on flight research. NASA History Division (1999) Google Scholar
  10. 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) MathSciNetCrossRefMATHGoogle Scholar
  11. 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) MathSciNetMATHGoogle Scholar
  12. Caflisch, R., Sammartino, M.: Existence and singularities for the Prandtl boundary layer equations. Z. Angew. Math. Mech. 80(11–12), 733–744 (2000) MathSciNetCrossRefMATHGoogle Scholar
  13. Chemetov, N.V., Antontsev, S.N.: Euler equations with non-homogeneous Navier slip boundary condition. Physica D 237(1), 92–105 (2008) MathSciNetCrossRefMATHGoogle Scholar
  14. 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) MathSciNetCrossRefMATHGoogle Scholar
  15. Constantin, P.: Euler and Navier–Stokes equations. Publ. Mat. 52(2), 235–265 (2008) MathSciNetCrossRefMATHGoogle Scholar
  16. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989) MathSciNetCrossRefMATHGoogle Scholar
  17. Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92(1), 102–163 (1970) MathSciNetCrossRefMATHGoogle Scholar
  18. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations. Springer, Berlin (2001) MATHGoogle Scholar
  19. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986) CrossRefMATHGoogle Scholar
  20. 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) MathSciNetCrossRefMATHGoogle Scholar
  21. Jager, W., Mikelic, A.: On the roughness-induced effective boundary conditions for a viscous flow. J. Differ. Equ. 170, 96–122 (2001) MathSciNetCrossRefGoogle Scholar
  22. Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28 (1984) MathSciNetCrossRefMATHGoogle Scholar
  23. 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) MathSciNetCrossRefMATHGoogle Scholar
  24. Kruzkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sb. 10, 217–243 (1970) CrossRefGoogle Scholar
  25. Kufner, A., John, O., Fučík, S.: Function Spaces. Academia Publishing House of the Czechoslovak Academia of Sciences, Prague (1977) MATHGoogle Scholar
  26. Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Math., vol. 14. AMS, Providence (2001) MATHGoogle Scholar
  27. Lions, P.-L.: Mathematical Topics in Fluid Mechanics, vol. 1. Clarendon Press, Oxford University Press, New York (1996) MATHGoogle Scholar
  28. Lions, J.L., Magenes, E.: Problèmes aux limites non Homogénes et Applications, vol. 2. Dunod, Paris (1968) MATHGoogle Scholar
  29. 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) MathSciNetCrossRefMATHGoogle Scholar
  30. Malek, J., Necas, J., Rokyta, M., Ruzicka, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996) MATHGoogle Scholar
  31. 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) Google Scholar
  32. Mucha, P.: On the inviscid limit of the Navier–Stokes equations for flows with large flux. Nonlinearity 16, 1715–1732 (2003) MathSciNetCrossRefMATHGoogle Scholar
  33. Necas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (2010) Google Scholar
  34. Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Chapman & Hall/CRC, London (1999) MATHGoogle Scholar
  35. Otto, F.: Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729–734 (1996) MathSciNetMATHGoogle Scholar
  36. 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) CrossRefMATHGoogle Scholar
  37. 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) CrossRefGoogle Scholar
  38. 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) MathSciNetCrossRefMATHGoogle Scholar
  39. Schlichting, H., Gersten, K.: Boundary-Layer Theory. Springer, Berlin (2003) Google Scholar
  40. Simon, J.: Compact sets in the space L p(0,T;B). Ann. Mat. Pura Appl. 146(1), 65–96 (1986) CrossRefGoogle Scholar
  41. Temam, R., Wang, X.: Boundary layers associated with incompressible Navier–Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 179(2), 647–686 (2002) MathSciNetCrossRefMATHGoogle Scholar
  42. 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) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CMAFUniversidade de LisboaLisboaPortugal
  2. 2.GFM e Dep. de Matemática FCTUNLLisboaPortugal

Personalised recommendations