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Boundary Layers for the Navier–Stokes Equations Linearized Around a Stationary Euler Flow

  • Gung-Min Gie
  • James P. Kelliher
  • Anna L. Mazzucato
Article

Abstract

We study the viscous boundary layer that forms at small viscosity near a rigid wall for the solution to the Navier–Stokes equations linearized around a smooth and stationary Euler flow (LNSE for short) in a smooth bounded domain \(\Omega \subset \mathbb {R}^3\) under no-slip boundary conditions. LNSE is supplemented with smooth initial data and smooth external forcing, assumed ill-prepared, that is, not compatible with the no-slip boundary condition. We construct an approximate solution to LNSE on the time interval [0, T], \(0<T<\infty \), obtained via an asymptotic expansion in the viscosity parameter, such that the difference between the linearized Navier–Stokes solution and the proposed expansion vanishes as the viscosity tends to zero in \(L^2(\Omega )\) uniformly in time, and remains bounded independently of viscosity in the space \(L^2([0,T];H^1(\Omega ))\). We make this construction both for a 3D channel domain and a smooth domain with a curved boundary. The zero-viscosity limit for LNSE, that is, the convergence of the LNSE solution to the solution of the linearized Euler equations around the same profile when viscosity vanishes, then naturally follows from the validity of this asymptotic expansion. This article generalizes and improves earlier works, such as Temam and Wang (Indiana Univ Math J 45(3):863–916, 1996), Xin and Yanagisawa (Commun Pure Appl Math 52(4):479–541, 1999), and Gie (Commun Math Sci 12(2):383–400, 2014).

Keywords

Boundary layers singular perturbations vanishing viscosity limit 

Mathematics Subject Classification

35B25 35C20 76D10 35K05 

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Notes

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflicts of interest.

References

  1. 1.
    Batchelor, G.K.: An Introduction to Fluid Dynamics, paperback edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  2. 2.
    Cannon, J.R.: The One-Dimensional Heat Equation, volume 23 of Encyclopedia of Mathematics and Its Applications. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, With a foreword by Felix E. Browder (1984)Google Scholar
  3. 3.
    Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23(2), 591–609 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gie, G.-M.: Asymptotic expansion of the Stokes solutions at small viscosity: the case of non-compatible initial data. Commun. Math. Sci. 12(2), 383–400 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gie, G.-M., Hamouda, M., Temam, R.: Boundary layers in smooth curvilinear domains: parabolic problems. Discrete Contin. Dyn. Syst. 26(4), 1213–1240 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gie, G.-M., Jung, C.-Y., Temam, R.: Recent progresses in boundary layer theory. Discrete Contin. Dyn. Syst. 36(5), 2521–2583 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gie, G.-M., Kelliher, J.P.: Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions. J. Differ. Equ. 253(6), 1862–1892 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grenier, E., Guo, Y., Nguyen, T.T.: Spectral stability of Prandtl boundary layers: an overview. Analysis (Berlin) 35(4), 343–355 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Guo, Y., Nguyen, T.: A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64(10), 1416–1438 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983), Vol. 2 of Math. Sci. Res. Inst. Publ., pp. 85–98. Springer, New York (1984)Google Scholar
  11. 11.
    Klingenberg, W.: A course in differential geometry. Springer, New York, Translated from the German by David Hoffman, Graduate Texts in Mathematics, Vol. 51 (1978)Google Scholar
  12. 12.
    Koch, H.: Transport and instability for perfect fluids. Math. Ann. 323(3), 491–523 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lions, J.-L.: Perturbations singulières dans les Problèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, vol. 323. Springer, Berlin (1973)zbMATHGoogle Scholar
  14. 14.
    Lombardo, M.C., Sammartino, M.: Zero viscosity limit of the Oseen equations in a channel. SIAM J. Math. Anal. 33(2), 390–410 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Maekawa, Y., Mazzucato, A.: The Inviscid Limit and Boundary Layers for Navier–Stokes Flows, pp. 1–48. Springer International Publishing, Cham (2016)Google Scholar
  16. 16.
    Prandtl, L.: Verber flüssigkeiten bei sehr kleiner reibung. Verk. III Intem. Math. Kongr. Heidelberg, pp. 484–491, (1905), Teuber, LeibzigGoogle Scholar
  17. 17.
    Sohr, H.: The Navier–Stokes equations. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2001. An elementary functional analytic approach, [2013 reprint of the 2001 original] [MR1928881]Google Scholar
  18. 18.
    Temam, R.: Navier–Stokes Equations. AMS Chelsea Publishing, Providence, RI, Theory and numerical analysis, Reprint of the 1984 edition (2001)Google Scholar
  19. 19.
    Temam, R., Wang, X.: Asymptotic analysis of Oseen type equations in a channel at small viscosity. Indiana Univ. Math. J. 45(3), 863–916 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Temam, R., Wang, X.: Boundary layers for Oseen’s type equation in space dimension three. Russian J. Math. Phys. 5(2), 227–246 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Temam, R., Wang, X.M.: Asymptotic analysis of the linearized Navier–Stokes equations in a channel. Differ. Integral Equ. 8(7), 1591–1618 (1995)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Xin, Z., Yanagisawa, T.: Zero-viscosity limit of the linearized Navier–Stokes equations for a compressible viscous fluid in the half-plane. Commun. Pure Appl. Math. 52(4), 479–541 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Gung-Min Gie
    • 1
  • James P. Kelliher
    • 2
  • Anna L. Mazzucato
    • 3
  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  2. 2.Department of MathematicsUniversity of CaliforniaRiversideUSA
  3. 3.Department of MathematicsPenn State UniversityUniversity ParkUSA

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