Skip to main content
SpringerLink
Log in

The Navier-Stokes Equations in a Periodic Channel

  • Chapter
  • First Online:
Book cover Singular Perturbations and Boundary Layers

Part of the book series: Applied Mathematical Sciences ((AMS,volume 200))

  • 948 Accesses

Abstract

The Navier-Stokes equations appear as a singular perturbation of the Euler equations in which the small parameter ɛ is the viscosity or inverse of the Reynolds number. In many cases the convergence of the solutions of the Navier-Stokes equations to those of the Euler equations remains an outstanding open problem of mathematical physics. The result is not known in the case of the no-slip boundary condition, even in space dimension 2 for which the existence, uniqueness, and regularity of solution for all time is known for both the Navier-Stokes and Euler equations; see, e.g., [Kat84, Kat86, FT79, Tem75, Tem76, Tem01]. Fortunately, and this is the object of Chapters 6 and 7, this problem has been solved in a number of particular situations: special symmetries or boundary conditions other than the no-slip boundary condition.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The set \(\\mathcal{L}(H,\,D(A))\) denotes the space of bounded linear operators from H into D(A).

References

  1. S. N. Antontsev, J. I. Dí az, and S. Shmarev. Energy methods for free boundary problems, volume 48 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 2002. Applications to nonlinear PDEs and fluid mechanics.

    Google Scholar 

  2. S. N. Antontsev, A.V. Kazhikhov, and V. N. Monakhov. Boundary value problems in mechanics of nonhomogeneous fluids, Studies in Mathematics and its Applications 22, North-Holland Publishing Co., Amsterdam, 1990. Translated from the Russian.

    MATH  Google Scholar 

  3. Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer-Verlag, New York, 2011. xiv+599 pp.

    Google Scholar 

  4. M. Cannone, M. C. Lombardo, and M. Sammartino. Well-posedness of Prandtl equations with non-compatible data. Nonlinearity, 26(12):3077–3100, 2013.

    Article  MathSciNet  Google Scholar 

  5. M. Cannone, M. C. Lombardo, and M. Sammartino. On the Prandtl boundary layer equations in presence of corner singularities. Acta Appl. Math., 132:139–149, 2014.

    Article  MathSciNet  Google Scholar 

  6. P. Constantin and C. Foias. Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.

    MATH  Google Scholar 

  7. N. El Berdan, J. I. Dí az, and J. M. Rakotoson. The uniform Hopf inequality for discontinuous coefficients and optimal regularity in BMO for singular problems. J. Math. Anal. Appl., 437(1):350–379, 2016.

    Google Scholar 

  8. C. Foias and R. Temam. Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations. J. Math. Pures Appl. (9), 58(3):339–368, 1979.

    MathSciNet  MATH  Google Scholar 

  9. K.O. Friedrichs, The mathematical structure of the boundary layer problem in Fluid Dynamics (R. von Mises and K.O. Friedrichs, eds), Brown Univ., Providence, RI (reprinted by Springer-Verlag, New York, 1971). pp. 171–174.

    Google Scholar 

  10. F. Gargano, M. Sammartino and V. Sciacca. Singularity formation for Prandtl’s equations. Phys. D 238 (2009), no. 19, 1975–1991.

    Article  MathSciNet  Google Scholar 

  11. D. Gerard-Varet and N. Masmoudi. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 6, 1273–1325.

    Article  MathSciNet  Google Scholar 

  12. D. Gérard-Varet and T. Nguyen. Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77 (2012), no. 1–2, 71–88.

    MathSciNet  MATH  Google Scholar 

  13. Gung-Min Gie. Asymptotic expansion of the Stokes solutions at small viscosity: the case of non-compatible initial data. Commun. Math. Sci., 12(2):383–400, 2014.

    Article  MathSciNet  Google Scholar 

  14. Gung-Min Gie, Makram Hamouda, and Roger Temam. Asymptotic analysis of the Stokes problem on general bounded domains: the case of a characteristic boundary. Appl. Anal., 89(1):49–66, 2010.

    Article  MathSciNet  Google Scholar 

  15. Gung-Min Gie and James P. Kelliher. Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions. J. Differential Equations, 253(6):1862–1892, 2012.

    Article  MathSciNet  Google Scholar 

  16. M. Hamouda, C.-Y. Jung, and R. Temam. Existence and regularity results for the inviscid primitive equations with lateral periodicity. Appl. Math. Optim., 73(3):501–522, 2016.

    Article  MathSciNet  Google Scholar 

  17. Makram Hamouda and Roger Temam. Some singular perturbation problems related to the Navier-Stokes equations. In Advances in deterministic and stochastic analysis, pages 197–227. World Sci. Publ., Hackensack, NJ, 2007.

    Google Scholar 

  18. Makram Hamouda and Roger Temam. Boundary layers for the Navier-Stokes equations. The case of a characteristic boundary. Georgian Math. J., 15(3):517–530, 2008.

    Google Scholar 

  19. A. Haraux. Systèmes dynamiques dissipatifs et applications, volume 17 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris, 1991.

    Google Scholar 

  20. Dragoş Iftimie and Gabriela Planas. Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity, 19(4):899–918, 2006.

    Article  MathSciNet  Google Scholar 

  21. Dragoş Iftimie and Franck Sueur. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch. Rational Mech. Anal., 199(1): 145–175, 2011

    Article  MathSciNet  Google Scholar 

  22. Tosio Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. In Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), volume 2 of Math. Sci. Res. Inst. Publ., pages 85–98. Springer, New York, 1984.

    Google Scholar 

  23. Tosio Kato. Remarks on the Euler and Navier-Stokes equations in R 2. In Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983), volume 45 of Proc. Sympos. Pure Math., pages 1–7. Amer. Math. Soc., Providence, RI, 1986.

    Google Scholar 

  24. I. Kukavica, N. Masmoudi, V. Vicol, Vlad and T.K. Wong. On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J. Math. Anal. 46 (2014), no. 6, 3865–3890.

    Article  MathSciNet  Google Scholar 

  25. J.-L. Lions. Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, Vol. 323. Springer-Verlag, Berlin-New York, 1973.

    Book  Google Scholar 

  26. M. C. Lombardo, M. Cannone and M. Sammartino. Well-posedness of the boundary layer equations. SIAM J. Math. Anal. 35 (2003), no. 4, 987–1004.

    Article  MathSciNet  Google Scholar 

  27. C-J. Liu, Y-G. Wang, and T. Yang. On the Ill-Posedness of the Prandtl Equations in Three-Dimensional Space. Arch. Ration. Mech. Anal. 220 (2016), no. 1, 83–108.

    Article  MathSciNet  Google Scholar 

  28. Nader Masmoudi and Frédéric Rousset. Uniform regularity for the Navier-Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal., 203(2):529–575, 2012.

    Article  MathSciNet  Google Scholar 

  29. O. A. Oleĭ nik. On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid. J. Appl. Math. Mech., 30:951–974 (1967), 1966.

    Google Scholar 

  30. Robert E. O’Malley, Jr. Singular perturbation analysis for ordinary differential equations, volume 5 of Communications of the Mathematical Institute, Rijksuniversiteit Utrecht. Rijksuniversiteit Utrecht, Mathematical Institute, Utrecht, 1977.

    Google Scholar 

  31. Madalina Petcu, Euler equation in a 3D channel with a noncharacteristic boundary, Differential Integral Equations, 19 (2006), 297–326.

    MathSciNet  MATH  Google Scholar 

  32. Ludwig Prandtl. Über flüssigkeitsbewegung bei sehr kleiner reibung. In 3 Internationalen Mathematischen Kongress, 1904, pages 484–491.

    Google Scholar 

  33. L. Prandtl. Verber flüssigkeiten bei sehr kleiner reibung. Verk. III Intem. Math. Kongr. Heidelberg, pages 484–491, 1905, Teuber, Leibzig.

    Google Scholar 

  34. Ludwig Prandtl. Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- und Aerodynamik. Herausgegeben von Walter Tollmien, Hermann Schlichting, Henry Görtler. Schriftleitung: F. W. Riegels. In 3 Teilen. Springer-Verlag, Berlin, 1961.

    Google Scholar 

  35. M. Sammartino and R. E. Caflisch. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm. Math. Phys., 192(2):433–461, 1998.

    Article  MathSciNet  Google Scholar 

  36. M. Sammartino and R. E. Caflisch. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm. Math. Phys., 192(2):463–491, 1998.

    Article  MathSciNet  Google Scholar 

  37. H. Schlichting and K. Gersten. Boundary-layer theory. Springer-Verlag, Berlin, ninth edition, 2017. With contributions from Egon Krause and Herbert Oertel Jr., Translated from the German by Katherine Mayes.

    Google Scholar 

  38. Roger Temam. On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32–43.

    Article  MathSciNet  Google Scholar 

  39. R. Temam. Local existence of C ∞ solutions of the Euler equations of incompressible perfect fluids. In Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975), pages 184–194. Lecture Notes in Math., Vol. 565. Springer, Berlin, 1976.

    Google Scholar 

  40. R. Temam. Navier-Stokes equations and nonlinear functional analysis, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 1995.

    Google Scholar 

  41. R. Temam. Navier-Stokes equations. AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis, Reprint of the 1984 edition.

    Google Scholar 

  42. Roger Temam and Xiaoming Wang. Asymptotic analysis of the linearized Navier-Stokes equations in a channel. Differential Integral Equations, 8(7):1591–1618, 1995.

    MathSciNet  MATH  Google Scholar 

  43. Roger Temam and Xiaoming Wang. Asymptotic analysis of the linearized Navier-Stokes equations in a general 2D domain. Asymptot. Anal., 14(4):293–321, 1997.

    MathSciNet  MATH  Google Scholar 

  44. R. Temam and X. Wang. Remarks on the Prandtl equation for a permeable wall. ZAMM Z. Angew. Math. Mech., 80(11–12):835–843, 2000. Special issue on the occasion of the 125th anniversary of the birth of Ludwig Prandtl.

    Article  MathSciNet  Google Scholar 

  45. Roger Temam and Xiaoming Wang. Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case. J. Differential Equations, 179 (2002), no. 2, 647–686.

    Article  MathSciNet  Google Scholar 

  46. M. I. Višik and L. A. Lyusternik. Regular degeneration and boundary layer for linear differential equations with small parameter. Uspehi Mat. Nauk (N.S.), 12(5(77)):3–122, 1957.

    Google Scholar 

  47. M. I. Višik and L. A. Ljusternik. Regular degeneration and boundary layer for linear differential equations with small parameter. Amer. Math. Soc. Transl. (2), 20:239–364, 1962.

    MathSciNet  Google Scholar 

  48. Xie, Xiaoqiang and Li, Changmin. Boundary layers of the incompressible fluids for a permeable wall. ZAMM Z. Angew. Math. Mech. 91 (2011), no. 1, 68–84.

    Article  MathSciNet  Google Scholar 

  49. Zhouping Xin and Taku Yanagisawa. Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane. Comm. Pure Appl. Math., 52(4):479–541, 1999.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics, University of Louisville, Louisville, KY, USA

    Gung-Min Gie

  2. Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia

    Makram Hamouda

  3. Indiana University, Bloomington, IN, USA

    Makram Hamouda

  4. Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea

    Chang-Yeol Jung

  5. Mathematics, Indiana University, Institute for Scientific Computing and Applied Mathematics, Bloomington, IN, USA

    Roger M. Temam

Authors
  1. Gung-Min Gie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Makram Hamouda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang-Yeol Jung
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Roger M. Temam
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gie, GM., Hamouda, M., Jung, CY., Temam, R.M. (2018). The Navier-Stokes Equations in a Periodic Channel. In: Singular Perturbations and Boundary Layers. Applied Mathematical Sciences, vol 200. Springer, Cham. https://doi.org/10.1007/978-3-030-00638-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation