Advertisement

Journal of Mathematical Fluid Mechanics

, Volume 16, Issue 3, pp 571–594 | Cite as

Compressible Navier–Stokes Equations on Thin Domains

  • David MalteseEmail author
  • Antonín Novotný
Article

Abstract

We consider the barotropic Navier–Stokes system describing the motion of a compressible viscous fluid confined to a straight layer \({\Omega_\varepsilon = \omega\times (0, \varepsilon)}\) , where ω is a particular 2-D domain (a periodic cell, bounded domain or the whole 2-D space). We show that the weak solutions in the 3D domain converge to a (strong) solutions of the 2-D Navier–Stokes system on ω as \({\varepsilon \to 0}\) on the maximal life time of the strong solution.

Keywords

Compressible Navier–Stokes system dimension reduction thin domains 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams R.A.: Sobolev spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Bella, P., Feireisl, E., Novotny, A.: Dimensional reduction for compressible viscous fluids. Preprint IM-2013-21. http://www.math.cas.cz
  3. 3.
    Dafermos C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dain S.: Generalized Korn’s inequality and conformal killing vectors. Calc. Var. Part. Differ. Equ. 25, 535–540 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Feireisl E.: Dynamics of viscous compressible fluids. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  6. 6.
    Feireisl E., Jin B., Novotný A.: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech. 14, 712–730 (2012)Google Scholar
  7. 7.
    Feireisl E., Novotny A.: Singular limits in thermodynamics of viscous fluids. Birkhauser, Basel (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Feireisl E., Novotný A., Sun Y.: Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60, 611–631 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Germain P.: Weak-strong uniqueness for the isentropic compressible Navier–Stokes system. J. Math. Fluid Mech. 13(1), 137–146 (2011)CrossRefzbMATHMathSciNetADSGoogle Scholar
  10. 10.
    Iftimie D., Raugel G., Sell G.R.: Navier–Stokes equations in thin 3D domains with Navier boundary conditions. Indiana Univ. Math. J. 56, 1083–1156 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Jesslé, D., Jin, B.J., Novotny, A.: Navier–Stokes–Fourier system on unbounded domains: weak solutions, relative entropies, weak-strong uniqueness. SIAM J. Math. Anal. 45(3), 1003–1026 (2013)Google Scholar
  12. 12.
    Kazhikhov A.V.: Correctness “in the large” of mixed boundary value problems for a model of equations of a viscous gas (in Russian). Din. Sphlosn. Sredy 21, 18–47 (1975)Google Scholar
  13. 13.
    Lewicka M., Müller S.: The uniform Korn Poincaré inequality in thin domains. Ann. I. H. Poincaré 28, 443–469 (2011)CrossRefzbMATHADSGoogle Scholar
  14. 14.
    Lions P.-L.: Mathematical topics in fluid dynamics, vol. 1, Incompressible models. Oxford Science Publication, Oxford (1996)Google Scholar
  15. 15.
    Lions P.-L.: Mathematical topics in fluid dynamics, vol. 2, Compressible models. Oxford Science Publication, Oxford (1998)Google Scholar
  16. 16.
    Mellet A., Vasseur A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier–Stokes equations. SIAM J. Math. Anal. 39(4), 1344–1365 (2007/08)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Novotny A., Straskraba I.: Introduction to the mathematical theory of compressible fluids. Oxford Science Publication, Oxford (2004)Google Scholar
  18. 18.
    Rajagopal, K. R.: A new development and interpretation of the Navier–Stokes fluid which reveals why the “Stokes assumption” is inapt. Int. J. Non-Linear Mech. 50, 141–151 (2013)Google Scholar
  19. 19.
    Raugel G., Sell G.R.: Navier–Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions. J. Am. Math. Soc. 6, 503–568 (1993)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Reshetnyak Yu. G.: Stability theorems in geometry and analysis, vol. 304 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1994). Translated from the 1982 Russian original by N. S. Dairbekov and V. N. Dyatlov, and revised by the author, Translation edited and with a foreword by S. S. Kutateladze.Google Scholar
  21. 21.
    Valli A., Zajaczkowski W.: Navier–Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986)CrossRefzbMATHMathSciNetADSGoogle Scholar
  22. 22.
    Vodák R.: Asymptotic analysis of steady and nonsteady Navier–Stokes equations for barotropic compressible flow. Acta Appl. Math. 110, 991–1009 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.IMATH, EA 2134Université du Sud Toulon-VarLa GardeFrance

Personalised recommendations