Skip to main content
SpringerLink
Log in

On the low Mach number limit of compressible flows in exterior moving domains

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

We study the incompressible limit of solutions to the compressible barotropic Navier–Stokes system in the exterior of a bounded domain undergoing a simple translation. The problem is reformulated using a change of coordinates to fixed exterior domain. Using the spectral analysis of the wave propagator, the dispersion of acoustic waves is proved by means of the RAGE theorem. The solution to the incompressible Navier–Stokes equations is identified as a limit.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. van Baalent, G., Wittwer, P.: Time periodic solutions of the Navier–Stokes equations with nonzero constant boundary conditions at infinity. SIAM J. Math. Anal. 43 (2011), 4:1787–1809

  2. Bogovskii, M.E.: Solution of some vector analysis problems connected with operators div and grad. (in Russian) Trudy Sem. S.L. Sobolev, 80 (1980):5–40

  3. Cycon, H.L., Froese, R.G., Kirschi, W., Simon, G: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer (1987)

  4. Danchin R.: Low Mach number limit for viscous compressible flows. M2AN Math. Model Numer. Anal. 39, 459–475 (2005)

    Article  MathSciNet  Google Scholar 

  5. Desjardins B., Grenier E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455, 2271–2279 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Desjardins B., Grenier E., Lions P.-L., Masmoudi N.: Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Feireisl E.: Flows of viscous compressible fluids under strong stratification: incompressible limits for long-range potential forces. Mathematical Models and Methods in Applied Sciences 21, 7–27 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feireisl E., Karper T., Kreml O., Stebel J.: Stability with respect to domain of the low Mach number limit of compressible viscous fluids. Mathematical Models and Methods in Applied Sciences 23, 2465–2493 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feireisl, E., Kreml, O., Nečasová, Š., Neustupa, J., Stebel, J.: Weak solutions to the barotropic Navier–Stokes system with slip boundary conditions in time dependent domains. J. Differ. Equ. 254 (2013), no. 1, 125–140

  10. Feireisl, E., Kreml, O., Nečasová, Š., Neustupa, J., Stebel, J.: Incompressible limits of fluids excited by moving boundaries. SIAM J. Math. Anal. 46 (2014), No. 2, 1456–1471

  11. Feireisl E., Novotný A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhaüser, Berlin (2009)

    Book  MATH  Google Scholar 

  12. Fujita H., Sauer N.: On existence of weak solutions of the Navier–Stokes equations in regions with moving boundaries. J. Fac. Sci. Univ. Tokyo Sect. I 17, 403–420 (1970)

    MathSciNet  MATH  Google Scholar 

  13. Galdi G.P.: An introduction to the mathematical theory of the Navier–Stokes equations, I. Springer, New York (1994)

    MATH  Google Scholar 

  14. Lighthill J.: On sound generated aerodynamically I. General theory. Proc. of the Royal Society of London A 211, 564–587 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lighthill J.: On sound generated aerodynamically II. General theory. Proc. of the Royal Society of London A 222, 1–32 (1954)

    MathSciNet  MATH  Google Scholar 

  16. Lions P.-L.: Mathematical topics in fluid dynamics, Vol.2, Compressible models. Oxford Science Publication, Oxford (1998)

    MATH  Google Scholar 

  17. Lions P.-L., Masmoudi N.: Incompressible Limit for a Viscous Compressible Fluid. J. Math. Pures Appl. 77, 585–627 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Masmoudi, N.: Examples of singular limits in hydrodynamics In Handbook of Differential Equations, III, C. Dafermos, E. Feireisl Eds., Elsevier, Amsterdam (2006)

  19. Neustupa, J., Penel, P.: A Weak Solution to the Navier–Stokes System with Navier’s Boundary Condition in a Time-Varying Domain In Recent Developments of Mathematical Fluid Mechanics, eds. Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M., Springer, Basel (2016).

  20. Reed, M., Simon, B.: Methods of modern mathematical physics. III. Scattering theory. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.

  21. Schochet S.: The mathematical theory of low Mach number flows. M2ANMath. Model Numer. anal. 39, 441–458 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sýkora, P.: Compressible fluid motion in time dependent domains. Diploma Thesis, Charles University (2012)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Eduard Feireisl, Ondřej Kreml, Václav Mácha & Šárka Nečasová

Authors
  1. Eduard Feireisl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ondřej Kreml
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Václav Mácha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Šárka Nečasová
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Šárka Nečasová.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feireisl, E., Kreml, O., Mácha, V. et al. On the low Mach number limit of compressible flows in exterior moving domains. J. Evol. Equ. 16, 705–722 (2016). https://doi.org/10.1007/s00028-016-0338-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-016-0338-2

Keywords

Search

Navigation