Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Asymptotic Behavior of the Linearized Semigroup at Space-Periodic Stationary Solution of the Compressible Navier–Stokes Equation

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

The asymptotic behavior of the linearized semigroup at spatially periodic stationary solution of the compressible Navier–Stokes equation in a periodic layer of \(\mathbb {R}^n\) \((n=2,3)\) is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the linearized semigroup is decomposed into two parts; one behaves like a solution of an \(n-1\) dimensional linear heat equation as time goes to infinity and the other one decays exponentially.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Aoyama, R.: Decay estimates on solutions of the linearized compressible Navier-Stokes equation around a parallel flow in a cylindrical domain. Kyushu J. Math. 69, 293–343 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aoyama, R., Kagei, Y.: Spectral properties of the semigroup for the linearized compressible Navier-Stokes equation around a parallel flow in a cylindrical domain. Adv. Differ. Equ. 21, 265–300 (2016)

    MATH  MathSciNet  Google Scholar 

  3. Aoyama, R., Kagei, Y.: Large time behavior of solutions to the compressible Navier-Stokes equations around a parallel flow in a cylindrical domain. Nonlinear Anal. 127, 362–396 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bogovskii, M.E.: Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Soviet Math. Dokl. 20, 1094–1098 (1979)

    Google Scholar 

  5. Březina, J.: Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a time-periodic parallel flow. SIAM J. Math. Anal. 45, 3514–3574 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Březina, J., Kagei, Y.: Spectral properties of the linearized compressible Navier-Stokes equation around time-periodic parallel flow. J. Differ. Equ. 255, 1132–1195 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Deckelnick, K.: Decay estimates for the compressible Navier-Stokes equations in unbounded domains. Math. Z. 209, 115–130 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. 1. Springer-Verlag, New York (1994)

    MATH  Google Scholar 

  9. Hoff, D., Zumbrun, K.: Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44, 604–676 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Heywood, J.G., Padula, M.: On the steady transport equation. In: Galdi, G.P., Heywood, J.G., Rannacher, R. (eds.) Fundamental Directions in Mathematical Fluid Mechanics, pp. 149–170. Birkhäuser, Basel (2000)

  11. Iooss, G., Padula, M.: Structure of the linearized problem for compressible parallel fluid flows. Ann. Univ. Ferrara Sez. VII 43, 157–171 (1998)

    MATH  MathSciNet  Google Scholar 

  12. Kagei, Y.: Asymptotic behavior of solutions of the compressible Navier-Stokes equation around a parallel flow. Arch. Ration. Mech. Anal. 205, 585–650 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kagei, Y., Nagafuchi, Y., Sudou, T.: Decay estimates on solutions of the linearized compressible Navier-Stokes equation around a Poiseuille type flow, J. Math-for-Industry 2, 39–56 (2010A). Correction to “Decay estimates on solutions of the linearized compressible Navier-Stokes equation around a Poiseuille type flow” in J. Math-for-Ind., 2, 39–56 (2010A), J. Math-for-Industry 2, 235 (2010B)

  14. Kagei, Y., Kobayashi, T.: Asymptotic behavior of solutions of the compressible Navier-Stokes equation on the half space. Arch. Ration. Mech. Anal. 177, 231–330 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kagei, Y., Makio, N.: Spectral properties of the linearized semigroup of the compressible Navier-Stokes equation on a periodic layer. Publ. Res. Inst. Math. Sci. 51, 337–372 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kato, T.: Perturbation theory for linear operators. Springer-Verlag, Berlin, Heidelberg, New York (1980)

    MATH  Google Scholar 

  17. Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Ph. D. Thesis, Kyoto University (1983)

  18. Kobayashi, T.: Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in \({ R}^3\). J. Differ. Equ. 184, 587–619 (2002)

    Article  MATH  ADS  Google Scholar 

  19. Kobayashi, T., Shibata, Y.: Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in \({ R}^3\). Commun. Math. Phys. 200, 621–659 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Matsumura, A.: An energy method for the equations of motion of compressible viscous and heat-conductive fluids, University of Wisconsin-Madison, MRC Technical Summary Report # 2194, pp. 1–16 (1981)

  21. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn. Acad. Ser. A 55, 337–342 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  22. Matsumura, A., Nishida, T.: Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Reed, M., Simon, B.: Methods of modern mathematical physics IV. Academic Press, London (1979)

    MATH  Google Scholar 

  24. Shibata, Y., Tanaka, K.: Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid. Comput. Math. Appl. 53, 605–623 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Valli, A.: On the existence of stationary solutions to compressible Navier-Stokes equations. Ann. Inst. H Poincaré 4, 99–113 (1987)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, Nishi-ku, Motooka 744, Fukuoka, 819-0395, Japan

    Shota Enomoto

  2. Faculty of Mathematics, Kyushu University, Nishi-ku, Motooka 744, Fukuoka, 819-0395, Japan

    Yoshiyuki Kagei

Authors
  1. Shota Enomoto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshiyuki Kagei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshiyuki Kagei.

Additional information

Communicated by Y. Shibata.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Enomoto, S., Kagei, Y. Asymptotic Behavior of the Linearized Semigroup at Space-Periodic Stationary Solution of the Compressible Navier–Stokes Equation. J. Math. Fluid Mech. 19, 739–772 (2017). https://doi.org/10.1007/s00021-016-0304-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-016-0304-3

Keywords

Search

Navigation