Skip to main content
SpringerLink
Log in

On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted L ∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L ∞ norm of the perturbation decays as time goes to infinity.

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. Brezina 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 

  2. Feireisl E., Matušu-Necasová Š., Petzeltová H., Straškrava.: On the motion of a viscous compressible fluid driven by a time-periodic external force. Arch. Rational Mech. Anal. 149, 69–96 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Feireisl E., Mucha P.B., Novotný A., Pokorný M.: Time-periodic solutions to the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 745–786 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kagei Y., Kawashima S.: Stability of planar stationary solutions to the compressible Navier–Stokes equation on the half space. Commun. Math. Phys. 266, 401–430 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. 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 

  6. Kagei Y., Tsuda K.: Existence and stability of time periodic solution to the compressible Navier–Stokes equation for time periodic external force with symmetry. J. Differ. Equ. 258, 399–444 (2015)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Kaniel S., Shinbrot M.: A reproductive property of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 24, 363–369 (1967)

    MATH  MathSciNet  Google Scholar 

  9. Kozono H., Nakao M.: Periodic solutions of the Navier–Stokes equations in unbounded domains. Tohoku Math. J. 48, 33–50 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Okita M.: On the convergence rates for the compressible Navier–Stokes equations with potential force. Kyushu J. Math. 68, 261–286 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Serrin J.: A note on the existence of periodic solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 3, 120–122 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  13. Shibata Y., Shimizu S.: A decay property of the Fourier transform and its application to the Stokes problem. J. Math. Fluid Mech. 3, 213–230 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Shibata Y., Tanaka K.: On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance. J. Math. Soc. Japan 55, 797–826 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ma H., Ukai S., Yang T.: Time periodic solutions of compressible Navier–Stokes equations. J. Differ. Equ. 248, 2275–2293 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Valli A.: Periodic and stationary solutions for compressible Navier–Stokes equations via a stability method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, 607–647 (1983)

    MATH  MathSciNet  Google Scholar 

  17. Yamazaki M.: The Navier–Stokes equations in the weak-L n space with timedependent external force. Math. Ann. 317, 635–675 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Kazuyuki Tsuda

Authors
  1. Kazuyuki Tsuda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuyuki Tsuda.

Additional information

Communicated by T.-P. Liu

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsuda, K. On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space. Arch Rational Mech Anal 219, 637–678 (2016). https://doi.org/10.1007/s00205-015-0902-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0902-x

Keywords

Search

Navigation