Skip to main content
SpringerLink
Log in

On L 1-summability and asymptotic profiles for smooth solutions to Navier–Stokes equations in a 3D exterior domain

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

The exterior nonstationary problem is studied for the 3D Navier-Stokes equations. The L 1-summability is proved for smooth solutions which correspond to initial data satisfying certain symmetry and moment conditions. The result is then applied to show that such solutions decay in time more rapidly than observed in general. Furthermore, an asymptotic expansion is deduced and a lower bound estimate is given for the rates of decay in time.

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. Borchers, W., Sohr, H.: On the semigroup of the Stokes operator for exterior domains in L q spaces. Math. Z. 196, 415–425 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Borchers, W., Miyakawa, T.: Algebraic L 2 decay for Navier-Stokes flows in exterior domains. Acta Math 165, 189–227 (1990)

    MathSciNet  MATH  Google Scholar 

  3. Borchers, W., Miyakawa, T.: Algebraic L 2 decay for Navier-Stokes flows in exterior domains, II. Hiroshima Math. J. 21, 621–640 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Borchers, W., Miyakawa, T.: On stability of exterior stationary Navier-Stokes flows. Acta Math. 174, 1–72 (1995)

    Google Scholar 

  5. Brandolese, L.: On the localization of symmetric and asymmetric solutions of the Navier-Stokes equations in ℝn. C. R. Acad. Sci. Paris, Sér. I Math. 332(2), 125–130 (2001)

    Google Scholar 

  6. Chen, Zh.-M.: Solutions of the stationary and nonstationary Navier-Stokes equations in exterior domains. Pacific J. Math. 159, 227–240 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Constantin, P.: Navier-Stokes equations and area of interfaces. Commun. Math. Phys. 129, 241–266 (1990)

    MathSciNet  MATH  Google Scholar 

  8. Fujigaki, Y., Miyakawa, T.: Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space. SIAM J. Math. Anal. 33, 523–544 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Fujigaki, Y., Miyakawa, T.: Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the half-space. Methods Appl. Anal. 8, 121–158 (2002)

    MATH  Google Scholar 

  10. Fujigaki, Y., Miyakawa, T.: On solutions with fast decay of nonstationary Navier-Stokes system in the half-space. To appear in Nonlinear Problems in Mathematical Physics and Related Topics, I. Kluwer/Plenum, New York, 2002

  11. Giga, Y., Sohr, H.: On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 103–130 (1989)

    MathSciNet  Google Scholar 

  12. Giga, Y., Sohr, H.: Abstract L p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)

    MathSciNet  MATH  Google Scholar 

  13. He, C., Miyakawa, T.: On weighted norm estimates for solutions to nonstationary incompressible Navier-Stokes equations in an exterior domain. Preprint, Kobe University, 2002

  14. Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    MathSciNet  MATH  Google Scholar 

  15. Iwashita, H.: L q –L r estimates for solutions of nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problem in L q spaces. Math. Ann. 285, 265–288 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Kozono, H.: L 1–solutions of the Navier-Stokes equations in exterior domains. Math. Ann. 312, 319–340 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Kozono, H., Ogawa, T.: Some L p estimate for the exterior Stokes flow and an application to the non-stationary Navier-Stokes equations. Indiana Univ. Math. J. 41, 789–808 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Kozono, H., Ogawa, T., Sohr, H.: Asymptotic behavior in L r for weak solutions of the Navier-Stokes equations in exterior domains. Manuscripta Math. 74, 253–275 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Lions, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 1, Incompressible Models, Oxford University Press, 1996

  20. Miyakawa, T.: On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115–140 (1982)

    MathSciNet  MATH  Google Scholar 

  21. Miyakawa, T.: Hardy spaces of solenoidal vector fields, with applications to the Navier-Stokes equations. Kyushu J. Math. 50, 1–64 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Miyakawa, T.: Application of Hardy space techniques to the time-decay problem for incompressible Navier-Stokes flows in ℝn. Funkcial. Ekvac. 41, 383–434 (1998)

    MathSciNet  Google Scholar 

  23. Miyakawa, T.: On upper and lower bounds of rates of decay for nonstationary Navier-Stokes equations in the whole space. To appear in Hiroshima Math. J. (2002)

  24. Miyakawa, T., Schonbek, M.E.: On optimal decay rates for weak solutions to the Navier-Stokes equations in ℝn. Math. Bohem. 126, 443–455 (2001)

    MathSciNet  MATH  Google Scholar 

  25. Miyakawa, T., Sohr, H.: On energy inequality, smoothness and large time behavior in L 2 for weak solutions of the Navier-Stokes equations in exterior domains. Math. Z. 199, 455–478 (1988)

    MathSciNet  MATH  Google Scholar 

  26. Mizumachi, R.: On the asymptotic behavior of incompressible viscous fluid motions past bodies. J. Math. Soc. Japan 36, 498–522 (1984)

    Google Scholar 

  27. Stein, E.M.: Harmonic Analysis. Princeton University Press, Princeton, 1993

  28. von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Friedrich Vieweg & Sohn, Braunschweig/Wiesbaden, 1985

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe, 657-8501, Japan

    Cheng He & Tetsuro Miyakawa

  2. Insitute of Applied Mathematics, Academy of Mathematics and System Sciences, Academia Sinica, Beijing, 100080, China

    Cheng He

  3. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japn

    Tetsuro Miyakawa

Authors
  1. Cheng He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tetsuro Miyakawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tetsuro Miyakawa.

Additional information

Mathematics Subject Classifications (1991): 35Q30, 76D05.

On leave of absence from Institute of Applied Mathematics, Academy of Mathematics and System Sciences. Academia Sinica, Beijing 100080, People’s Republic of China. Supported by JSPS

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, C., Miyakawa, T. On L 1-summability and asymptotic profiles for smooth solutions to Navier–Stokes equations in a 3D exterior domain. Math. Z. 245, 387–417 (2003). https://doi.org/10.1007/s00209-003-0551-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0551-x

Keywords

Search

Navigation