Skip to main content
SpringerLink
Log in

The 3D Time-Dependent Oseen System: Link Between \(L^p\)-Integrability in Time and Pointwise Decay in Space of the Velocity

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

Abstract

A representation formula without pressure term is derived for regular solutions to the 3D time-dependent Oseen system in exterior Lipschitz domains. This formula is valid even if no boundary conditions are imposed. It is used in order to exhibit how the velocity decays pointwise in space. It turns out that the rate of this decay depends on \(L^p\)-integrability in time of the velocity. In addition, this work is the basis for successor papers dealing with spatial decay of \(L^q\)-weak solutions and mild solutions to the time-dependent Oseen system, and with \(L^2\)-strong solutions to the stability problem related to the Navier-Stokes system with Oseen term.

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.

Similar content being viewed by others

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)

    MATH  Google Scholar 

  2. Bae, H.-O., Jin, B.J.: Estimates of the wake for the 3D Oseen equations. Discrete Contin. Dyn. Syst. Ser. B 10, 1–18 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Bae, H.-O., Roh, J.: Stability for the 3D Navier–Stokes equations with nonzero far field velocity on exterior domains. J. Math. Fluid Mech. 14, 117–139 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Borchers, W., Sohr, H.: On the equations \(\text{ rot }\, v =g\) and \(\text{ div }\, u =f\) with zero boundary conditions. Hokkaido Math. J. 19, 67–87 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Deuring, P.: The single-layer potential associated with the time-dependent Oseen system. In: Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics. Chalkida, Greeece, May 11–13, pp. 117–125 (2006)

  6. Deuring, P.: On volume potentials related to the time-dependent Oseen system. WSEAS Trans. Math. 5, 252–259 (2006)

    MathSciNet  Google Scholar 

  7. Deuring, P.: On boundary driven time-dependent Oseen flows. Banach Center Publ. 81, 119–132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deuring, P.: A potential theoretic approach to the time-dependent Oseen system. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics. Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday, pp. 191–214. Springer, Berlin (2010)

    Google Scholar 

  9. Deuring, P.: Spatial decay of time-dependent Oseen flows. SIAM J. Math. Anal. 41, 886–922 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deuring, P.: A representation formula for the velocity part of 3D time-dependent Oseen flows. J. Math. Fluid Mech. 16, 1–39 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Deuring, P.: The Cauchy problem for the homogeneous time-dependent Oseen system in \( \mathbb{R}^3 \): spatial decay of the velocity. Math. Bohemica 138, 299–324 (2013a)

    MathSciNet  MATH  Google Scholar 

  12. Deuring, P.: Pointwise spatial decay of time-dependent Oseen flows: the case of data with noncompact support. Discrete Contin. Dyn. Syst. Ser. A 33, 2757–2776 (2013b)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deuring, P.: Spatial decay of time-dependent incompressible Navier–Stokes flows with nonzero velocity at infinity. SIAM J. Math. Anal. 45, 1388–1421 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deuring, P.: Pointwise spatial decay of weak solutions to the Navier–Stokes system in 3D exterior domains. J. Math. Fluid Mech. 17, 199–232 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Deuring, P.: Pointwise decay in space and in time for incompressible flow around a rigid body moving with constant velocity. J. Math. Fluid Mech. 21, article 11 (2019) (35 pages)

  16. Deuring, P.: \(L^q\)-weak solutions to the time-dependent Oseen system: decay estimates. To appear in Math. Nachr. https://hal.archives-ouvertes.fr/hal-02465651

  17. Deuring, P.: Time-dependent incompressible viscous flows around a rigid body: estimates of spatial decay independent of boundary conditions. Submitted. https://hal.archives-ouvertes.fr/hal-02508815

  18. Deuring, P.: Spatial asymptotics of mild solutions to the time-dependent Oseen system. To appear in Commun. Pure Appl. Anal. https://hal.archives-ouvertes.fr/hal-02546017

  19. Deuring, P., Kračmar, S.: Exterior stationary Navier–Stokes flows in 3D with non-zero velocity at infinity: approximation by flows in bounded domains. Math. Nachr. 269–270, 86–115 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Deuring, P., Kračmar, S.: Nečasová, Š: On pointwise decay of linearized stationary incompressible viscous flow around rotating and tranlating bodies. SIAM J. Math. Anal. 43, 705–738 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deuring, P., Varnhorn, W.: On Oseen resolvent estimates. Diff. Int. Equat. 23, 1139–1149 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Enomoto, Y., Shibata, Y.: Local energy decay of solutions to the Oseen equation in the exterior domain. Indiana Univ. Math. J. 53, 1291–1330 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Enomoto, Y., Shibata, Y.: On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier–Stokes equation. J. Math. Fluid Mech. 7, 339–367 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Farwig, R.: The stationary exterior 3D-problem of Oseen and Navier–Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211, 409–447 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fučik, S., John, O., Kufner, A.: Function Spaces. Noordhoff, Leyden (1977)

    MATH  Google Scholar 

  26. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York e (2011)

    MATH  Google Scholar 

  27. Galdi, G.P., Heywood, J.G., Shibata, Y.: On the global existence and convergence to steady state of Navier–Stokes flow past an obstacle that is started from rest. Arch. Rational Mech. Anal. 138, 307–318 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Heywood, J.G.: The exterior nonstationary problem for the Navier–Stokes equations. Acta Math. 129, 11–34 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Heywood, J.G.: The Navier–Stokes equations. On the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups, vol. 31. American Mathematical Society, Colloquium Publications, Providence, RI (1957)

    MATH  Google Scholar 

  31. Hishida, T.: Large time behavior of a generalized Oseen evolution operator, with applications to the Navier–Stokes flow past a rotating obstacle. Math. Ann. 372, 915–949 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hishida, T.: Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains. Arch. Rational Mech. Anal. 238, 215–254 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Knightly, G. H.: Some decay properties of solutions of the Navier–Stokes equations. In: Rautmann, R. (ed.): Approximation Methods for Navier–Stokes Problems. Lecture Notes in Math. 771, Springer, pp. 287–298 (1979)

  34. Kobayashi, T., Shibata, Y.: On the Oseen equation in three-dimensional exterior domains. Math. Ann. 310, 1–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kozono, H.: Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains. Math. Ann. 320, 709–730 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Masuda, K.: On the stability of incompressible viscous fluid motions past bodies. J. Math. Soc. Jpn. 27, 294–327 (1975)

    Article  MATH  Google Scholar 

  37. 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 

  38. Mizumachi, R.: On the asymptotic behaviour of incompressible viscous fluid motions past bodies. J. Math. Soc. Jpn. 36, 497–522 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shen, Zongwei: Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier–Stokes equations in Lipschitz cylinders. Am. J. Math. 113, 293–373 (1991)

    Article  MATH  Google Scholar 

  40. Simader, C.G., Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics, vol. 321. CRC Press, London (1996)

    MATH  Google Scholar 

  41. Shibata, Y.: On an exterior initial boundary value problem for Navier–Stokes equations. Q. Appl. Math. 57, 117–155 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sohr, H.: The Navier–Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  43. Solonnikov, V., A.: A priori estimates for second order parabolic equations. Trudy Mat. Inst. Steklov. 70, : 133–212 (Russian). English translation, AMS Translations 65(1967), 51–137 (1964)

  44. Solonnikov, V., A.: Estimates for solutions of nonstationary Navier-Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38, : 153–231 (Russian); English translation. J. Soviet Math. 8(1977), 467–529 (1973)

  45. Takahashi, S.: A weighted equation approach to decay rate estimates for the Navier–Stokes equations. Nonlinear Anal. 37, 751–789 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  46. Teman, R.: Navier–Stokes Equations. Theory and Numerical Analysis. AMS Chelsea Publishing, Providence R.I. (2001)

    Google Scholar 

  47. Weis, L.: Operator-valued Fourier multiplier theorems and maximal regularity. Math. Ann. 319, 735–758 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Yoshida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. EA 2797 – LMPA – Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Université du Littoral Côte d’Opale, LMPA, CS 80699, 62226, Calais, France

    Paul Deuring

Authors
  1. Paul Deuring
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Deuring.

Ethics declarations

Conflict of interest

There is no conflict of interest related to this article.

Additional information

Communicated by G. P. Galdi

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deuring, P. The 3D Time-Dependent Oseen System: Link Between \(L^p\)-Integrability in Time and Pointwise Decay in Space of the Velocity. J. Math. Fluid Mech. 23, 46 (2021). https://doi.org/10.1007/s00021-021-00562-6

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-021-00562-6

Keywords

Mathematics Subject Classification

Search

Navigation