A Potential-Theoretic Approach to the Time-Dependent Oseen System



We consider an initial-boundary value problem for the time-dependent Oseen system in a 3D exterior domain. This problem is reduced to an integral equation for the single layer potential related to the the Oseen system. The resolution of this integral equation, in turn, is reduced to a result by Shen, American Journal of Mathematics, 113, 293–373, 1991 on the nonstationary Stokes system.


Time dependent Oseen system Single layer potential Integral equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brown, R. M.: Layer Potentials and Boundary Value Problems for the Heat Equation in Lipschitz Domains. Thesis, University of Minnesota (1987)Google Scholar
  2. 2.
    Deuring, P.: The Stokes System in an Infinite Cone. Akademie Verlag, Berlin (1994)Google Scholar
  3. 3.
    Deuring, P.: The single-layer potential associated with the time-dependent Oseen system. Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics. Chalkida, Greeece, May 11–13, 117–125 (2006)Google Scholar
  4. 4.
    Deuring, P.: On volume potentials related to the time-dependent Oseen system. WSEAS Trans. Math. 5, 252–259 (2006)MathSciNetGoogle Scholar
  5. 5.
    Deuring, P.: On boundary-driven time-dependent Oseen flows. Banach Center Publ. 81, 119–132 (2008)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Deuring, P.: Spatial decay of time-dependent Oseen flows. SIAM J. Math. Anal. 41, 886–922 (2009)CrossRefMathSciNetGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    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)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Farwig, R.: The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211, 409–447 (1992)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fučik, S., John, O., Kufner, A.: Function Spaces. Noordhoff, Leyden (1977)Google Scholar
  12. 12.
    Galdi, G. P., Silvestre, A. S.: Strong solutions to the Navier-Stokes equations around a rotating obstacle. Arch. Rat. Mech. Appl. 176, 331–350 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Galdi, G. P., Silvestre, S. A.: The steady motion of a Navier-Stokes liquid around a rigid body. Arch. Rat. Mech. Appl. 184, 371–400 (2007)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, Berlin e.a. (1996)Google Scholar
  15. 15.
    Knightly, G. H.: Some decay properties of solutions of the Navier-Stokes equations. In: R. Rautmann (ed.), Approximation Methods for Navier-Stokes Problems. Lecture Notes in Math. 771, 287–298, Springer (1979)Google Scholar
  16. 16.
    Kobayashi, T., Shibata, Y.: On the Oseen equation in three dimensional exterior domains. Math. Ann. 310, 1–45 (1998)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Mikhlin, S. G., Prössdorf, S.: Singular Integral Operators. Springer, Berlin e.a. (1986)Google Scholar
  18. 18.
    Nečcas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris (1967)Google Scholar
  19. 19.
    Shen, Z.: Boundary value problems for parabolic Lamée systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders. Am. J. Math. 113, 293–373 (1991)MATHCrossRefGoogle Scholar
  20. 20.
    Shibata, Y.: On an exterior initial boundary value problem for Navier-Stokes equations. Quarterly Appl. Math. 57, 117–155 (1999)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Univ Lille Nord de FranceLilleFrance
  2. 2.ULCO, LMPACalaisFrance

Personalised recommendations