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Journal of Mathematical Sciences

, Volume 130, Issue 4, pp 4893–4898 | Cite as

Nonuniqueness of a Solution to the Problem on Motion of a Rigid Body in a Viscous Incompressible Fluid

  • V. N. Starovoitov
Article

Abstract

This paper is devoted to the problem on motion of a rigid body in a viscous incompressible fluid. It is proved that there exist at least two weak solutions of this problem if collisions of the body with the boundary of the flow domain are allowed. These solutions have different behavior of the body after the collision. Namely, for the first solution, the body goes away from the boundary after the collision. In the second solution, the body and the boundary remain in contact. Bibliography 15 titles.

Keywords

Weak Solution Rigid Body Incompressible Fluid Flow Domain Viscous Incompressible Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    N. V. Yudakov, “Solvability of the problem on motion of a rigid body is a viscous incompressible fluid,” Dinamika Sploshnoy Sredy, 18, 249–253 (1974).Google Scholar
  2. 2.
    K.-H. Hoffmann and V. N. Starovoitov, “On a motion of a solid body in a viscous fluid,” Preprint M9617, Technische Universitat Munchen (1996).Google Scholar
  3. 3.
    K.-H. Hoffmann and V. N. Starovoitov, “On a motion of a solid body in a viscous fluid. Two-dimensional case,” Adv. Math. Sci. Appl., 9, 633–648 (1999).Google Scholar
  4. 4.
    G. P. Galdi, “On the steady self-propelled motion of a body in a viscous incompressible fluid,” Arch. Rat. Mech. Anal., 148, 53–88 (1999).CrossRefGoogle Scholar
  5. 5.
    B. Desjardins and M. J. Esteban, “Existence of weak solutions for the motion of rigid bodies in a viscous fluid,” Arch. Rat. Mech. Anal., 146, 59–71 (1999).CrossRefGoogle Scholar
  6. 6.
    K.-H. Hoffmann and V. N. Starovoitov, “Zur Bewegung einer Kugel in einer zahen Flussigkeit,” Documenta Mathematica, 5, 15–21 (2000).Google Scholar
  7. 7.
    V. N. Starovoitov, “Irregular problems of hydrodynamics,” Doctoral dissertation, Novosibirsk State University (2000).Google Scholar
  8. 8.
    C. Conca, J. A. San Martin, and M. Tucsnak, “Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid,” Comm. Partial Differential Equations, 25, 1019–1042 (2000).Google Scholar
  9. 9.
    M. D. Gunzburger, H.-C. Lee, and G. Seregin, “Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions,” J. Math. Fluid Mech., 2, 219–266 (2000).Google Scholar
  10. 10.
    J. A. San Martin, V. N. Starovoitov, and M. Tucsnak, “Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid,” Arch. Rat. Mech. Anal., 161, 113–147 (2002).CrossRefGoogle Scholar
  11. 11.
    V. N. Starovoitov, “Behavior of a rigid body in an incompressible viscous fluid near a boundary,” in: Proceedings of the International Conference “Free Boundary Problems: Theory and Applications,” Trento, Italy (2002) (to appear).Google Scholar
  12. 12.
    E. Feireisl, “On the motion of rigid bodies in a viscous fluid,” Applications of Mathematics, 47, 463–484 (2002).CrossRefGoogle Scholar
  13. 13.
    T. Takahashi, “Existence of strong solutions for the problem of a rigid-fluid system,” C. R. Acad. Sci. Paris, Ser. I, 336, 453–458 (2003).Google Scholar
  14. 14.
    R. Temam, Problemes Mathematiques en Plasticite, Methodes Mathematiques de l’Informatique, 12. Gauthier-Villars (1983).Google Scholar
  15. 15.
    O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid [in Russian], Moscow (1970).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. N. Starovoitov
    • 1
  1. 1.Lavrentyev Institute of HydrodynamicsNovosibirskRussia

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