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.
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REFERENCES
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).
K.-H. Hoffmann and V. N. Starovoitov, “On a motion of a solid body in a viscous fluid,” Preprint M9617, Technische Universitat Munchen (1996).
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).
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).
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).
K.-H. Hoffmann and V. N. Starovoitov, “Zur Bewegung einer Kugel in einer zahen Flussigkeit,” Documenta Mathematica, 5, 15–21 (2000).
V. N. Starovoitov, “Irregular problems of hydrodynamics,” Doctoral dissertation, Novosibirsk State University (2000).
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).
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).
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).
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).
E. Feireisl, “On the motion of rigid bodies in a viscous fluid,” Applications of Mathematics, 47, 463–484 (2002).
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).
R. Temam, Problemes Mathematiques en Plasticite, Methodes Mathematiques de l’Informatique, 12. Gauthier-Villars (1983).
O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid [in Russian], Moscow (1970).
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To Vsevolod Alekseevich Solonnikov on the occasion of his jubilee
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 199–209.
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Starovoitov, V.N. Nonuniqueness of a Solution to the Problem on Motion of a Rigid Body in a Viscous Incompressible Fluid. J Math Sci 130, 4893–4898 (2005). https://doi.org/10.1007/s10958-005-0384-8
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DOI: https://doi.org/10.1007/s10958-005-0384-8