Geodesic
Non-uniqueness of the solution to the problem of a motion of a rigid body in a viscous incompressible fluid
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 199-209 
V. N. Starovoitov. Non-uniqueness of the solution to the problem of a motion of a rigid body in a viscous incompressible fluid. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 199-209. http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a9/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

This paper is devoted to the investigation of the problem on a 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 the 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.

[1] N. V. Yudakov, “Razreshimost zadachi o dvizhenii tverdogo tela v vyazkoi neszhimaemoi zhidkosti”, Dinamika sploshnoi sredy, 18, 1974, 249–253

[2] K.-H. Hoffmann, V. N. Starovoitov, On a motion of a solid body in a viscous fluid, Preprint M9617, Technische Universität München, 1996

[3] K.-H. Hoffmann, V. N. Starovoitov, “On a motion of a solid body in a viscous fluid. Two–dimensional case”, Adv. Math. Sci. Appl., 9:2 (1999), 633–648 | MR | Zbl

[4] G. P. Galdi, “On the steady self-propelled motion of a body in a viscous incompressible fluid”, Arch. Ration. Mech. Anal., 148:1 (1999), 53–88 | DOI | MR | Zbl

[5] B. Desjardins, M. J. Esteban, “Existence of weak solutions for the motion of rigid bodies in a viscous fluid”, Arch. Ration. Mech. Anal., 146:1 (1999), 59–71 | DOI | MR | Zbl

[6] K.-H. Hoffmann, V. N. Starovoitov, “Zur Bewegung einer Kugel in einer zähen Flüssigkeit”, Documenta Mathematica, 5 (2000), 15–21 | MR | Zbl

[7] V. N. Starovoitov, Neregulyarnye zadachi gidrodinamiki, Doktorskaya dissertatsiya, Novosibirskii gosudarstvennyi universitet, 2000

[8] C. Conca, J. A. San Martin, M. Tucsnak, “Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid”, Comm. Partial Differential Equations, 25:5–6 (2000), 1019–1042 | MR | Zbl

[9] M. D. Gunzburger, H.-C. Lee, G. Seregin, “Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions”, J. Math. Fluid Mech., 2:3 (2000), 219–266 | DOI | MR | Zbl

[10] J. A. San Martin, V. N. Starovoitov, M. Tucsnak, “Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid”, Arch. Ration. Mech. Anal., 161:2 (2002), 113–147 | DOI | MR | Zbl

[11] V. N. Starovoitov, “Behavior of a rigid body in an incompressible viscous fluid near a boundary”, Free boundary problems: theory and applications, Proceedings of the International Conference (Trento, Italy, 2002) (to appear)

[12] E. Feireisl, “On the motion of rigid bodies in a viscous fluid”, Applications of Mathematics, 47:6 (2002), 463–484 | DOI | MR | Zbl

[13] T. Takahashi, “Existence of strong solutions for the problem of a rigid–fluid system”, C. R. Acad.Sci.Paris, Ser. I, 336:5 (2003), 453–458 | MR | Zbl

[14] R. Temam, Problèmes mathématiques en plasticité, Méthodes Mathématiques de l'Informatique, 12, Gauthier-Villars, Montrouge, 1983 | MR | Zbl

[15] O. A. Ladyzhenskaya, Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Nauka, M., 1970 | MR