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Blow up and grazing collision in viscous fluid solid interaction systems. (English) Zbl 1187.35290

Summary: We investigate qualitative properties of strong solutions to a classical system describing the fall of a rigid ball under the action of gravity inside a bounded cavity filled with a viscous incompressible fluid. We prove contact between the ball and the boundary of the cavity implies blow up of strong solutions and such a contact has to occur in finite time under symmetry assumptions on the initial data.

MSC:

35R35 Free boundary problems for PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35B44 Blow-up in context of PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
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[1] Conca, C.; San Martín, J.; Tucsnak, M., Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. partial differential equations, 25, 5-6, 1019-1042, (2000) · Zbl 0954.35135
[2] Cooley, M.; O’Neill, M., On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere, Mathematika, 16, 37-49, (1969) · Zbl 0174.27703
[3] Desjardins, B.; Esteban, M.J., Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. ration. mech. anal., 146, 1, 59-71, (1999) · Zbl 0943.35063
[4] Desjardins, B.; Esteban, M.J., On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. partial differential equations, 25, 7-8, 1399-1413, (2000) · Zbl 0953.35118
[5] Feireisl, E., On the motion of rigid bodies in a viscous incompressible fluid, J. evol. equ., 3, 3, 419-441, (2003), dedicated to Philippe Bénilan · Zbl 1039.76071
[6] D. Gérard-Varet, M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., in press · Zbl 1192.35131
[7] Grandmont, C.; Maday, Y., Existence for an unsteady fluid – structure interaction problem, M2AN math. model. numer. anal., 34, 3, 609-636, (2000) · Zbl 0969.76017
[8] Gunzburger, M.D.; Lee, H.-C.; Seregin, G.A., Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. math. fluid mech., 2, 3, 219-266, (2000) · Zbl 0970.35096
[9] M. Hillairet, Interactive features in fluid mechanics, PhD thesis, Ecole normale supérieure de Lyon, 2005
[10] Hillairet, M., Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. partial differential equations, 32, 7-9, 1345-1371, (2007) · Zbl 1221.35279
[11] Hillairet, M.; Takahashi, T., Collisions in three-dimensional fluid structure interaction problems, SIAM J. math. anal., 40, 6, 2451-2477, (2009) · Zbl 1178.35291
[12] Houot, J.; Munnier, A., On the motion and collisions of rigid bodies in an ideal fluid, Asymptot. anal., 56, 3-4, 125-158, (2008) · Zbl 1165.35300
[13] O’Neill, M.E.; Stewartson, K., On the slow motion of a sphere parallel to a nearby plane wall, J. fluid mech., 27, 705-724, (1967) · Zbl 0147.45302
[14] San Martín, J.A.; Starovoitov, V.; Tucsnak, M., Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. ration. mech. anal., 161, 2, 113-147, (2002) · Zbl 1018.76012
[15] Starovoĭtov, V.N., On the nonuniqueness of the solution of the problem of the motion of a rigid body in a viscous incompressible fluid, Zap. nauchn. sem. S.-Petersburg. otdel. mat. inst. Steklov. (POMI), Kraev. zadachi mat. fiz. i smezh. vopr. teor. funktsii, 34, 231-232, (2003)
[16] Starovoitov, V.N., Behavior of a rigid body in an incompressible viscous fluid near a boundary, (), 313-327 · Zbl 1060.76038
[17] Takahashi, T., Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. differential equations, 8, 12, 1499-1532, (2003) · Zbl 1101.35356
[18] Temam, R., Problèmes mathématiques en plasticité, (1983), Gauthier-Villars Montrouge · Zbl 0547.73026
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