no. 1
Blow up and grazing collision in viscous fluid solid interaction systems
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 291-313

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.

DOI : 10.1016/j.anihpc.2009.09.007
Classification : 35R35, 76D03, 76D05
Keywords: Fluid–structure interaction, Navier–Stokes equations, Rigid body, Cauchy theory, Qualitative properties, Collisions 
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     author = {Hillairet, Matthieu and Takahashi, Tak\'eo},
     title = {Blow up and grazing collision in viscous fluid solid interaction systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {291--313},
     year = {2010},
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     zbl = {1187.35290},
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}
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Hillairet, Matthieu; Takahashi, Takéo. Blow up and grazing collision in viscous fluid solid interaction systems. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 291-313. doi: 10.1016/j.anihpc.2009.09.007

[1] C. Conca, J. San Martín, M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations 25 no. 5–6 (2000), 1019-1042 | Zbl

[2] M. Cooley, M. O'Neill, On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere, Mathematika 16 (1969), 37-49 | Zbl

[3] B. Desjardins, M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal. 146 no. 1 (1999), 59-71 | MR | Zbl

[4] B. Desjardins, M.J. Esteban, On weak solutions for fluid-rigid structure interaction: Compressible and incompressible models, Comm. Partial Differential Equations 25 no. 7–8 (2000), 1399-1413 | MR | Zbl

[5] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ. 3 no. 3 (2003), 419-441 | MR | Zbl

[6] D. Gérard-Varet, M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., in press | MR

[7] C. Grandmont, Y. Maday, Existence for an unsteady fluid–structure interaction problem, M2AN Math. Model. Numer. Anal. 34 no. 3 (2000), 609-636 | MR | EuDML | Zbl | Numdam

[8] M.D. Gunzburger, H.-C. Lee, G.A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech. 2 no. 3 (2000), 219-266 | Zbl

[9] M. Hillairet, Interactive features in fluid mechanics, PhD thesis, Ecole normale supérieure de Lyon, 2005

[10] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 no. 7–9 (2007), 1345-1371 | MR | Zbl

[11] M. Hillairet, T. Takahashi, Collisions in three-dimensional fluid structure interaction problems, SIAM J. Math. Anal. 40 no. 6 (2009), 2451-2477 | MR | Zbl

[12] J. Houot, A. Munnier, On the motion and collisions of rigid bodies in an ideal fluid, Asymptot. Anal. 56 no. 3–4 (2008), 125-158 | MR | Zbl

[13] M.E. O'Neill, K. Stewartson, On the slow motion of a sphere parallel to a nearby plane wall, J. Fluid Mech. 27 (1967), 705-724 | MR | Zbl

[14] J.A. San Martín, V. 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 no. 2 (2002), 113-147 | MR | Zbl

[15] V.N. Starovoĭtov, 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) 306 (2003), 199-209, Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsii 34 (2003), 231-232 | MR

[16] V.N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near a boundary, Free Boundary Problems, Trento, 2002, Internat. Ser. Numer. Math. vol. 147, Birkhäuser, Basel (2004), 313-327 | MR | Zbl

[17] T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations 8 no. 12 (2003), 1499-1532 | MR | Zbl

[18] R. Temam, Problèmes mathématiques en plasticité, Gauthier–Villars, Montrouge (1983) | MR | Zbl

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