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.
Mots-clés : Fluid–structure interaction, Navier–Stokes equations, Rigid body, Cauchy theory, Qualitative properties, Collisions
@article{AIHPC_2010__27_1_291_0, 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}, publisher = {Elsevier}, volume = {27}, number = {1}, year = {2010}, doi = {10.1016/j.anihpc.2009.09.007}, mrnumber = {2580511}, zbl = {1187.35290}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.007/} }
TY - JOUR AU - Hillairet, Matthieu AU - Takahashi, Takéo TI - Blow up and grazing collision in viscous fluid solid interaction systems JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 291 EP - 313 VL - 27 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.007/ DO - 10.1016/j.anihpc.2009.09.007 LA - en ID - AIHPC_2010__27_1_291_0 ER -
%0 Journal Article %A Hillairet, Matthieu %A Takahashi, Takéo %T Blow up and grazing collision in viscous fluid solid interaction systems %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 291-313 %V 27 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.007/ %R 10.1016/j.anihpc.2009.09.007 %G en %F AIHPC_2010__27_1_291_0
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. http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.007/
[1] 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] 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] 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] 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] 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] Existence for an unsteady fluid–structure interaction problem, M2AN Math. Model. Numer. Anal. 34 no. 3 (2000), 609-636 | EuDML | Numdam | MR | Zbl
, ,[8] 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] 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] Collisions in three-dimensional fluid structure interaction problems, SIAM J. Math. Anal. 40 no. 6 (2009), 2451-2477 | MR | Zbl
, ,[12] On the motion and collisions of rigid bodies in an ideal fluid, Asymptot. Anal. 56 no. 3–4 (2008), 125-158 | MR | Zbl
, ,[13] On the slow motion of a sphere parallel to a nearby plane wall, J. Fluid Mech. 27 (1967), 705-724 | MR | Zbl
, ,[14] 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] 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] 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] 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] Problèmes mathématiques en plasticité, Gauthier–Villars, Montrouge (1983) | MR | Zbl
,Cité par Sources :