In this talk, we will present a recent result on fluid solid interaction problem. We consider the system formed by the incompressible Navier Stokes equations coupled with Newton’s laws to describe the motion of a finite number of homogeneous rigid disks within a viscous homogeneous incompressible fluid in the whole space R2 . The motion of the rigid bodies inside the fluid makes the fluid domain time dependent and unknown a priori. First, we generalize the existence and uniqueness of strong solutions result of the considered system in the case of a single rigid body moving in a bounded cavity in , and then by careful analysis of how elliptic estimates for the Stokes operator depend on the geometry of the fluid domain, we extend these solutions up to collision. Finally, we prove contact between rigid bodies cannot occur for almost arbitrary configurations by studying the distance between solids by a multiplier approach . This talk is based on the results of the preprint .
 Gérard-Varet, D., Hillairet, M., Regularity issues in the problem of fluid structure interaction, Arch. For ration. Mech. Anal., page 375-407 (2010).
 Sabbagh, L., On the motion of several disks in an unbounded viscous incompressible fluid, in progress.
 Takahashi, T., Analysis of strong solutions for the equation modelling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, page 1499-1532 (2003).