Existence of Weak Solutions for the Motion of Rigid Bodies in a Viscous Fluid

Abstract

. We study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of $\R^d$ $(d=2$ or $3)$ with Dirichlet boundary conditions. By introducing an appropriate weak formulation for the complete problem, we prove existence of solutions for initial velocities in $H^1_0(\Omega)$. In the absence of collisions, solutions exist for all time in dimension 2, whereas in dimension 3 the lifespan of solutions is infinite only for small enough data.