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