Regularity Issues in the Problem of Fluid Structure Interaction

We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier–Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have Hölder regularity C ^1,α , 0 < α ≦ 1. First, we show the existence and uniqueness of strong solutions up to the collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard H ² estimate for smoother domains. Then, we study the asymptotic behaviour of one C ^1,α body falling over a flat surface. We show that a collision is possible in finite time if and only if α < 1/2.