Abstract
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 C1,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 H2 estimate for smoother domains. Then, we study the asymptotic behaviour of one C1,α body falling over a flat surface. We show that a collision is possible in finite time if and only if α< 1/2.
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Gérard-Varet, D., Hillairet, M. Regularity Issues in the Problem of Fluid Structure Interaction. Arch Rational Mech Anal 195, 375–407 (2010). https://doi.org/10.1007/s00205-008-0202-9
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DOI: https://doi.org/10.1007/s00205-008-0202-9