On the Asymptotic Behavior in Time of the Kinetic Energy in a Rigid Body-Liquid Problem

We give sufficient conditions on the initial data for the decay in time of the kinetic energy, E, of solutions to the system of equations describing the motion of a rigid body in a Navier-Stokes liquid. More precisely, assuming the initial data ``small" in appropriate norm, we show that if, in addition, the initial velocity field of the liquid, v₀, is in Lq, q∈(1,2), then $E(t)$ vanishes as t→∞ with a specific order of decay. The order remains, however, unspecified if v₀∈ L².