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, $v_0$, is in $L^q$, $q\in (1,2)$, then $E(t)$ vanishes as $t\to \mathrm{\infty }$ with a specific order of decay. The order remains, however, unspecified if $v_0\in L^2$.