On the asymptotic behavior in time of the kinetic energy in a rigid body–liquid problem

Sufficient conditions on the initial data are given 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, under the assumption the initial data are “small” in an appropriate norm, it is shown 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 $t\to\infty$ with a specific order of decay. The order remains, however, unspecified if $v_0\in L^2$.