  We study inertial motions of the coupled system, S, constituted by a rigid
body containing a cavity that is completely filled with a viscous liquid. We
show that for data of arbitrary size (initial kinetic energy and total angular
momentum) every weak solution (a la Leray-Hopf) converges, as time goes to
infinity, to a uniform rotation, thus proving a famous "conjecture" of
Zhukovskii. Moreover we show that, in a wide range of initial data, this
rotation must occur along the central axis of inertia of S that has the largest
moment of inertia. Furthermore, necessary and sufficient conditions for the
rigorous nonlinear stability of permanent rotations are provided, which improve
and/or generalize results previously given by other authors under different
types of approximation of the original equations and/or suitable symmetry
assumptions on the shape of the cavity. Finally, we present a number of results
obtained by a targeted numerical simulation that, on the one hand, complement
the analytical findings, whereas, on the other hand, point out new features
that the analysis is yet not able to catch, and, as such, lay the foundation
for interesting and challenging future investigation.
