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A rigid body, with an interior cavity entirely filled with a Navier-Stokesliquid, moves in absence of external torques relative to the center of mass ofthe coupled system body-liquid (inertial motions). The only steady-statemotions allowed are then those where the system, as a whole rigid body, rotatesuniformly around one of the central axes of inertia (permanent rotations).Objective of this article is two-fold. On the one hand, we provide sufficientconditions for the asymptotic, exponential stability of permanent rotations, aswell as for their instability. On the other hand, we study the asymptoticbehavior of the generic motion in the class of weak solutions and show thatthere exists a time $t_0$ after that all such solutions must decayexponentially fast to a permanent rotation. This result provides a {\em full}and rigorous explanation of Zhukovsky's conjecture, and explains, likewise,other interesting phenomena that are observed in both lab and numericalexperiments.

Stability of Permanent Rotations and Long-Time Behavior of Inertial Motions of a Rigid Body with an Interior Liquid-Filled Cavity