Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid

We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. Zhukovskiy's Theorem states that in the limit of time going to infinity, the relative fluid velocity tends to 0 and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. We give a rigorous proof of this result. In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.