  We consider the inertial motion of a system constituted by a rigid body with
an interior cavity entirely filled with a viscous incompressible fluid. Navier
boundary conditions are imposed on the cavity surface. We prove the existence
of weak solutions and determine the critical spaces for the governing evolution
equation. Using parabolic regularization in time-weighted spaces, we establish
regularity of solutions and their long-time behavior. We show that every weak
solution \`a la Leray-Hopf to the equations of motion converges to an
equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid
configuration. A nonlinear stability analysis shows that equilibria associated
with the largest moment of inertia are asymptotically (exponentially) stable,
whereas all other equilibria are normally hyperbolic and unstable in an
appropriate topology.
