  We consider the inertial motion of a rigid body with an interior cavity that
is completely filled with a viscous incompressible fluid. The equilibria of the
system are characterized and their stability properties are analyzed. It is
shown that equilibria associated with the largest moment of inertia are
normally stable, while all other equilibria are normally hyperbolic. We show
that every Leray-Hopf weak solution converges to an equilibrium at an
exponential rate. In addition, we determine the critical spaces for the
governing evolution equation, and we demonstrate how parabolic regularization
in time-weighted spaces affords great flexibility in establishing regularity of
solutions and their convergence to equilibria.
