We study inertial motions of the coupled system, (Formula presented.) , constituted by a rigid body containing a cavity entirely filled with a viscous liquid. We show that for arbitrary initial data having only finite kinetic energy, every corresponding weak solution (à la Leray–Hopf) converges, as time goes to infinity, to a uniform rotation, unless two central moments of inertia of (Formula presented.) coincide and are strictly greater than the third one. This corroborates a famous “conjecture” of N.Ye. Zhukovskii in several physically relevant cases. Moreover, we show that, in a known range of initial data, this rotation may only occur along the central axis of inertia of (Formula presented.) with the larger moment of inertia. We also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations, which improve and/or generalize results previously given by other authors under different types of approximation. Finally, we present 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

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.2010 Mathematics Subject Classification. Primary: 35Q35, 35Q30, 35B40, 35K58, 76D05.

In this note we announce a number of analytical and numerical results related to the motion of a system S constituted by a rigid body with a cavity that is completely filled with a Navier-Stokes liquid, and that moves in absence of external forces (inertial motions). Our investigation shows, in particular, that the ultimate motion of S about its center of mass is a permanent rotation, thus proving a longstanding conjecture of N.Ye. Zhukovskii. We also present other interesting features of inertial motions that are emphasized by our numerical tests, but that still lack a rigorous mathematical proof. © 2013 Académie des sciences

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