Inertial motions of a rigid body with a cavity filled with a viscous liquid

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. (C) 2013 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.