On the stability of non-symmetric equilibrium figures of a rotating viscous incompressible liquid

We consider a classical problem of stability of equilibrium figures of a liquid rotating uniformly as a rigid body about a fixed axis. We connect the problem of stability with the behavior for large t of solutions of an evolution problem governing the motion of an isolated liquid mass whose initial data are slight perturbations of the regime of a rigid rotation. The main attention is given to the case when the figure is not rotationally symmetric; in this case the regime of a rigid rotation defines a periodic solution of the above-mentioned nonstationary problem. It is proved that a sufficient condition of stability is the positivity of the second variation of the energy functional in an appropriate function space.