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

Abstract

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 a motion of an isolated liquid mass whose initial data are slight perturbations of the regime of 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 non-stationary problem. It is proved that the sufficient condition of stability is the positivity of the second variation of the energy functional in an appropriate space of functions.