On the Problem of Evolution of an Isolated Liquid Mass

The paper is concerned with the problem of stability of equilibrium figures of a uniformly rotating, viscous, incompressible, self-gravitating liquid subjected to capillary forces at the boundary. It is shown that a rotationally symmetric equilibrium figure $F$ is exponentially stable if the functional $G$ defined on the set of domains $\Omega$ close to $F$ and satisfying the conditions of volume invariance ($|\Omega|=|F|$) and the barycenter position attains its minimum for $\Omega=F$. The proof is based on the direct analysis of the corresponding evolution problem with initial data close to the regime of a rigid rotation.