Stability of the rotation of a two-phase drop with self-gravity

A uniformly rotating finite mass consisting of two immiscible viscous incompressible self-gravitating fluids is governed by interface problem for Navier–Stokes system with mass forces in the right-hand side. Surface tension acts on the interface as well as on the exterior free boundary. The proof of stability is based on the analysis of an evolutionary problem for small perturbations of the equilibrium state of a rotating two-phase fluid with self-gravity. It is proved that under sufficient smallness of initial data, exponentially decreasing mass forces and angular velocity, as well as the positivity of the second variation of energy functional, the perturbation of the axisymmetric equilibrium figure exponentially tends to zero as $t\to\infty $, the motion of the drop going over to the rotation of liquid mass as a solid.