On nonlinear instability of capillary equilibrium figures

Problem of nonlinear instability of equilibrium figures F* of incompressible fluids, under potential external forces, is studied. First, the problem of stability of an equilibrium figure F* for an abstract system is reduced to the sign of the difference between the energy of the perturbed motion, and that of F*, say energy of perturbation. All control conditions are only sufficient conditions. Second, employing the local character of the nonlinear stability, some nonlinear instability theorems are proven by direct method. Third, may be less important, the definition of loss of control from initial data for F* it introduced. It is constructed a class of equilibrium figures F* such that: F* is nonlinearly stable; the motions, corresponding to initial data sufficiently far from F*, cannot be controlled from their initial data for all time. We compute a lower bound for the norms of initial data above which it occurs the loss of control from initial data.