Inversion of the lagrange theorem in the problem of stability of rotating viscous incompressible liquid

Abstract

The paper contains analysis of the spectrum of a linear problem arising in the study of the stability of a finite isolated mass of uniformly rotating viscous incompressible self-gravitating liquid. It is assumed that the capillary forces on the free boundary of the liquid are not taken into account. It is proved that when the second variation of the energy functional can take negative values, then the spectrum of the problem contains finite number of points with positive real parts, which means instability of the rotating liquid in a linear approximation. The proof relies on the theorem on the invariant subspaces of dissipative operators in the Hilbert space with an indefinite metrics.