Other Operator Approaches to Hydrodynamics Problems of Ideal Fluids

Classical Results and Modern Approaches to Nonconservative Stability

“…The number of pairs of complex-conjugate eigenvalues of B (counting multiplicities) does not exceed the number of negative squares of the quadratic form Q(R, R), which can be equal only to one when L < 0. Hence, for L < 0 an unstable solution R = e iλ 0 t R 0 can exist with Imλ 0 < 0; all real eigenvalues are simple except for maybe one (Kopachevskii and Krein 2001).…”

“…If the real and imaginary part of the complex number Z describe the deviation of the unit vector of the symmetry axis of the top in the coordinates x, y, and z, then these equations are, see e.g. Kopachevskii and Krein (2001); Kirillov (2013a):…”

“…Therefore, in the case of the ellipsoidal shapes of the shell and the cavity, the Hilbert space {R} = {Z , W, v} of the Sobolev's problem endowed with the indefinite metric (L < 0) decomposes into the three-dimensional space of the reduced model (79), where the self-adjoint operator B can have complex eigenvalues and real defective eigenvalues, and a complementary infinite-dimensional space, whichis free of these complications. The very idea that the signature of the indefinite metric can serve for counting unstable eigenvalues of an operator that is self-adjoint in a functional space equipped with such a metric, turned out to be a concept of a rather universal character possessing powerful generalizations that were initiated by Pontryagin in 1944(Yakubovich and Starzhinskii 1975;Kopachevskii and Krein 2001).…”