Spectral analysis of viscous static compressible fluid equilibria

It is generally assumed that the study of the spectrum of the linearized Na vier-Stokes equations around a static state will provide information about the stability of the equilibrium. This is obvious for inviscid barotropic c ompressible fluids by the self-adjoint character of the relevant operator, and rather easy for viscous incompressible fluids by the compact character of the resolvent. The viscous compressible linearized system, both for peri odic and homogeneous Dirichlet boundary problems, satisfies neither conditi on, but it does turn out to be the generator of an immediately continuous, almost stable semigroup, which justifies the analysis of the spectrum as pr edictive of the initial behaviour of the flow. As for the spectrum itself, except for a unique negative finite accumulation point, it is formed by eig envalues with negative real part, and nonreal eigenvalues are confined to a certain bounded subset of complex numbers.