Spectral analysis of linearized stationary equations of viscous compressible fluid in $\mathbb {R}^3$, with periodic boundary conditions

Abstract

The operator whose spectrum is studied corresponds to linearized stationary equations of viscous compressible fluid in $\mathbb {R}^3$, with periodic boundary conditions. The equations are obtained by linearization of the nonlinear model equations of viscous compressible fluid near an arbitrary solution depending on the variable $x$. It is proved that the operator in question is sectorial and that its spectrum is discrete. Also, a subset of the complex plane that contains the spectrum is described. The resolvent is estimated off a sector in the complex plane that is symmetric with respect to the real axis.