Spectral Analysis and Hydrodynamic Manifolds for the Linearized Shakhov Model

We perform a complete spectral analysis of the linearized Shakhov model involving two relaxation times ${\tau }_{\mathrm{f}\mathrm{a}\mathrm{s}\mathrm{t}}$ and ${\tau }_{\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w}}$. Our results are based on spectral functions derived from the theory of finite-rank perturbations, which allows us to infer the existence of a critical wave number $k_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ limiting the number of discrete eigenvalues above the essential spectrum together with the existence of a finite-dimensional slow manifold defining non-local hydrodynamics. We discuss the merging of hydrodynamic modes as well as the existence of second sound and the appearance of ghost modes beneath the essential spectrum in dependence of the Prandtl number.