Non-Hydrodynamic Solutions to the linear Density-dependent BGK equation

We prove the existence of non-hydrodynamic solutions to the linear density-dependent BGK equation in $d$ dimensions. Specifically, we show the existence of an initial condition for any Knudsen number $τ$ for which the dissipation rate of the macroscopic mass density diverges $\sim 1/τ$. Our results rely on a detailed spectral analysis of the linear BGK operator, an explicit solution formula for the time-dependent problem using a combination of Fourier series with the Laplace transform and subsequent contour integration arguments from complex analysis.