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

TL;DR

This paper proves the existence of non-hydrodynamic solutions to the linear density-dependent BGK equation across dimensions, revealing divergence in dissipation rates for any Knudsen number through spectral and complex analysis techniques.

Contribution

It introduces a rigorous proof of non-hydrodynamic solutions for the density-dependent BGK equation using spectral analysis and explicit solution formulas.

Findings

Existence of non-hydrodynamic solutions for all Knudsen numbers.

Dissipation rate of mass density diverges as 1/Knudsen number.

Spectral analysis confirms the solutions' properties.

Abstract

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 $\tau$ for which the dissipation rate of the macroscopic mass density diverges $\sim 1/\tau$. 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.