Exact Hydrodynamic Manifolds for the Linear Boltzmann BGK Equation I: Spectral Theory

TL;DR

This paper provides a detailed spectral analysis of the linear BGK operator in the Boltzmann equation, establishing the existence of a finite-dimensional hydrodynamic manifold and identifying a critical wave number that limits hydrodynamic modes.

Contribution

It offers an explicit spectral characterization of the BGK operator, confirming the finite number of hydrodynamic modes and introducing a basis for spectral closure methods.

Findings

Existence of a critical wave number $k_{crit}$ limiting hydrodynamic modes

Finite number of isolated eigenvalues above the essential spectrum

Validation benchmark for hydrodynamic closure theories

Abstract

We perform a complete spectral analysis of the linear three-dimensional Boltzmann BGK operator resulting in an explicit transcendental equation for the eigenvalues. Using the theory of finite-rank perturbations, we confirm the existence of a critical wave number $k_{\rm crit}$ which limits the number of hydrodynamic modes in the frequency space. This implies that there are only finitely many isolated eigenvalues above the essential spectrum at each wave number, thus showing the existence of a finite-dimensional, well-separated linear hydrodynamic manifold as a combination of invariant eigenspaces. The obtained results can serve as a benchmark for validating approximate theories of hydrodynamic closures and moment methods and provides the basis for the spectral closure operator.