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We perform a complete spectral analysis of the linear three-dimensionalBoltzmann BGK operator resulting in an explicit transcendental equation for theeigenvalues. Using the theory of finite-rank perturbations, we prove that thereexists a critical wave number $k_{crit}$ which limits the number ofhydrodynamic modes in the frequency space. This implies that there are onlyfinitely many isolated eigenvalues above the essential spectrum, thus showingthe existence of a finite-dimensional, well-separated linear hydrodynamicmanifold as a combination of invariant eigenspaces. The obtained results canserve as a benchmark for validating approximate theories of hydrodynamicclosures and moment methods.

Exact Hydrodynamic Manifolds for the Linearized Three-Dimensional Boltzmann BGK Equation