An Infinite System of Diffusion Equations Arising in Transport Theory: The Coupled Spherical Harmonics Expansion Model

In this paper, we derive a diffusion model of "SHE" type from a diffusion approximation of the Boltzmann equation. SHE (or Spherical Harmonics Expansion) models consist of a diffusion equation in the position-energy space. In the present model, the diffusion operator couples the various energy levels, while they stay uncoupled in the standard SHE model, at least in the force-free case. This new model is expected to give a better description of particle transport when inelastic collisions are important. The duality structure of the diffusion operator is underlined. A uniqueness result for the diffusion model is shown. A rigorous proof that the solutions of the Boltzmann equation converge (in a weak sense) towards those of the SHE model is developed in the force-free case.