The Fredholm Alternative with Application to the Derivation of Hilbert Expansion Based Fluid Models for Plasma Edge Modelling

In the multiscale context of many-particle systems, it is often desired to derive macroscopic fluid models from kinetic equations describing physical phenomena at a mesoscopic level. In nuclear fusion applications, the macroscopic fluid models for describing the neutral particles in the so-called plasma edge are typically based on a phenomenological basis. In this paper, we explore a more rigorous alternative: Hilbert expansion based fluid models. In the derivation of Hilbert expansion based fluid models from linear kinetic equations, the Fredholm alternative gives rise to solvability conditions that result in the desired macroscopic fluid equations. We derive a Hilbert expansion based fluid model in the so-called steady-state scaling of a linear kinetic equation describing neutral particles in the plasma edge, with special attention for the use of the Fredholm alternative, and illustrate the performance of the resulting macroscopic fluid model numerically.