Diffusive limits for the Knudsen gas in a thin channel with accommodation on the boundary

We consider a model for a free molecular flow in a thin channel bounded by two parallel plates on which Maxwellian boundary conditions with (fixed) accommodation and a generic scattering kernel apply. Using functional analytic tools, we show that as the width of the channel vanishes, and on a suitable temporal scale, the evolution of the density is described by a diffusion problem. We distinguish two classes of temporal scalings (normal and anomalous) and we show that an infinitesimal amount of grazing collisions with the walls of the channel is responsible for the anomalous diffusion. The method employed is adapted from the original work of F. Golse in Asymptot. Anal. 17 (1998), 1-12.