Analysis of the M1 model: Well-posedness and diffusion asymptotics

This paper is devoted to the analysis of the M1 model which arises in radiative transfer theory. The derivation of the model is based on the entropy minimization principle, which leads to a hyperbolic system of balance laws with relaxation. In the multi-dimensional case, we establish the existence-uniqueness of a globally defined smooth solution under a suitable smallness condition on the initial data. In the one-dimensional case we show that the smallness condition does not depend on the particles mean free path so that we can also rigorously justify the consistency of the model with the diffusion asymptotics. The result extends the analysis of Coulombel et al. [J.-F. Coulombel, F. Golse T. Goudon, Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptotic Analysis, 45 (2005) 1-39] to the case where the entropy functional accounts for relaxation towards the Planckian state, which is physically more relevant.