Some new results from the theory of the radiative transfer equation

In this talk, I will give an overview about some interesting results concerning the radiative transfer equation (RTE), which is the kinetic equation describing the interaction of a body with radiation and which can be used for the study of radiative heat transfer. After having introduced this kinetic model, we will study the well-posedness problem for the stationary RTE. In particular I will present the existence result for the case in which the problem reduces to a non-local non-linear integral equation. We will then turn to the diffusion approximation of the RTE, i.e. to the case in which the mean free path of the photons is much smaller than the characteristic length of the domain. Under these assumptions, one can derive, using matched asymptotic expansions, specific diffusion equations solved by the radiation intensity in the bulk of the domain. Finally, I will introduce an ongoing project about a new version of the classical one-dimensional two-phase Stefan problem. Specifically, we will consider a model describing the melting of ice in which the heat is transferred by conduction in the liquid and by both conduction and radiation in the solid. This is based on joint work with Juan. J. L. Velázquez and Jin Woo Jang.