On the Diffusion Approximation of the Stationary Radiative Transfer Equation with Absorption and Emission

Abstract

In this paper, we study the distribution of temperature of a body due to the transfer of radiation. Specifically, the boundary value problem for the stationary radiative transfer equation is considered. In all the analysis, we assume the so-called local thermal equilibrium (LTE), i.e., there is a well-defined temperature of the body at each point. We consider the limit in which the mean free path of the photons is much smaller than the characteristic length of the domain. In this case, we can approximate the solution by means of the so-called diffusion approximation. The analysis of this paper is restricted to the case in which the absorption coefficient is independent of the frequency \( \nu \) (the so-called Grey approximation). We ignore also scattering effects. Under these assumptions, we show that the density of radiative energy u, which is proportional to the fourth power of the temperature, solves in the limit an elliptic equation. The boundary values for that limit equation can be determined uniquely analyzing a suitable boundary layer problem. The method developed here allows to prove all the results using maximum principle arguments for a class of non-local elliptic equations.