On the diffusive limits of radiative heat transfer system I: well prepared initial and boundary conditions

TL;DR

This paper proves that solutions to a nonlinear radiative heat transfer system converge to a diffusion model in the diffusive limit, under well-prepared initial and boundary conditions, using advanced mathematical techniques.

Contribution

It establishes the diffusive limit of a nonlinear radiative heat transfer system with various boundary conditions, including convergence rates under regularity assumptions.

Findings

Weak solutions converge to a nonlinear diffusion model

Convergence rate is obtained under regularity conditions

No initial or boundary layers due to well-prepared conditions

Abstract

We study the diffusive limit approximation for a nonlinear radiative heat transfer system that arises in the modeling of glass cooling, greenhouse effects and in astrophysics. The model is considered with the reflective radiative boundary conditions for the radiative intensity, and periodic, Dirichlet and Robin boundary conditions for the temperature. The global existence of weak solutions for this system is given by using a Galerkin method with a careful treatment of the boundary conditions. Using the compactness method, averaging lemma and Young measure theory, we prove our main result that the weak solution converges to a nonlinear diffusion model in the diffusive limit. Moreover, under more regularity conditions on the limit system, the diffusive limit is also analyzed by using a relative entropy method. In particular, we get a rate of convergence. The initial and boundary conditions are assumed to be well-prepared in the sense that no initial and boundary layer exist.