Diffusive limits of the steady state radiative heat transfer system: Boundary layers

TL;DR

This paper analyzes the diffusive limit of the steady state radiative heat transfer system with boundary layers, constructing an approximate solution and proving convergence using fixed point methods, extending previous work to more general boundary conditions.

Contribution

It introduces a new approach to handle boundary layers in the diffusive limit of radiative heat transfer, including a spectral assumption for stability and extending results to ill-prepared boundary data.

Findings

Constructed a composite approximate solution considering boundary layers.

Proved convergence to the approximate solution in the diffusive limit.

Extended previous results to cases with boundary layers and ill-prepared data.

Abstract

In this paper, we study the diffusive limit of the steady state radiative heat transfer system for non-homogeneous Dirichlet boundary conditions in a bounded domain with flat boundaries. A composite approximate solution is constructed using asymptotic analysis taking into account of the boundary layers. The convergence to the approximate solution in the diffusive limit is proved using a Banach fixed point theorem. The major difficulty lies on the nonlinear coupling between elliptic and kinetic transport equations. To overcome this problem, a spectral assumption ensuring the linear stability of the boundary layers is proposed. Moreover, a combined $L^2$-$L^\infty$ estimate and the Banach fixed point theorem are used to obtain the convergence proof. This results extend our previous work \cite{ghattassi2020diffusive} for the well-prepared boundary data case to the ill-prepared case when boundary layer exists.