Diffusive limits of the steady state radiative heat transfer system: Curvature effects

TL;DR

This paper investigates the diffusive limit of a nonlinear radiative heat transfer system in a curved boundary domain, introducing geometric corrections to boundary layer solutions to achieve accurate $L^\infty$ approximations.

Contribution

It develops a geometric correction to boundary layer problems for curved domains, ensuring valid $L^\infty$ approximations in the diffusive limit, extending previous flat boundary results.

Findings

Successful construction of a geometric correction for curved boundaries.

Validation of spectral assumptions for stability in curved domains.

Extension of convergence results to corrected approximate solutions.

Abstract

This paper is devoted to the diffusive limit of the nonlinear radiative heat transfer system with curved boundary domain (\textit{two dimensional disk}). The solution constructed in \cite{ghattassi2022convergence} by the leading order interior solution and the boundary layer corrections fails here to approximate the solutions in $L^\infty$ sense for the diffusive limit. The present paper aims to construct a geometric correction to the boundary layer problem and obtain a valid approximate solution in $L^\infty$ sense. The main tools to overcome the convergence problem, are to use matched asymptotic expansion techniques, fixed-point theorems, linear and nonlinear stability analysis of the boundary layer problem. In particular, the spectral assumption on the leading order interior solution, which was proposed for the flat case in \cite{Bounadrylayer2019GHM2}, is shown to be still valid which guarantee the stability of the boundary layer expansion with geometric corrections. Moreover, the convergence result established in \cite[Lemma 10]{ghattassi2022convergence} remain applicable for the approximate solution with geometric corrections.