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arXiv · 2210.16681

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

Mohamed Ghattassi·Xiaokai Huo·Nader Masmoudi
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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.

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Mohamed Ghattassi, Xiaokai Huo, Nader Masmoudi. 2022-10-29. Diffusive limits of the steady state radiative heat transfer system: Boundary layers. https://arxiv.org/abs/2210.16681

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