| Abstract: |
| This work investigates the diffusive limit of nonlinear radiative heat transfer systems, focusing on boundary layers under various conditions, including reflective radiative, Dirichlet, Robin, and curved boundaries. The global existence of weak solutions is demonstrated using the Galerkin method, and the convergence of these solutions to a nonlinear diffusion model in the diffusive limit is established through compactness techniques, Young measure theory, and the Banach fixed-point theorem. This work also addresses the nonlinear Milne problem, where the nonlinearity of the Stefan-Boltzmann law introduces additional mathematical challenges. Existence, exponential decay, and uniqueness of solutions are proven using uniform estimates, monotonicity properties, and spectral assumptions. Furthermore, the coupling between elliptic and kinetic transport equations is resolved via combined \( L^2 \)-\( L^\infty \) estimates. The extension to curved boundary domains includes a novel geometric correction for boundary layers, ensuring stability and convergence of solutions. These results significantly extend the existing mathematical framework for radiative heat transfer systems, providing a rigorous analysis of diffusive limits in complex geometries. |
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