Stability of the nonlinear Milne Problem for radiative heat transfer system

TL;DR

This paper investigates the existence, uniqueness, and stability of solutions to the nonlinear Milne problem in radiative heat transfer, addressing the mathematical challenges posed by the nonlinear Stefan-Boltzmann law.

Contribution

It extends the theory of the Milne problem to nonlinear systems with the Stefan-Boltzmann law, establishing existence, uniqueness, and stability results using monotonicity and spectral analysis.

Findings

Existence of solutions on finite intervals proven using monotonic convergence.

Solutions extend to the half-space with uniform weighted estimates.

Solutions converge to constants as x approaches infinity.

Abstract

This paper focuses on the nonlinear Milne problem of the radiative heat transfer system on the half-space. The nonlinear model is described by a second order ODE for temperature coupled to transport equation for radiative intensity. The nonlinearity of the fourth power Stefan-Boltzmann law of black body radiation, bring additional difficulty in mathematical analysis, compared to the well-developed theory for Milne problem of linear transport equation. With the help of the monotonicity property of the second order ODE, we prove the existence of the nonlinear Milne problem on a finite interval using monotonic convergence theorems. Then the solution is extended to the half-space using a uniform weighted estimate and the compactness method. Moreover, the solutions are proved to converge to constants as $x\to \infty$. Therefore, the linear stability analysis is used to study the uniqueness of the nonlinear Milne problem. The existence and uniqueness for the linearized system is established under a spectral assumption on the solution of the nonlinear problem. The spectral assumption is shown to be satisfied when the boundary data is close to the well-prepared case by using a generalized Hardy's inequality. The uniqueness of the solution to the half-line nonlinear Milne problem is established in a neighborhood of solutions satisfying a spectral assumption and the energy estimate.The current work extends the study of Milne problem for linear transport equations and provides a comprehensive study on the nonlinear Milne problem of radiative heat transfer systems.