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Equilibrium-diffusion limit of the radiation model

Lei Li

TL;DR

The paper rigorously derives the equilibrium-diffusion limit for a radiative transfer model on $\mathbb{T}^{3}$ by employing a Hilbert expansion to separate interior and initial-layer dynamics of the coupled kinetic-diffusion system with grey emission. It proves well-posedness of the equilibrium neutron transport equation in $L^{\infty}$, constructs comprehensive interior and initial-layer expansions, and demonstrates convergence to a diffusive limit with explicit error rates. The analysis combines linearized and nonlinear stability arguments, controlling remainders via energy estimates and a fixed-point framework to reach high-order convergence results. This advances the mathematical understanding of radiation diffusion in simplified geometries and provides precise rates for the radiation intensity and temperature in the low-$\varepsilon$ regime, with potential impact on radiation hydrodynamics modeling.

Abstract

We justify rigorously the equilibrium-diffusion limit of the model consists of a radiative transfer satisfied by the specific intensity of radiation coupled to a diffusion equation satisfied by the material temperature. For general initial data, we construct the existence of the solution to the coupled model in $\mathbb{T}^{3}$ by the Hilbert expansion and prove the convergence of the solutions to the limiting system in the equilibrium-diffusion regime. Moreover, the initial layer for the radiative density and the temperature are constructed to get the strong convergence in $L^\infty$ norm. We also get the convergence rates about the intensity of radiation and temperature in this paper.

Equilibrium-diffusion limit of the radiation model

TL;DR

The paper rigorously derives the equilibrium-diffusion limit for a radiative transfer model on by employing a Hilbert expansion to separate interior and initial-layer dynamics of the coupled kinetic-diffusion system with grey emission. It proves well-posedness of the equilibrium neutron transport equation in , constructs comprehensive interior and initial-layer expansions, and demonstrates convergence to a diffusive limit with explicit error rates. The analysis combines linearized and nonlinear stability arguments, controlling remainders via energy estimates and a fixed-point framework to reach high-order convergence results. This advances the mathematical understanding of radiation diffusion in simplified geometries and provides precise rates for the radiation intensity and temperature in the low- regime, with potential impact on radiation hydrodynamics modeling.

Abstract

We justify rigorously the equilibrium-diffusion limit of the model consists of a radiative transfer satisfied by the specific intensity of radiation coupled to a diffusion equation satisfied by the material temperature. For general initial data, we construct the existence of the solution to the coupled model in by the Hilbert expansion and prove the convergence of the solutions to the limiting system in the equilibrium-diffusion regime. Moreover, the initial layer for the radiative density and the temperature are constructed to get the strong convergence in norm. We also get the convergence rates about the intensity of radiation and temperature in this paper.

Paper Structure

This paper contains 10 sections, 13 theorems, 305 equations.

Table of Contents

  1. Introduction
  2. Well-posedness of equilibrium regime neutron transport equation
  3. $L^{\infty}$ well-posedness
  4. Hilbert Expansion
  5. Interior expansion
  6. Initial layer expansion.
  7. Construction of asymptotic expansion.
  8. Diffusive Limit
  9. Linearized system
  10. Nonlinear system

Key Result

Theorem 1.1

Assume that $\vec{u}\in C^{N+2}([0, \infty)\times\mathbb{T}^{3})$, $h(\vec{x}, \vec{w})\in L^{\infty}(\mathbb{S}^{2}; H^{N+2}(\mathbb{T}^{3}))$ and $\theta^{0}(\vec{x})\in H^{N+2}(\mathbb{T}^{3})$ with $h(\vec{x}, \vec{w})>0, \theta^{0}(\vec{x})\geq a$ and $\|h-(\theta^{0})^{4}\|_{L^{\infty}(\mathbb and where $(f_{0}, \theta_{0})\in (C([0, T]; H^{N+2}(\mathbb{T}^{3}))\cap L^{2}(0, T; H^{N+3}(\mat

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Theorem 3.2
  • ...and 6 more