Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compactness and existence theory for a general class of stationary radiative transfer equations

Elena Demattè, Jin Woo Jang, Juan J. L. Velázquez

Abstract

In this paper, we study the steady-states of a large class of stationary radiative transfer equations in a $C^1$ convex bounded domain. Namely, we consider the case in which both absorption-emission and scattering coefficients depend on the local temperature $T$ and the radiation frequency $ν.$ The radiative transfer equation determines the temperature of the material at each point. The main difficulty in proving existence of solutions is to obtain compactness of the sequence of integrals along lines that appear in several exponential terms. We prove a new compactness result suitable to deal with such a non-local operator containing integrals on a line segment. On the other hand, to obtain the existence theory of the full equation with both absorption and scattering terms we combine the compactness result with the construction of suitable Green functions for a class of non-local equations.

Compactness and existence theory for a general class of stationary radiative transfer equations

Abstract

In this paper, we study the steady-states of a large class of stationary radiative transfer equations in a convex bounded domain. Namely, we consider the case in which both absorption-emission and scattering coefficients depend on the local temperature and the radiation frequency The radiative transfer equation determines the temperature of the material at each point. The main difficulty in proving existence of solutions is to obtain compactness of the sequence of integrals along lines that appear in several exponential terms. We prove a new compactness result suitable to deal with such a non-local operator containing integrals on a line segment. On the other hand, to obtain the existence theory of the full equation with both absorption and scattering terms we combine the compactness result with the construction of suitable Green functions for a class of non-local equations.
Paper Structure (21 sections, 13 theorems, 276 equations)

This paper contains 21 sections, 13 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Summary of literature
  3. Main theorems
  4. Strategy of the proof and main estimates
  5. Outline of the paper
  6. Derivation of a non-local integral equation for the temperature and the regularization of the equation
  7. Derivation of a non-local integral equation
  8. Non-local integral equation in the Grey case
  9. Regularization of the non-local equation
  10. Existence theory for the pure emission-absorption case
  11. Existence of solutions to the regularized problem in the Grey case
  12. Compactness theory for operators defined by means of some line integrals
  13. Proof of Theorems \ref{['main theorem pseudo grey']} and \ref{['Grey thm']}
  14. Full equation with both scattering and emission-absorption
  15. Main result in the case of the Grey approximation
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\Omega\subset {\mathbb{R}^3}$ be bounded and open with $C^1$-boundary and strictly positive curvature. Suppose that the incoming boundary profile $g_\nu$ satisfies the bound and that $\alpha_\nu^a(T(x))=Q(\nu)\alpha(T(x))$ is bounded, strictly positive and $C^1$ in $T$, where $Q:\mathbb{R}_+\to\mathbb{R}_+$ and $\alpha:\mathbb{R}_+\to\mathbb{R}_+$. Then there exists a solution $(T,I_\nu) \in

Theorems & Definitions (26)

  • Theorem 1.1
  • Proposition 1.2: Compactness result for line integrals
  • Theorem 1.3: Full equations in the pseudo Grey case
  • Theorem 2.1
  • proof : Proof of Proposition \ref{['compact']}
  • Corollary 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['Grey thm']}
  • ...and 16 more