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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Milne problem for the radiative transfer equations (with frequency dependence)


Author: François Golse
Journal: Trans. Amer. Math. Soc. 303 (1987), 125-143
MSC: Primary 85A25; Secondary 35Q20
MathSciNet review: 896011
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Abstract: We study the following stationary frequency dependent transport equation:

\begin{displaymath}\begin{array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,... ... (\mu ,\,\nu ),\qquad \nu > 0,\;u \in ]0;\,1[,} \\ \end{array} \end{displaymath}

where $ {B_\nu }$ is the well-known Planck function appearing in astrophysics. We are able to describe the asymptotic behavior of $ f$ and $ T$ for $ x$ large, when $ \sigma (\nu ,\,T)$ is of the special form $ \sigma (\nu ,\,T) = \sigma (\nu )k(T)$. Our method relies mainly on the monotonicity of the nonlinearity. The proof does not use any linearization of the equation; in particular, no smallness assumption on the data $ \varphi $ (in any sense) is required. Résumé. Nous étudions l'équation de transport stationnaire avec dépendance en fréquence:

\begin{displaymath}\begin{array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,... ...\mu ,\,\nu );\qquad \nu > 0,\;\mu \in ]0;\,1[.} \\ \end{array} \end{displaymath}

Lorsque $ \sigma (\nu ,\,T)$ est de la forme particulière $ \sigma (\nu ,\,T) = \sigma (\nu )k(T)$, nous savons décrire le comportement asymptotique de $ f$ et $ T$ pour $ x$ grand. Notre méthode repose principalement sur la monotonie de la non-linéarité. La preuve n'utilise aucune linéarisation de l'équation; en particulier, nous n'avons besoin d'aucune hypothèse de petitesse (d'aucune sorte) sur la donnée $ \varphi $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0896011-0
PII: S 0002-9947(1987)0896011-0
Keywords: Radiative transfer, Milne problem, nonlinear transport equations, boundary layers, transfert radiatif, problème de Milne, equations de transport non-linéaires, couches limites
Article copyright: © Copyright 1987 American Mathematical Society