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

Abstract:

We study the following stationary frequency dependent transport equation: \[ \begin {array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu , T)[f - {B_\nu }(T)] = 0,\qquad x > 0,\;\nu > 0,\;\mu \in ] - 1; 1[,} \\ {\iint _{{{\mathbf {R}}^ + } \times ] - 1; 1[} {\sigma (\nu , T)[{B_\nu }(T) - f] d\nu \frac {{d\mu }} {2} = 0,}} \\ {f(0, \mu , \nu ) = \varphi (\mu , \nu ),\qquad \nu > 0,\;u \in ]0; 1[,} \\ \end {array} \] 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: \[ \begin {array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu , T)[f - {B_\nu }(T)] = 0,\qquad x > 0,\;\nu > 0,\;\mu \in ] - 1; 1[,} \\ {\iint _{{{\mathbf {R}}^ + } \times ] - 1; 1[} {\sigma (\nu , T)[{B_\nu }(T) - f] d\nu \frac {{d\mu }} {2} = 0,}} \\ {f(0, \mu , \nu ) = \varphi (\mu , \nu );\qquad \nu > 0,\;\mu \in ]0; 1[.} \\ \end {array} \] 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 $\varphi$.