The Milne problem for the radiative transfer equations (with frequency dependence)
Authors: Golse François 
Published: June 2010

We study the following stationary frequency dependent transport equation:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row mu partial-differential Subscript x Baseline f plus sigma left-parenthesis nu comma upper T right-parenthesis left-bracket f minus upper B Subscript nu Baseline left-parenthesis upper T right-parenthesis right-bracket equals 0 comma x greater-than 0 comma nu greater-than 0 comma mu element-of right-bracket negative 1 semicolon 1 left-bracket comma 2nd Row double-integral Underscript bold upper R Superscript plus Baseline times right-bracket negative 1 semicolon 1 left-bracket Endscripts sigma left-parenthesis nu comma upper T right-parenthesis left-bracket upper B Subscript nu Baseline left-parenthesis upper T right-parenthesis minus f right-bracket d nu StartFraction d mu Over 2 EndFraction equals 0 comma 3rd Row f left-parenthesis 0 comma mu comma nu right-parenthesis equals phi left-parenthesis mu comma nu right-parenthesis comma nu greater-than 0 comma u element-of right-bracket 0 semicolon 1 left-bracket comma EndLayout">

μ

∂
x

f
+
σ
(
ν
,

T
)
[
f
−

B
ν

(
T
)
]
=
0
,

x
>
0
,

ν
>
0
,

μ
∈
]
−
1
;

1
[
,

∬

R

+

×
]
−
1
;

1
[

σ
(
ν
,

T
)
[

B
ν

(
T
)
−
f
]

d
ν

d
μ

2

=
0
,

f
(
0
,

μ
,

ν
)
=
φ
(
μ
,

ν
)
,

ν
>
0
,

u
∈
]
0
;

1
[
,

\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
ν

{B_\nu }

is the well-known Planck function appearing in astrophysics. We are able to describe the asymptotic behavior of

f
f

and

T
T

for

x
x

large, when

σ
(
ν
,

T
)

\sigma (\nu ,\,T)

is of the special form

σ
(
ν
,

T
)
=
σ
(
ν
)
k
(
T
)

\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

φ
\varphi

(in any sense) is required. Résumé. Nous étudions l’équation de transport stationnaire avec dépendance en fréquence:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row mu partial-differential Subscript x Baseline f plus sigma left-parenthesis nu comma upper T right-parenthesis left-bracket f minus upper B Subscript nu Baseline left-parenthesis upper T right-parenthesis right-bracket equals 0 comma x greater-than 0 comma nu greater-than 0 comma mu element-of right-bracket negative 1 semicolon 1 left-bracket comma 2nd Row double-integral Underscript bold upper R Superscript plus Baseline times right-bracket negative 1 semicolon 1 left-bracket Endscripts sigma left-parenthesis nu comma upper T right-parenthesis left-bracket upper B Subscript nu Baseline left-parenthesis upper T right-parenthesis minus f right-bracket d nu StartFraction d mu Over 2 EndFraction equals 0 comma 3rd Row f left-parenthesis 0 comma mu comma nu right-parenthesis equals phi left-parenthesis mu comma nu right-parenthesis semicolon nu greater-than 0 comma mu element-of right-bracket 0 semicolon 1 left-bracket period EndLayout">

μ

∂
x

f
+
σ
(
ν
,

T
)
[
f
−

B
ν

(
T
)
]
=
0
,

x
>
0
,

ν
>
0
,

μ
∈
]
−
1
;

1
[
,

∬

R

+

×
]
−
1
;

1
[

σ
(
ν
,

T
)
[

B
ν

(
T
)
−
f
]

d
ν

d
μ

2

=
0
,

f
(
0
,

μ
,

ν
)
=
φ
(
μ
,

ν
)
;

ν
>
0
,

μ
∈
]
0
;

1
[
.

\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

σ
(
ν
,

T
)

\sigma (\nu ,\,T)

est de la forme particulière

σ
(
ν
,

T
)
=
σ
(
ν
)
k
(
T
)

\sigma (\nu ,\,T) = \sigma (\nu )k(T)

, nous savons décrire le comportement asymptotique de

f
f

et

T
T

pour

x
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

φ
\varphi

.