# Diffusive limits of the steady state radiative heat transfer system:
Boundary layers

Mohamed Ghattassi  Department of Mathematics, New York University in Abu
Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates
<EMAIL_ADDRESS>,  Xiaokai Huo  Department of Mathematics, Iowa State
University, 411 Morrill Rd, Ames, IA 50011, USA<EMAIL_ADDRESS>and  Nader
Masmoudi  Department of Mathematics, New York University in Abu Dhabi,
Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates– Courant
Institute of Mathematical Sciences, New York University, 251 Mercer Street,
New York, NY 10012, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the diffusive limit of the steady state radiative heat
transfer system for non-homogeneous Dirichlet boundary conditions in a bounded
domain with flat boundaries. A composite approximate solution is constructed
using asymptotic analysis taking into account of the boundary layers. The
convergence to the approximate solution in the diffusive limit is proved using
a Banach fixed point theorem. The major difficulty lies on the nonlinear
coupling between elliptic and kinetic transport equations. To overcome this
problem, a spectral assumption ensuring the linear stability of the boundary
layers is proposed. Moreover, a combined $L^{2}$-$L^{\infty}$ estimate and the
Banach fixed point theorem are used to obtain the convergence proof. This
results extend our previous work [6] for the well-prepared boundary data case
to the ill-prepared case when boundary layer exists.

Dans cet article, nous étudions la limite de diffusion du système de transfert
de chaleur radiatif en régime stationnaire pour des conditions aux limites de
type Dirichlet non homogènes dans un domaine borné avec frontière plate. Une
solution approchée composite est construite à l’aide d’une analyse
asymptotique prenant en compte les couches limites. La convergence vers la
solution approchée dans la limite de diffusion est démontrée à l’aide d’un
théorème de point fixe de Banach. La difficulté majeure réside dans le
couplage non linéaire entre l’équation elliptique et l’équation de transport
cinétique. Pour remédier à ce problème, une hypothèse spectrale assurant la
stabilité linéaire des couches limites est proposée. De plus, une estimation
combinée $L^{2}-L^{\infty}$ et le théorème du point fixe de Banach sont
utilisés pour obtenir la preuve de convergence. Ces résultats étendent nos
travaux précédents [6] pour le cas des données aux limites bien préparées au
cas mal préparé lorsque la couche limite existe.

###### Key words and phrases:

Radiative transfer system, Diffusive limits, Boundary layer, Milne problem

###### Contents

  1. 1 Introduction
    1. 1.1 Problem Statements
    2. 1.2 Main results
    3. 1.3 Related work
    4. 1.4 Plan of the paper
  2. 2 Asymptotic Analysis
    1. 2.1 Interior expansion
    2. 2.2 Boundary layer corrections
    3. 2.3 Construction of the approximate composition solution
    4. 2.4 Error of the composite expansion
  3. 3 Properties of the approximate solutions
    1. 3.1 Interior approximations
    2. 3.2 Boundary layer approximation
    3. 3.3 Residual Estimates
    4. 3.4 An inequality
  4. 4 Diffusive Limit
    1. 4.1 Linearized system
    2. 4.2 Nonlinear system
  5. A Existence of the steady state radiative transfer system

##  1\. Introduction

###  1.1. Problem Statements

We consider the following steady state radiative heat transfer system in the
space $x\in\Omega=[0,1]\times\mathbb{T}^{2}$ and $\beta\in\mathbb{S}^{2}$:

$\displaystyle\varepsilon^{2}\Delta
T^{\varepsilon}+\langle\psi^{\varepsilon}-(T^{\varepsilon})^{4}\rangle=0,$
(1.1)
$\displaystyle\varepsilon\beta\cdot\nabla\psi^{\varepsilon}+\psi^{\varepsilon}-(T^{\varepsilon})^{4}=0,$
(1.2)

with Dirichlet boundary conditions

$\displaystyle T^{\varepsilon}(x)=T_{b}(x),\text{ for
}x\in\partial\Omega=\\{0,1\\}\times\mathbb{T}^{2},$ (1.3)
$\displaystyle\psi^{\varepsilon}(x,\beta)=\psi_{b}(x,\beta),\text{ for
}(x,\beta)\in\Gamma_{-}.$ (1.4)

Here $T^{\varepsilon}=T^{\varepsilon}(x)$ is the temperature,
$\psi^{\varepsilon}=\psi^{\varepsilon}(x,\beta)$ is the radiation intensity.
The bracket $\langle\cdot\rangle$ denotes the momentum
$\langle\psi(\beta)\rangle=\int_{\mathbb{S}^{2}}\psi(\beta)d\beta$. We denote
the boundary set $\Gamma$ by

$\displaystyle\Gamma=\\{(x,\beta):x\in\partial\Omega,\beta\in\mathbb{S}^{2}\\},$

and $\Gamma_{+}=\Gamma\cap\\{(x,\beta):\beta\cdot n(x)>0\\}$ being the out-
flow boundary, and $\Gamma_{-}=\Gamma\cap\\{(x,\beta):\beta\cdot n(x)<0\\}$
being the in-flow boundary, where $n(x)$ is the exterior normal vector on the
boundary. Note that the boundary conditions are imposed on the in-flow
boundary and the values of $\psi$ on the out-flow boundary are determined from
the system.

When the boundary data is well-prepared, i.e. $\psi_{b}(x,\beta)=T_{b}^{4}(x)$
for $(x,\beta)\in\Gamma_{-}$, the solution of (1.1)-(1.2) was proved in [6] to
converge to the following nonlinear elliptic equation

$\displaystyle\Delta T_{0}+\frac{4\pi}{3}\Delta T_{0}^{4}=0,$ (1.5)
$\displaystyle\psi_{0}=T_{0}^{4},$ (1.6)

subject to boundary conditions

$\displaystyle T_{0}(x)=T_{b}(x),\quad\text{for any }x\in\partial\Omega.$
(1.7)

The proof is given by using two methods; the weak convergence method and the
relative entropy method. However, both methods rely on the assumption
$\psi_{b}=T_{b}^{4}$ to obtain the estimates that are needed for the
convergence estimates and fail for general boundary data, due to the presence
of boundary layers.

The main objective of this paper is to study the diffusive limit
($\varepsilon\to 0$) for the general boundary data. First, a boundary layer
problem is used to describe the behavior of solutions to system (1.1)-(1.2)
near the boundary as $\varepsilon\to 0$. Second, a composite approximate
solution is constructed by combining the solutions to the boundary layer
problem and solutions to system (1.5)-(1.6). In particular, higher order
approximate solutions are constructed via asymptotic analysis. Last, the
convergence of solutions to system (1.1)-(1.2) to the constructed approximate
solution is proved by using an $L^{2}$-$L^{\infty}$ type estimate.

The boundary layer problem has already been studied in [7]. Here for
simplicity of notations, we assume that the boundary data at the top boundary
$\\{x_{1}=1\\}$ is well-prepared, i.e. $\psi_{b}(x,\beta)=T_{b}^{4}(x)$ for
$(x,\beta)\in\Gamma_{-}\cap\\{x_{1}=1\\}$, so that boundary layer only exists
at the bottom $\\{x_{1}=0\\}$. Introducing $\eta=x_{1}/\varepsilon^{2}$, the
corresponding boundary layer problem for system (1.1)-(1.2) is

$\displaystyle\partial_{\eta}^{2}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{T}_{0}^{4}\rangle=0,$
(1.8)
$\displaystyle\mu\partial_{\eta}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4}=0,$
(1.9)

with boundary conditions

$\displaystyle\tilde{T}_{0}(\eta=0,x^{\prime})=T_{b}(x^{\prime}),\quad\text{for
any }x^{\prime}\in\mathbb{T}^{2},$ (1.10)
$\displaystyle\tilde{\psi}_{0}(\eta=0,x^{\prime},\beta)=\psi_{b}(0,x^{\prime},\beta),\quad\text{for
any }x^{\prime}\in\mathbb{T}^{2},\,(0,x^{\prime},\beta)\in\Gamma_{-},$ (1.11)

where $\mu=-n(x)\cdot\beta=\beta_{1}$ and
$\tilde{T}_{0}=\tilde{T}_{0}(\eta,x^{\prime})$,
$\tilde{\psi}_{0}=\tilde{\psi}_{0}(\eta,x^{\prime},\beta)$ with
$x^{\prime}=(x_{2},x_{3})\in\mathbb{T}^{2}$. The above problem is also called
_nonlinear Milne problem_ of the radiative heat transfer system. Existence of
weak solutions for this problem was proved in [7]. Moreover, the weak
solutions are shown to converge as $\eta\to\infty$ to some non-negative
constants
$\tilde{T}_{0,\infty}(x^{\prime}):=\lim_{\eta\to\infty}\tilde{T}_{0}(\eta,x^{\prime})$,
$\tilde{\psi}_{0,\infty}(x^{\prime},\beta):=\lim_{\eta\to\infty}\tilde{\psi}_{0}(\eta,x^{\prime},\beta)$,
which give the boundary conditions for the nonlinear limiting equation (1.5)
by

$\displaystyle
T_{0}(x_{1}=0,x^{\prime})=\tilde{T}_{0,\infty}(x^{\prime}),\quad\text{for any
}x^{\prime}\in\mathbb{T}^{2}.$ (1.12)

However, the linear stability for the nonlinear Milne problem holds under a
spectral assumption, which reads as

  1. (A)

There exists a constant $\tau>0$ such that the function $\tilde{T}_{0}\in
L^{2}_{\rm loc}(\mathbb{R}_{+}\times\mathbb{T}^{2})$ satisfies the inequality

$\displaystyle\quad M\int_{0}^{\infty}e^{2\tau
x}(2\tilde{T}_{0}^{\frac{3}{2}})^{2}|\partial_{x}f|^{2}dx\geq
4\int_{0}^{\infty}e^{2\tau
x}|\partial_{x}(2\tilde{T}_{0}^{\frac{3}{2}})|^{2}f^{2}dx$ (1.13)

for any measurable function $f\in C^{1}(\mathbb{R}_{+})$ with $f(0)=0$, for
some constant $M<1$.

A composite approximate solution can thus be constructed by combining the
interior approximate problem (1.5)-(1.6) with (1.12) and the solution to the
Milne problem (1.8)-(1.11). To do this, we introduce a cut-off function
$\chi=\chi(x_{1})=\chi(\varepsilon\eta)$ such that

$\displaystyle\chi(x_{1})=\left\\{\begin{array}[]{cl}1,&\text{for }0\leq
x_{1}\leq\frac{1}{4}\delta,\\\ 0,&\text{for }x_{1}>\frac{3}{8}\delta,\\\
\in(0,1),&\text{otherwise.}\end{array}\right.$ (1.17)

Here where $\delta>0$ is a small constant and will be chosen later (see
Theorem 9). The boundary layer corrections are given by

$\displaystyle\bar{T}_{0}=\chi(x_{1})(\tilde{T}_{0}-\tilde{T}_{0,\infty}),\quad\bar{\psi}_{0}=\chi(x_{1})(\tilde{\psi}_{0}-\tilde{T}_{0,\infty}^{4}).$
(1.18)

The major goal of this work is to prove that
$(T_{0}+\bar{T}_{0},\psi_{0}+\bar{\psi}_{0})$ provides an approximate solution
to system (1.1)-(1.2) when $\varepsilon$ is small.

###  1.2. Main results

The main result of this paper is the following theorem.

######  Theorem 1.

Let $T_{b}\in C^{2}(\partial\Omega)$ and $\psi_{b}\in C^{1}(\Gamma_{-})$ be
non-negative functions. Let $(T_{0},\psi_{0})$ be the smooth solution to
system (1.5)-(1.6) with boundary condition (1.12) and
$(\tilde{T}_{0},\tilde{\psi}_{0})$ be the smooth solution to the nonlinear
Milne problem (1.8)-(1.11). Let the boundary layer correction
$(\bar{T}_{0},\bar{\psi}_{0})$ be given by (1.18). Assume $\tilde{T}_{0}$
satisfies the spectral assumption (A) and has a lower bound $\tilde{T}_{0}\geq
a$ for some constant $a>0$. Then for $\varepsilon>0$ sufficiently small, there
exists a unique solution $(T^{\varepsilon},\psi^{\varepsilon})\in
L^{\infty}(\Omega)\times L^{\infty}(\Omega\times\mathbb{S}^{2})$ to system
(1.1)-(1.2) with boundary conditions (1.3)-(1.4), satisfying

$\displaystyle\|T^{\varepsilon}-T_{0}-\bar{T}_{0}\|_{L^{\infty}(\Omega)}=O(\varepsilon),\quad\|\psi^{\varepsilon}-T_{0}^{4}-\bar{\psi}_{0}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}=O(\varepsilon).$
(1.19)

The above theorem rigorously shows the convergence of solutions of system
(1.1)-(1.2) to the approximate solution $(T_{0}+\bar{T}_{0})$ as
$\varepsilon\to 0$. In order to prove the above theorem, we need higher order
approximate expansions beyond (1.5)-(1.6) and (1.8)-(1.9). Let $N\geq 1$ be an
integer, we take the ansatz for the composite approximate solution
$(T^{a},\psi^{a})$ as

$T^{\varepsilon}\sim
T^{a}:=\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k}),\quad\psi^{\varepsilon}\sim\psi^{a}:=\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}+\bar{\psi}_{k}),$

where $(T_{k},\psi_{k})$ is the $k$-th order interior approximation
(satisfying system (2.40)-(2.41)) and $(\bar{T}_{k},\bar{\psi}_{k})$ is the
$k$-th order boundary layer correction (defined by (2.39) and system
(2.35)-(2.36)), see Section 2 for details of the derivations. Note that due to
the nonlinearity of $(T^{\varepsilon})^{4}$, the interior expansion and
boundary layer expansion are coupled. In particular, Taylor’s expansions of
the interior approximations are used for the construction of the boundary
layer corrections. The composite approximate solution could be shown to
satisfy

$\displaystyle\varepsilon^{2}\Delta
T^{a}+\langle\psi^{a}-(T^{a})^{4}\rangle=\mathcal{R}_{1}(T^{a},\psi^{a}),$
(1.20)
$\displaystyle\varepsilon\beta\cdot\nabla\psi^{a}+\psi^{a}-(T^{a})^{4}=\mathcal{R}_{2}(T^{a},\psi^{a}),$
(1.21)

with boundary conditions

$\displaystyle T^{a}(x)=T_{b}(x),\quad\text{for any }x\in\partial\Omega,$
(1.22) $\displaystyle\psi^{a}(x,\beta)=\psi_{b}(x,\beta),\text{ for
}(x,\beta)\in\Gamma_{-},$ (1.23)

where $\mathcal{R}_{1}(T^{a},\psi^{a})$ and $\mathcal{R}_{2}(T^{a},\psi^{a})$
are the approximation errors and are of order $\varepsilon^{N+1}$ (see Theorem
9).

The main idea for the proof of Theorem 1 is to show the existence and
uniqueness of system (1.1)-(1.2) in a small neighborhood of the approximate
solution $(T^{a},\psi^{a})$. In order to achieve this, we construct a sequence
of functions $\\{T^{k},\psi^{k}\\}_{k=0}^{\infty}$ solving

$\displaystyle\varepsilon^{2}\Delta
T^{k}+\langle\psi^{k}-4(T^{a})^{3}T^{k}\rangle=\langle(T^{k-1})^{4}-4(T^{a})^{3}T^{k-1}\rangle,$
(1.24)
$\displaystyle\varepsilon\beta\cdot\nabla\psi^{k}+\psi^{k}-4(T^{a})^{3}T^{k}=(T^{k-1})^{4}-4(T^{a})^{3}T^{k-1},$
(1.25)

with boundary conditions

$\displaystyle T^{k}(x)=T_{b}(x),\quad\text{for any }x\in\partial\Omega,$
$\displaystyle\psi^{k}(x,\beta)=\psi_{b}(x,\beta),\text{ for
}(x,\beta)\in\Gamma_{-}.$

This defines a mapping $\mathcal{T}:L^{\infty}(\Omega)\times
L^{\infty}(\Omega\times\mathbb{S}^{2})\to L^{\infty}(\Omega)\times
L^{\infty}(\Omega\times\mathbb{S}^{2})$ with
$(T^{k},\psi^{k})=\mathcal{T}((T^{k-1},\psi^{k-1}))$. For $\varepsilon>0$
sufficiently small and $N\geq 5$, $s\geq 3$, the mapping $\mathcal{T}$ can be
shown to be a contraction in the space

$\displaystyle\mathcal{O}_{s}:=$ $\displaystyle\\{(T,\psi)\in L^{2}\cap
L^{\infty}(\Omega)\times L^{2}\cap L^{\infty}(\Omega\times\mathbb{S}^{2}):$
$\displaystyle\qquad\|T-T^{a}\|_{L^{2}\cap L^{\infty}(\Omega)}\leq
K\varepsilon^{s},\|\psi-\psi^{a}\|_{L^{2}\cap
L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq K\varepsilon^{s}\\},$ (1.26)

where $K>0$ is a positive constant. Then by the Banach fixed point theorem,
there exists a fixed point $(T^{\varepsilon},\psi^{\varepsilon})$ of
$\mathcal{T}$ such that
$\mathcal{T}((T^{\varepsilon},\psi^{\varepsilon}))=(T^{\varepsilon},\psi^{\varepsilon})\in\mathcal{O}_{s}$.
By the definition of $\mathcal{T}$, $(T^{\varepsilon},\psi^{\varepsilon})$
solves system (1.1)-(1.2) with boundary conditions (1.3)-(1.4). For example,
we can take $s=3$ and $N=5$ in the above arguments and conclude that there
exists a unique solution
$(T^{\varepsilon},\psi^{\varepsilon})\in\mathcal{O}_{s=3}$, i.e.

$\displaystyle\left\|T^{\varepsilon}-\sum_{k=0}^{5}\varepsilon^{k}T_{k}-\sum_{k=0}^{5}\varepsilon^{k}\bar{T}_{k}\right\|_{L^{2}\cap
L^{\infty}(\Omega)}\leq C\varepsilon^{3},$ (1.27)
$\displaystyle\left\|\psi^{\varepsilon}-\sum_{k=0}^{5}\varepsilon^{k}\psi_{k}-\sum_{k=0}^{5}\varepsilon^{k}\bar{\psi}_{k}\right\|_{L^{2}\cap
L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq C\varepsilon^{3},$ (1.28)

which leads to (1.19) and proves Theorem 1.

One of the most elusive and difficult issues to prove Theorem 1 is to show
that $\mathcal{T}$ is a contraction mapping. This is solved by using an
$L^{2}$-$L^{\infty}$ estimates on system (1.24)-(1.25). In order to get an
$L^{2}$ estimate on this system, the following coercivity inequality

$\displaystyle-\int_{\Omega}4(T^{a})^{3}g\Delta
gdx\geq-C\|g\|_{L^{2}(\Omega)}^{2},$ (1.29)

for any function $g\in H^{1}(\Omega)$, is used, which can be shown to hold
under the spectral assumption (A) (see Lemma 10). The $L^{\infty}$ estimate is
derived based on the elliptic regularity and the maximum principle for the
radiative transport equation.

The spectral assumption (A) plays a key role in the proof of Theorem 1. First,
it’s required to show the well-posedness of the nonlinear Milne problem
(1.8)-(1.11) and equations for higher order boundary corrections
$(\bar{T}_{k},\bar{\psi}_{k})$, $k\geq 1$. Second, under this assumption, the
exponential decay of $\bar{T}_{k},\bar{\psi}_{k}$, $k\geq 0$ can be shown,
which is needed in order to find the boundary condition for the interior
approximations. Third, the spectral assumption is used to show the inequality
(1.29), which is crucial in order to get the suitable $L^{2}$ estimate on
system (1.24)-(1.25) and prove $\mathcal{T}$ is a contraction mapping. Note
since
$\bar{T}_{k}=\bar{T}_{k}(\eta,x^{\prime})=\bar{T}(x_{1}/\varepsilon,x^{\prime})$,
$\partial^{2}_{\eta}\bar{T}_{k}$ may be of order $1/\varepsilon^{2}$ and thus
the left term of (1.29) may be singular as $\varepsilon\to 0$. Thanks to the
spectral assumption (A), inequality (1.29) holds and overcome the above
singularity.

Our work also provides another approach to justify the diffusive limit in the
well-prepared case, which was already done in [6] using different methods.
Indeed, when the boundary data is well-prepared, $\psi_{b}=T_{b}^{4}$, and no
boundary layer exists. In this case, the boundary layer corrections
$\bar{T}_{k}\equiv 0,\bar{\psi}_{k}\equiv 0$. We can thus take
$T^{a}=\sum_{k=0}^{N}\varepsilon^{k}T_{k}$,
$\psi^{a}=\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}$. Since $\nabla T^{a}$ is
bounded, inequality (1.29) holds obviously. Therefore, Theorem 1 holds and
(1.19) implies the convergence of $(T^{\varepsilon},\psi^{\varepsilon})$ to
$(T_{0},\psi_{0})$, which is the solution to the problem (1.5)-(1.7).

###  1.3. Related work

When the diffusion operator is not considered, system (1.1)-(1.2) reduces to
the linear transport equation $\varepsilon\beta\cdot\nabla
U^{\varepsilon}+U^{\varepsilon}-\langle U^{\varepsilon}\rangle/(4\pi)=0$. As
$\varepsilon\to 0$, its solution convergences to $U_{0}+\bar{U}_{0}$, where
$U_{0}=U_{0}(x)$ is the solution of the Laplacian equation $\Delta U_{0}=0$
and $\bar{U}_{0}=\bar{U}_{0}(x,\beta)$ is a boundary layer correction defined
by $\bar{U}_{0}=\chi(x_{1})(\tilde{U}_{0}-\tilde{U}_{0,\infty})$ where
$\tilde{U}_{0}=\tilde{U}(\eta,x^{\prime},\beta)$ is the solution to
$\mu\partial_{\eta}\tilde{U}_{0}+\tilde{U}_{0}-\langle\tilde{U}_{0}\rangle/(4\pi)=0$.
This convergence was first rigorously proved in [4] whereas the study of this
linear Milne problem was done in [3]. However, when the boundary is not flat,
$\|U^{\varepsilon}-U_{0}-\bar{U}_{0}\|_{L^{\infty}}$ is shown to not converge
to zero as $\varepsilon\to 0$ in [15]. This can be overcome by introducing a
geometric correction $U_{0}^{\varepsilon},\bar{U}_{0}^{\varepsilon}$ with
considerations of the curvature effects. They proved that
$\|U^{\varepsilon}-U_{0}^{\varepsilon}-\bar{U}_{0}^{\varepsilon}\|_{L^{\infty}}$
converges to zero as $\varepsilon\to 0$ in the 2D unit disk [15], in the
annulus [16], in the 2D convex domain [9, 13] and in the 3D convex domain
[14]. For more references on the diffusive limit of the linear transport
equation, we refer the reader to [12, 13] and references therein.

When the Laplacian term $\varepsilon^{2}\Delta T^{\varepsilon}$ is neglected,
the diffusive limit for system (1.1)-(1.2) has been studied in many works [8,
1, 2, 5, 11]. For example in [1], the authors derived the Rosseland
approximation on a different radiative transfer equation depending on the
frequency variable. The strong convergence of the solution of the radiative
transfer equation to the solution of the Rosseland equation for well-prepared
boundary data is proved using the so-called Hilbert’s expansion method.
Therefore, in [2], under some weak hypotheses on the various parameters of the
radiative transfer equation, the Rosseland approximation was justified in a
weak sense. Moreover, the boundary layer problem for system (1.1)-(1.2)
without the Laplacian term is constructed in [11] and the boundary condition
for the limiting system (1.5) is shown to satisfy a mixed Robin boundary
condition. Such a method is extended in [10] to construct the boundary layer
approximations for system (1.1)-(1.2). However, the method only provides the
zeroth and first order approximations near the boundary and could not extend
to get higher order approximations. Since our estimates (1.27)-(1.28) need
higher order expansions, here we provide a different way to obtain the
approximation boundary layer solutions up to any order.

###  1.4. Plan of the paper

The paper is organized as follows: In the next section, we construct the
composite approximate solutions and show they satisfies the radiative heat
transfer system in the perturbative sense, in Lemma 2. In section 3, the
properties of the approximate solutions are studied and the approximation
errors are shown in Theorem 9, and inequality (1.29) is proved in Lemma 10.
Finally, the proof of Theorem 1 is given, which consists of showing the
linearized stability and nonlinear stability of (1.1)-(1.2) in the
neighborhood of the approximate solution.

Notations. Throughout the paper, some standard notations are used. The norm
$\|\cdot\|_{L^{2}(\Omega)}$ and
$\|\cdot\|_{L^{2}(\Omega\times\mathbb{S}^{2})}$ are defined by
${\|f\|_{L^{2}(\Omega)}^{2}=\int_{\Omega}f^{2}dx,\,\,\forall f\in
L^{2}(\Omega)}$ and
$\|g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}=\int_{\Omega\times\mathbb{S}^{2}}g^{2}d\beta
dx,\,\,\,\forall g\in L^{2}(\Omega\times\mathbb{S}^{2})$. The norm
$\|\cdot\|_{L^{2}(\Gamma_{+})}$ and $\|\cdot\|_{L^{2}(\Gamma_{-})}$ are
defined respectively by
$\|g\|_{L^{2}(\Gamma_{+})}^{2}=\int_{\Gamma_{+}}\beta\cdot n(x)g^{2}d\beta
d\sigma_{x}$, and $\|g\|_{L^{2}(\Gamma_{-})}^{2}=\int_{\Gamma_{-}}|\beta\cdot
n(x)|g^{2}d\beta d\sigma_{x}$, where $\sigma_{x}$ is the surface element.

##  2\. Asymptotic Analysis

In this section, an approximate solution to system (1.1)-(1.2) is constructed
via asymptotic analysis. An interior expansion is first constructed which is
valid in the interior of the domain and then boundary layer corrections are
constructed accounting for the boundary layer effects. Finally, we combine the
results to get a composite approximate solution to system (1.1)-(1.2). We here
recall the simplifying assumption that the boundary layer only occurs near the
bottom $\\{x\in\partial\Omega,x_{1}=0\\}$.

###  2.1. Interior expansion

We take the interior expansion to be

$\displaystyle
T^{\varepsilon}(x)\sim\sum_{k=0}^{N}\varepsilon^{k}T_{k}(x),\quad\psi^{\varepsilon}(x,\beta)\sim\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}(x,\beta).$
(2.1)

Define

$\displaystyle\mathcal{R}_{1}(T,\psi):=\varepsilon^{2}\Delta
T+\langle\psi-T^{4}\rangle,\quad\mathcal{R}_{2}(T,\psi):=\varepsilon\beta\cdot\nabla\psi+(\psi-T^{4}).$
(2.2)

Plugging (2.1) into the above formulas gives

$\displaystyle\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)$
$\displaystyle\quad=\sum_{k=0}^{N}\varepsilon^{k}(\Delta
T_{k-2}+\langle\psi_{k}-\mathcal{C}(T,k))\rangle+\varepsilon^{N+1}\Delta
T_{N-1}+\varepsilon^{N+2}\Delta
T_{N}-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T,k)\rangle,$ (2.3)
$\displaystyle\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)$
$\displaystyle\quad=\sum_{k=0}^{N}\varepsilon^{k}(\beta\cdot\nabla\psi_{k-1}+\psi_{k}-\mathcal{C}(T,k))+\varepsilon^{N+1}\beta\cdot\nabla\psi_{N}-\sum_{k=N+1}^{4N}\varepsilon^{k}\mathcal{C}(T,k),$
(2.4)

where

$\displaystyle\mathcal{C}(T,k):=\sum_{\begin{subarray}{c}i+j+l+m=k\\\
i,j,l,m\geq 0\end{subarray}}T_{i}T_{j}T_{l}T_{m},$ (2.5)

and $(T_{k},\psi_{k}),k<0$ are taken to be zero.

Collecting terms with the same order, we take

$\displaystyle\Delta T_{k-2}+\langle\psi_{k}-\mathcal{C}(T,k)\rangle=0,$ (2.6)
$\displaystyle\beta\cdot\nabla\psi_{k-1}+\psi_{k}-\mathcal{C}(T,k)=0,$ (2.7)

for any $k=0,\ldots,N$. Therefore from the above two equations we obtain

$\displaystyle\Delta T_{k-2}=\langle\beta\cdot\nabla\psi_{k-1}\rangle.$

By (2.7), an iterative process on the above equation leads to

$\displaystyle\Delta T_{k-2}$
$\displaystyle=\langle\beta\cdot\nabla(-\beta\cdot\nabla\psi_{k-2}+\mathcal{C}(T,k-1))\rangle=-\langle\beta\cdot\nabla(\beta\cdot\nabla\psi_{k-2})\rangle$
$\displaystyle=-\langle(\beta\cdot\nabla)^{2}(-\beta\cdot\nabla\psi_{k-3}+\mathcal{C}(T,k-2))\rangle=-\frac{4\pi}{3}\Delta\mathcal{C}(T,k-2)\rangle+\langle(\beta\cdot\nabla)^{3}\psi_{k-3}\rangle.$

Consequently, equations (2.6)-(2.7) can be rewritten as

$\displaystyle\Delta
T_{k}+\frac{4\pi}{3}\Delta\mathcal{C}(T,k)=\langle(\beta\cdot\nabla)^{3}\psi_{k-1}\rangle,$
(2.8) $\displaystyle\psi_{k}=-\beta\cdot\nabla\psi_{k-1}+\mathcal{C}(T,k),$
(2.9)

for any $k=0,\ldots,N$.

Note that here $T_{k}$ is solved by (2.8) and the the solution could be
plugged into (2.9) to get $\psi_{k}$. In addition, for $k=0$, equation (2.8)
becomes

$\displaystyle\Delta T_{0}+\frac{4\pi}{3}\Delta T_{0}^{4}=0,$

which is a nonlinear elliptic equation. For $k\geq 1$, equation (2.8) can be
rewritten as

$\displaystyle\Delta
T_{k}+\frac{4\pi}{3}\Delta(4T_{0}^{3}T_{k})=-\frac{4\pi}{3}\Delta\mathcal{E}(T,k-1)+\langle(\beta\cdot\nabla)^{3}\psi_{k-1}\rangle,$

which is a linear elliptic equation, where

$\displaystyle\mathcal{E}(T,k-1):=\sum_{\begin{subarray}{c}i+j+l+m=k\\\
i,j,l,m\geq 1\end{subarray}}T_{i}T_{j}T_{l}T_{m}.$

From the above equations, the residuals (2.3)-(2.4) is given by

$\displaystyle\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)=\varepsilon^{N+1}\Delta
T_{N-1}+\varepsilon^{N+2}\Delta
T_{N}-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T,k)\rangle,$
$\displaystyle\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)=\varepsilon^{N+1}\beta\cdot\nabla\psi_{N}-\sum_{k=N+1}^{4N}\varepsilon^{k}\mathcal{C}(T,k),$

where the right hand side are both formally of order $\varepsilon^{N+1}$.

###  2.2. Boundary layer corrections

We next find the approximation of system (1.1)-(1.2) near the boundary. Let
$\eta=x_{1}/\varepsilon$, we take the ansatz

$\displaystyle
T^{\varepsilon}(x)\sim\sum_{k=0}^{N}\varepsilon^{k}(\bar{T}_{k}(\eta,x^{\prime})+T_{k}(x)),\quad\psi^{\varepsilon}(x,\beta)\sim\sum_{k=0}^{N}\varepsilon^{k}(\bar{\psi}_{k}(\eta,x^{\prime},\beta)+\psi_{k}(x,\beta)),$

where $\bar{T}_{k},\bar{\psi}_{k}$ are the correction terms near the boundary
and $(T_{k},\psi_{k})$ are the interior expansions derived in the previous
subsection. Taking the above ansatz into (2.2) gives

$\displaystyle\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k}),\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}+\bar{\psi}_{k})\right)$
$\displaystyle\quad=\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)-\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)\rangle-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)\rangle$
$\displaystyle\qquad+\varepsilon^{N+1}\Delta_{x^{\prime}}\bar{T}_{N-1}+\varepsilon^{N+2}\Delta_{x^{\prime}}\bar{T}_{N}+\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(P,k)\rangle$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\partial_{\eta}^{2}\bar{T}_{k}+\Delta_{x^{\prime}}\bar{T}_{k-2}+\langle\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle).$
(2.10)

and

$\displaystyle\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k}),\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}+\bar{\psi}_{k})\right)$
$\displaystyle\quad=\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)-\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k))-\sum_{k=N+1}^{4N}\varepsilon^{k}(\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k))$
$\displaystyle\qquad+\varepsilon^{N+1}\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{N}+\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(P,k))$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\mu\partial_{\eta}\bar{\psi}_{k}+\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}+\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)).$
(2.11)

Here $\mu=\beta_{1}$, $x^{\prime}=(x_{2},x_{3})$,
$\beta^{\prime}=(\beta_{2},\beta_{3})$, $T_{k}(0)=T_{k}(x_{1}=0)$ and
$P_{k}=P_{k}(\eta,x^{\prime}),k=0,\ldots,N$ are the Taylor’s expansion of
$T_{k}(\varepsilon\eta,x^{\prime})$ around $\eta=0$, which is given by

$\displaystyle
P_{k}(\eta,x^{\prime})=\sum_{l=0}^{k}\frac{\eta^{l}}{l!}\frac{\partial^{l}}{\partial
x_{1}^{l}}T_{k-l}(0,x^{\prime}).$ (2.12)

Collecting terms of the same order in (2.2)-(2.2), we take

$\displaystyle\partial_{\eta}^{2}\bar{T}_{k}+\Delta_{x^{\prime}}\bar{T}_{k-2}+\langle\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle=0,$
(2.13)
$\displaystyle\mu\partial_{\eta}\bar{\psi}_{k}+\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}+\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)=0,$
(2.14)

for $k=0,\ldots,N$. Let
$\tilde{T}_{k}=\bar{T}_{k}+P_{k}(0)=\bar{T}_{k}+T_{k}(0)$ for any
$k=0,\ldots,N$, and $\tilde{\psi}_{0}=\bar{\psi}_{0}+{T}_{0}^{4}(0)$ and
$\tilde{\psi}_{k}=\bar{\psi}_{k}+4T_{0}^{3}(0)T_{k}(0)$, then

$\displaystyle-(\bar{T}_{0}+P_{0})^{4}+P_{0}^{4}=-\tilde{T}_{0}^{4}+T^{4}(0),$

and for $k\geq 1$,

$\displaystyle-4(\bar{T}_{0}+P_{0})^{3}(\bar{T}_{k}+P_{k})+4P_{0}^{3}P_{k}$
$\displaystyle=-4\tilde{T}_{0}^{3}\tilde{T}_{k}+4T_{0}^{3}(0)T_{k}(0)-4(\tilde{T}_{0}^{3}-T_{0}^{3}(0))(P_{k}-P_{k}(0)).$

Therefore, $\tilde{T}_{k}$, $\tilde{\psi}_{k}$ satisfies the equations

$\displaystyle\partial_{\eta}^{2}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{T}_{0}^{4}\rangle=0,$
(2.15)
$\displaystyle\mu\partial_{\eta}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4}=0,$
(2.16)

and for $k=1,\ldots,N$,

$\displaystyle\partial_{\eta}^{2}\tilde{T}_{k}+\Delta_{x^{\prime}}\bar{T}_{k-2}+\langle\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}\rangle$
$\displaystyle\quad+\langle-4(\tilde{T}_{0}^{3}-T_{0}^{3}(0))(P_{k}-P_{k}(0))-\mathcal{E}(\bar{T}+{P}(0),k-1)+\mathcal{E}(P(0),k-1)\rangle=0,$
(2.17)
$\displaystyle\mu\partial_{\eta}\tilde{\psi}_{k}+\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}+\tilde{\psi}_{k}$
$\displaystyle\quad-4\tilde{T}_{0}^{3}\tilde{T}_{k}-4(\tilde{T}_{0}^{3}-T_{0}^{3}(0))(P_{k}-P_{k}(0))-\mathcal{E}(\bar{T}+{P}(0),k-1)+\mathcal{E}(P(0),k-1)=0.$
(2.18)

The boundary conditions for the above equations are

$\displaystyle\tilde{T}_{0}(\eta=0,x^{\prime})=T_{b}(0,x^{\prime}),\quad\text{for
any }x^{\prime}\in\mathbb{T}^{2},$ (2.19)
$\displaystyle\tilde{\psi}_{0}(\eta=0,x^{\prime},\beta)=\psi_{b}(0,x^{\prime},\beta),\quad\text{for
any }(0,x^{\prime},\beta)\in\Gamma_{-},$ (2.20)

and for $k=1,\ldots,N$,

$\displaystyle\tilde{T}_{k}(\eta=0,x^{\prime})=0,\quad\text{for any
}x^{\prime}\in\mathbb{T}^{2},$ (2.21)
$\displaystyle\tilde{\psi}_{k}(\eta=0,x^{\prime},\beta)=\psi_{k}(0)-4T_{0}^{3}(0)T_{k}(0),\quad\text{for
any }(0,x^{\prime},\beta)\in\Gamma_{-}.$ (2.22)

The boundary conditions above are taken to be consistency with boundary
conditions (1.3)-(1.4) such that

$\displaystyle\sum_{k=0}^{N}(\bar{T}_{k}(0,x^{\prime})+T_{k}(0,x^{\prime}))=T_{b}(0,x^{\prime}),\quad\text{for
}x^{\prime}\in\mathbb{T}^{2},$ (2.23)
$\displaystyle\sum_{k=0}^{N}(\bar{\psi}_{k}(0,x^{\prime},\beta)+\psi_{k}(0,x^{\prime},\beta))=\psi_{b}(0,x^{\prime},\beta),\quad\text{for}(0,x^{\prime},\beta)\in\Gamma_{-}.$

###  2.3. Construction of the approximate composition solution

In order to combine the interior expansions and boundary corrections, we use
the cut-off function $\chi(x_{1})$ defined in (1.17) and we also introduc
another cut-off function $\chi_{0}(x_{1})$ by

$\displaystyle\chi_{0}(x_{1})=\left\\{\begin{array}[]{cl}1,&\text{for }0\leq
x_{1}\leq\frac{1}{2}\delta,\\\ 0,&\text{for }x\geq\frac{3}{4}\delta,\\\
\in(0,1),&\text{otherwise.}\end{array}\right.$ (2.27)

The construction of the composite approximate solution is done via the
following procedure.

_Step 1. Construction of $(\bar{T}_{0},\bar{\psi}_{0})$ and
$(T_{0},\psi_{0})$._ We first solve (2.17)-(2.18) when $k=0$ :

$\displaystyle\partial_{\eta}^{2}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{T}_{0}^{4}\rangle=0,$
(2.28)
$\displaystyle\mu\partial_{\eta}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4}=0,$
(2.29)

with boundary conditions

$\displaystyle\tilde{T}_{0}(\eta=0,x^{\prime})=T_{b}(0,x^{\prime}),\quad\text{for
any }x^{\prime}\in\mathbb{T}^{2},$ (2.30)
$\displaystyle\tilde{\psi}_{0}(\eta=0,x^{\prime},\beta)=\psi_{b}(0,x^{\prime},\beta),\quad\text{for
any }x^{\prime}\in\mathbb{T}^{2},\beta\in\mathbb{S}^{2}\text{ and
}\mu=\beta_{1}>0.$ (2.31)

This is the nonlinear Milne problem (1.8)-(1.11). It has been shown in [7]
that the above problem has a global weak solution
$(\tilde{T}_{0},\tilde{\psi}_{0})\in L^{2}_{\rm
loc}(\mathbb{R}_{+}\times\mathbb{T}^{2})\cap L^{2}_{\rm
loc}(\mathbb{R}_{+}\times\mathbb{T}^{2}\times\mathbb{S}^{2})$ and as
$\eta\to\infty$, the solution converges to some bounded functions
$(\tilde{T}_{0,\infty},\tilde{\psi}_{0,\infty})$ independent of $\eta$. We
define the lowest order boundary layer correction
$(\bar{T}_{0},\bar{\psi}_{0})$ by

$\displaystyle\bar{T}_{0}(\eta,x^{\prime})=\chi(\varepsilon\eta)(\tilde{T}_{0}(\eta,x^{\prime})-\tilde{T}_{0,\infty}(x^{\prime})),\quad\bar{\psi}_{0}(\eta,x^{\prime},\beta)=\chi(\varepsilon\eta)(\tilde{\psi}_{0}(\eta,x^{\prime},\beta)-\tilde{\psi}_{0,\infty}(x^{\prime},\beta)).$
(2.32)

Note due to the property of the nonlinear Milne problem, we have
$\tilde{\psi}_{0,\infty}=\tilde{T}_{0,\infty}^{4}.$ Here and below,
$\bar{T}_{k},\bar{\psi}_{k}$ are redefined by using a cut-off and is different
from the same notation in section 2.2.

We next give the leading order of the interior expansion $(T_{0},\psi_{0})$,
which is obtained by solving (2.8)-(2.9) for $k=0$:

$\displaystyle\Delta T_{0}+\frac{4\pi}{3}\Delta
T_{0}^{4}=0,\quad\psi_{0}=T_{0}^{4}$ (2.33)

with boundary conditions

$\displaystyle T_{0}(0,x^{\prime})=\tilde{T}_{0,\infty}(x^{\prime}),\quad
T_{0}(1,x^{\prime})=T_{b}(1,x^{\prime}),\quad\text{for any
}x^{\prime}\in\mathbb{T}^{2}.$ (2.34)

_Step 2. Construction of $(\bar{T}_{k},\bar{\psi}_{k})$ and $(T_{k},\psi_{k})$
for $k=1,\ldots,N$._ We solve (2.17)-(2.18) for $1\leq k\leq N$:

$\displaystyle\partial_{\eta}^{2}\tilde{T}_{k}+\langle\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}\rangle=-\chi_{0}\Delta_{x^{\prime}}\bar{T}_{k-2}+\langle
4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))\rangle$
$\displaystyle\quad+\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle,$
(2.35)
$\displaystyle\mu\partial_{\eta}\tilde{\psi}_{k}+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}=-\chi_{0}\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}+4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))$
$\displaystyle\quad+(\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)),$ (2.36)

with boundary conditions

$\displaystyle\tilde{T}_{k}(\eta=0,x^{\prime})=0,\quad\text{for any
}x^{\prime}\in\mathbb{T}^{2},$ (2.37)
$\displaystyle\tilde{\psi}_{k}(\eta=0,x^{\prime},\beta)=\beta\cdot\nabla\psi_{k-1}(0)-\mathcal{E}(T(0),k-1),\quad\text{for
any }x^{\prime}\in\mathbb{T}^{2},\beta\in\mathbb{S}^{2}\text{ and
}\mu=\beta_{1}>0.$ (2.38)

The above problem was studied in [7] where existence and uniqueness of
solutions are proved and the solutions are also shown to converge to some
bounded functions as $\eta\to\infty$. We thus obtain
$(\bar{T}_{k},\bar{\psi}_{k})$ by

$\displaystyle\bar{T}_{k}(\eta,x^{\prime})=\chi(\varepsilon\eta)(\tilde{T}_{k}(\eta,x^{\prime})-\tilde{T}_{k,\infty}(x^{\prime})),\quad\bar{\psi}_{0}(\eta,x^{\prime},\beta)=\chi(\varepsilon\eta)(\tilde{\psi}_{k}(\eta,x^{\prime},\beta)-\tilde{\psi}_{k,\infty}(x^{\prime},\beta))$
(2.39)

where

$\tilde{T}_{k,\infty}(x^{\prime})=\lim_{\eta\to\infty}\tilde{T}_{k}(\eta,x^{\prime}),\quad\tilde{\psi}_{k,\infty}(x^{\prime},\beta)=\lim_{\eta\to\infty}\tilde{\psi}_{k}(\eta,x^{\prime},\beta).$

The interior expansions $(T_{k},\psi_{k})$ are given, according to
(2.8)-(2.9), by solving the system

$\displaystyle\Delta
T_{k}+\frac{4\pi}{3}\Delta(4T_{0}^{3}T_{k})=\langle(\beta\cdot\nabla)^{3}\psi_{k-3}\rangle-\frac{4\pi}{3}\Delta\mathcal{E}(T,k-1),$
(2.40)
$\displaystyle\psi_{k}=-\beta\cdot\nabla\psi_{k-1}+4T_{0}^{3}T_{k}+\mathcal{E}(T,k-1)$
(2.41)

with boundary conditions

$\displaystyle T_{k}(0,x^{\prime})=\tilde{T}_{k,\infty}(x^{\prime}),\quad
T_{k}(1,x^{\prime})=T_{b}(1,x^{\prime}),\quad\text{for any
}x^{\prime}\in\mathbb{T}^{2}.$ (2.42)

_Step 3: The composite approximate solution._ With the above results, the
composite approximate solution is given by

$\displaystyle
T^{a}=\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k}),\quad\psi^{a}=\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}+\bar{\psi}_{k}).$
(2.43)

From (2.23), $(T^{a},\psi^{a})$ also satisfies the boundary conditions
(1.3)-(1.4).

###  2.4. Error of the composite expansion

In this section, we give the approximation errors. By the definition of
$(\bar{T}_{0},\bar{\psi}_{0})$ of (2.32) and equations (2.28)-(2.29), as well
as the relation (2.34) and $\tilde{\psi}_{0,\infty}=\tilde{T}_{0,\infty}^{4}$,
a direct computation gives

$\displaystyle E_{0}^{0}:$
$\displaystyle=\partial_{\eta}^{2}\bar{T}_{0}+\langle\bar{\psi}_{0}-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)\rangle$
$\displaystyle=(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}+\chi\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle-\langle(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}\rangle,$
(2.44)

and

$\displaystyle E_{0}^{1}:$
$\displaystyle=\mu\partial_{\eta}\bar{\psi}_{0}+\bar{\psi}_{0}-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)$
$\displaystyle=\mu(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})\partial_{\eta}\chi+\chi(\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4})-((\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}),$
(2.45)

Using the definition of $(\bar{T}_{k},\bar{\psi}_{k})$ in (2.39) and equations
(2.35)-(2.36) as well as the relation (2.37)-(2.38) and
$\tilde{\psi}_{k,\infty}=4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}$, we
get

$\displaystyle E_{k}^{0}:$
$\displaystyle=\partial_{\eta}^{2}\bar{T}_{k}+\Delta_{x^{\prime}}\bar{T}_{k-2}+\langle\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle$
$\displaystyle=(\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k}+\chi(\langle
4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}+4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))\rangle$
$\displaystyle\quad+\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle)-\langle\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(P,k)\rangle+(1-\chi\chi_{0})\Delta_{x^{\prime}}\bar{T}_{k-2},$
(2.46)

and

$\displaystyle E_{k}^{1}:$
$\displaystyle=\mu\partial_{\eta}\bar{\psi}_{k}+\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}+\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)$
$\displaystyle=\mu(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})\partial_{\eta}\chi+\chi(4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}+4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))$
$\displaystyle\quad+\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1))-\langle\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(P,k)\rangle+(1-\chi\chi_{0})\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}.$
(2.47)

Combining formulas (2.2) and (2.2), we get from the above equations

$\displaystyle\mathcal{R}_{1}(T^{a},\psi^{a})$
$\displaystyle=\varepsilon^{N+1}\Delta T_{N-1}+\varepsilon^{N+2}\Delta
T_{N}-\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle$
$\displaystyle\quad-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)\rangle+\varepsilon^{N+1}\Delta_{x^{\prime}}\bar{T}_{N-1}+\varepsilon^{N+2}\Delta_{x^{\prime}}\bar{T}_{N}+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{0},$
(2.48)

and

$\displaystyle\mathcal{R}_{2}(T^{a},\psi^{a})$
$\displaystyle=\varepsilon^{N+1}\beta\cdot\nabla\psi_{N}-\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k))$
$\displaystyle\quad-\sum_{k=N+1}^{4N}\varepsilon^{k}(\mathcal{C}(T+\bar{T},k))+\varepsilon^{N+1}\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{N}+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{1}.$
(2.49)

Therefore, we summarize the above results in the following lemma.

######  Lemma 2.

Let $(T^{a},\psi^{a})$ be given by (2.43). Then $(T^{a},\psi^{a})$ satisfies
the equations (1.20)-(1.21) with boundary conditions (1.22)-(1.23), where
$\mathcal{R}_{1}(T^{a},\psi^{a})$, $\mathcal{R}_{2}(T^{a},\psi^{a})$ are given
by (2.4) and (2.4), respectively.

##  3\. Properties of the approximate solutions

In this section, we study the properties of the approximate solution
$(T^{a},\psi^{a})$ constructed in the previous section. The properties of the
interior approximations are shown first, followed by the properties of the
boundary layer approximations. Then, the approximate errors
$\mathcal{R}_{1}(T^{a},\psi^{a})$ and $\mathcal{R}_{2}(T^{a},\psi^{a})$,
obtained in the previous section, are computed in Theorem 9. Finally, the
coercivity inequality (1.29) is shown in Lemma 10.

###  3.1. Interior approximations

The interior approximation $(T_{0},\psi_{0})$ is obtained by solving (2.33)
and $(T_{k},\psi_{k})$, $k=1,\ldots,N$ are obtained by solving (2.40). For
equation (2.33), the following lemma holds.

######  Lemma 3.

Assume $\tilde{T}_{0,\infty}\in L^{2}(\mathbb{T}^{2})$ and
$T_{b}(1,x^{\prime})\in L^{2}(\mathbb{T}^{2})$ satisfy
$\tilde{T}_{0,\infty}\geq 0$, $T_{b}(1,x^{\prime})\geq 0$ for any
$x^{\prime}\in\mathbb{T}^{2}$. Then, there exists a unique solution $T_{0}\in
C^{\infty}(\Omega)$ to equation (2.33) with boundary condition (2.34) and the
solution satisfies $T_{0}(x)\geq 0$ for any $x\in\Omega$.

###### Proof.

The proof of the above lemma follows directly from elliptic theory. Actually,
we may take $u=T_{0}+4\pi T_{0}^{4}/3$ and then $\Delta u=0$ in $\Omega$ and
$u(0,x^{\prime})=\tilde{T}_{0,\infty}(x^{\prime})+4\pi\tilde{T}^{4}_{0,\infty}(x^{\prime})/3$,
$u(1,x^{\prime})=T_{b}(1,x^{\prime})+4\pi T_{b}^{4}(1,x^{\prime})/3$ on the
boundary. We have $u\in C^{\infty}(\Omega)$ and so $T\in C^{\infty}(\Omega)$.
∎

We now provide in the following lemma a well-posedness result for the elliptic
equation (2.40).

######  Lemma 4.

Assume $\tilde{T}_{k,\infty}\in L^{2}(\mathbb{T}^{2})$ satisfy
$\tilde{T}_{k,\infty}\geq 0$. Given $T_{s},1\leq s\leq k-1$ and
$\psi_{s},1\leq s\leq k-1$ satisfy $T_{s}\in L^{2}(\Omega)$, $\psi_{s}\in
L^{2}(\Omega\times\mathbb{S}^{2}).$ Then there exists a unique solution
$T_{k}\in C^{\infty}(\Omega)$ to equation (2.40) with boundary conditions
(2.42).

###### Proof.

The proof of the above lemma also follows directly from elliptic theory by
taking $u=(1+16\pi T_{0}^{3}/3)T_{k}$. ∎

We finish this part by giving an $L^{\infty}$-estimate of the Taylor expansion
$P_{k}$ defined by (2.12).

######  Lemma 5.

Let $T_{s}$, $s=0,\ldots,N$ be the solution to equation (2.33) and (2.40), and
$P_{s}$, $s=0,\ldots,N$ be given by (2.12). Then

$\displaystyle\|P_{s}(\eta)\|_{L^{\infty}(\mathbb{T}^{2})}\leq
C(1+\eta^{s}),\quad\text{for any }s=0,\ldots,N,$ (3.1)

for some constant $C>0$.

###### Proof.

Since $T_{s}\in C^{\infty}(\Omega)$, $\partial_{x_{1}}^{l}T_{s}(0)$ are
bounded for any $l\geq 0$. Therefore,

$\displaystyle|P_{s}|$
$\displaystyle=\left|\sum_{l=0}^{s}\frac{\eta^{l}}{l!}\frac{\partial^{l}}{\partial{x}_{1}^{l}}T_{s-l}(0,x^{\prime})\right|\leq
C_{1}\sum_{l=0}^{s}\frac{\eta^{l}}{l!}\leq C(1+\eta^{s}),$ (3.2)

and thus the lemma holds. ∎

###  3.2. Boundary layer approximation

We now turn our attention to the half-space nonlinear and linear Milne
problems (2.28)-(2.29) and (2.35)-(2.36). Their analyses have been carried out
in [7]. Here we summarize the results in Lemma 6 and 7, for the proof we refer
the reader to [7]. First, we have the following lemma for system
(2.28)-(2.29).

######  Lemma 6 ([7]).

Given $(T_{b}(0,x^{\prime}),\psi_{b}(0,x^{\prime},\beta))\in
L^{\infty}(\mathbb{T}^{2})\times L^{\infty}(\Gamma_{-})$ satisfying
$T_{b}(0,x^{\prime})\geq 0$, $\psi_{b}(0,x^{\prime},\beta)\geq 0$ for any
$x^{\prime}\in\mathbb{T}^{2}$ and $\beta\in\mathbb{S}^{2}$ with $\beta_{1}\geq
0$. There exists a bounded solution $(\tilde{T}_{0},\tilde{\psi}_{0})\in
L^{2}_{\rm loc}(\mathbb{R}_{+}\times\mathbb{T}^{2})\times L^{2}_{\rm
loc}(\mathbb{R}_{+}\times\mathbb{T}^{2}\times\mathbb{S}^{2})$ to system
(2.28)-(2.29) with boundary conditions (2.30)-(2.31). Moreover, there exists a
constant function $\tilde{T}_{0,\infty}\in L^{\infty}(\mathbb{T}^{2})$ such
that

$\displaystyle|\tilde{T}_{0}(\eta,x^{\prime})-\tilde{T}_{0,\infty}(x^{\prime})|\leq
Ce^{-\lambda_{0}\eta},\quad|\tilde{\psi}(\eta,x^{\prime},\beta)-\tilde{T}_{0,\infty}^{4}(x^{\prime})|\leq
Ce^{-\lambda_{0}\eta},$ (3.3)

for any $\eta\in[0,\infty)$ and $\beta\in\mathbb{S}^{2}$. Here $C>0$ are
constants depending linearly on the constant
$\left(\int_{0}^{1}\mu(\psi_{b}-T_{b}^{4})^{2}d\mu\right)^{\frac{1}{2}}$ and
$\lambda_{0}$ is any fixed constant in $[0,1)$.

Furthermore, for sufficiently small value of
$\left(\int_{0}^{1}\mu(\psi_{b}-T_{b}^{4})^{2}d\mu\right)^{\frac{1}{2}}$, the
solution $(\tilde{T}_{0},\tilde{\psi}_{0})$ is unique.

Note that by the above lemma, $\tilde{\psi}_{0,\infty}:=\lim_{\eta\to
0}\tilde{\psi}_{0}(\eta)=\tilde{T}_{0,\infty}^{4}$.

For system (2.35)-(2.36), the spectral assumption (A) on $\tilde{T}_{0}$ is
needed in order to show the existence, uniqueness and exponential decay of
solutions. The following lemma holds.

######  Lemma 7 ([7]).

Given $\bar{T}_{s}\in L^{2}(\mathbb{R}_{+};C^{2}(\mathbb{T}^{2}))$,
$\bar{\psi}_{s}\in
L^{2}(\mathbb{R}_{+};C^{2}(\mathbb{T}^{2}\times\mathbb{S}^{2}))$, $0\leq s\leq
k-1$ such that

$\displaystyle|\bar{T}_{s}(\eta,x^{\prime})|\leq
Ce^{-\lambda_{k-1}\eta},\quad|\bar{\psi}_{s}(\eta,x^{\prime},\beta)|\leq
Ce^{-\lambda_{k-1}\eta}$ (3.4)

for some constants $C>0$ and for any $x^{\prime}\in\mathbb{T}^{2}$,
$\beta\in\mathbb{S}^{2}$, $\lambda_{k-1}>0$. Given $P_{s}\in
C([0,\infty)\times\mathbb{T}^{2})$ satisfying $|P_{s}|\leq C(1+\eta^{k-1})$
for some constant $C>0$ for any $0\leq s\leq k-1$. Assume $\tilde{T}_{0}$
satisfies the spectral assumption (A). Then there exists a unique bounded
solution $(\tilde{T}_{k},\tilde{\psi}_{k})\in L^{2}_{\rm
loc}(\mathbb{R}_{+}\times\mathbb{T}^{2})\times L^{2}_{\rm
loc}(\mathbb{R}_{+}\times\mathbb{T}^{2}\times\mathbb{S}^{2})$ to system
(2.35)-(2.36) with boundary conditions (2.37)-(2.38). Moreover, there exists
constants $(\tilde{T}_{k,\infty},\tilde{\psi}_{k,\infty})\in
L^{\infty}(\mathbb{T}^{2})\times
L^{\infty}(\mathbb{T}^{2}\times\mathbb{S}^{2})$ such that

$\displaystyle|\tilde{T}_{k}(\eta,x^{\prime})-\tilde{T}_{k,\infty}(x^{\prime})|\leq
Ce^{-\lambda^{\prime}\eta},\quad|\tilde{\psi}(\eta,x^{\prime},\beta)-\tilde{\psi}_{k,\infty}(x^{\prime},\beta)|\leq
Ce^{-\lambda^{\prime}\eta},$ (3.5)

for any constant $0<\lambda^{\prime}<\lambda_{k-1}$, where $C>0$ is a positive
constant independent of $k$ and the relation
$\tilde{\psi}_{k,\infty}=4\tilde{T}^{3}_{0,\infty}\tilde{T}_{k,\infty}$ holds.

###### Proof.

It has been proved in [7] that given functions $S_{1}=S_{1}(\eta,x^{\prime})$,
$S_{2}=S_{2}(\eta,x^{\prime},\beta)$ such that

$\displaystyle\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}S_{1}^{2}d\eta,\,\int_{0}^{\infty}\int_{\mathbb{S}^{2}}e^{2\lambda^{\prime}\eta}S_{2}^{2}d\eta
d\beta<\infty$ (3.6)

are bounded, the following equations

$\displaystyle\partial_{\eta}^{2}g+\langle\phi-4\tilde{T}_{0}^{3}g\rangle=S_{1},$
(3.7) $\displaystyle\mu\partial_{\eta}\phi+\phi-4\tilde{T}_{0}^{3}g=S_{2},$
(3.8)

with boundary conditions

$\displaystyle g(\eta=0,x^{\prime})=0,\quad\text{for any
}x^{\prime}\in\mathbb{T}^{2},$
$\displaystyle\phi(\eta=0,x^{\prime},\beta)=\phi_{b},\quad\text{for any
}(x,\beta)\in\Gamma_{-},$

have a unique bounded solution $(g,\phi)\in L^{2}_{\rm
loc}(\mathbb{R}_{+}\times\mathbb{T}^{2})\times L^{2}_{\rm
loc}(\mathbb{R}_{+}\times\mathbb{T}^{2}\times\mathbb{S}^{2})$. Moreover, there
exists a constant $g_{\infty}$ such that $|g-g_{\infty}|\leq
Ce^{-\lambda^{\prime}\eta}$, $|\phi-4\tilde{T}_{0,\infty}^{3}g_{\infty}|\leq
Ce^{-\lambda^{\prime}\eta}$, for the proof of this result we refer to the [7,
Theorem 2]. Therefore, we deduce that
$\tilde{\psi}_{k,\infty}=4\tilde{T}^{3}_{0,\infty}\tilde{T}_{k,\infty}$.

Taking $g=\tilde{T}_{k}$, $\phi=\tilde{\psi}_{k}$ and
$S_{1}=-\chi_{0}\Delta_{x^{\prime}}\bar{T}_{k-2}+\langle
4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))\rangle+\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle$,
$S_{2}=-\chi_{0}\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}+4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))+(\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1))$,
$\phi_{b}=\beta\cdot\nabla\psi_{k-1}(0)-\mathcal{E}(T(0),k-1)$, system
(3.7)-(3.8) becomes system (2.35)-(2.36) with boundary conditions
(2.37)-(2.38). Therefore, Lemma 7 holds if $S_{1}$, $S_{2}$ satisfy (3.6).

To show (3.6), from the assumption (3.4),

$\displaystyle\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}|\chi_{0}\Delta_{x^{\prime}}\bar{T}_{k-2}|^{2}d\eta\leq
C\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}e^{-2\lambda_{k-1}\eta}d\eta=\frac{C}{2(\lambda_{k-1}-\lambda^{\prime})},$
(3.9)
$\displaystyle\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}|-\chi_{0}\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{k-1}|^{2}d\eta\leq
C\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}e^{-2\lambda_{k-1}\eta}d\eta=\frac{C}{2(\lambda_{k-1}-\lambda^{\prime})}.$
(3.10)

Moreover, due to the assumption (3.4) and $|P_{s}|\leq C(1+\eta^{k})$, we have

$\displaystyle|4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))|$
$\displaystyle=|4(\tilde{T}_{0}-\tilde{T}_{0,\infty})(P_{k}-P_{k}(0)||(\tilde{T}_{0}^{2}+\tilde{T}_{0,\infty}^{2})|\leq
Ce^{-\lambda_{0}\eta}(1+\eta^{k}),$

and

$\displaystyle|\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)|$
$\displaystyle\quad=\Big{|}\sum_{\begin{subarray}{c}i+j+l+m=k\\\ i,j,l,m\geq
1\end{subarray}}(\bar{T}_{i}+P_{i})(\bar{T}_{j}+P_{j})(\bar{T}_{l}+P_{l})(\bar{T}_{m}+P_{m})-\sum_{\begin{subarray}{c}i+j+l+m=k\\\
i,j,l,m\geq 1\end{subarray}}P_{i}P_{j}P_{l}P_{m}\Big{|}$
$\displaystyle\quad=\Big{|}\sum_{\begin{subarray}{c}i+j+l+m=k\\\ i,j,l,m\geq
1\end{subarray}}(\bar{T}_{i}\bar{T}_{j}\bar{T}_{l}\bar{T}_{m}+3\bar{T}_{i}P_{j}P_{l}P_{m}+6\bar{T}_{i}\bar{T}_{j}P_{l}P_{m}+3\bar{T}_{i}\bar{T}_{j}\bar{T}_{l}P_{m})\Big{|}$
$\displaystyle\quad\leq
C(e^{-4\lambda_{k-1}\eta}+e^{-\lambda_{k-1}\eta}(1+\eta^{k-1})^{3}+e^{-2\lambda_{k-1}\eta}(1+\eta^{k-1})^{2}+e^{-3\lambda_{k-1}\eta}(1+\eta^{k-1}))$
$\displaystyle\quad\leq Ce^{-\lambda_{k-1}\eta}(1+\eta^{3k-3}).$ (3.11)

Therefore,

$\displaystyle\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}|4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))|^{2}d\eta$
$\displaystyle\leq
C\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}e^{-2\lambda_{0}\eta}(1+\eta^{k})d\eta$
$\displaystyle=\frac{C}{2(\lambda_{0}-\lambda^{\prime})}+\frac{C}{2^{k+1}(\lambda_{0}-\lambda^{\prime})^{k+1}}\Gamma(k+1),$
(3.12)

and

$\displaystyle\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}|\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)|^{2}d\eta$
$\displaystyle\leq
C\int_{0}^{\infty}e^{2\lambda^{\prime}\eta}e^{-2\lambda_{k-1}\eta}(1+\eta^{6k-6})d\eta$
$\displaystyle=\frac{C}{2(\lambda_{k-1}-\lambda^{\prime})}+\frac{C}{2^{6k-5}(\lambda_{k-1}-\lambda^{\prime})^{6k-5}}\Gamma(6k-5),$
(3.13)

where $\Gamma(n):=(n-1)!$ is the Gamma function. Combining the above
inequalities with (3.9)-(3.10) implies (3.6) and finishes the proof. ∎

In the above lemma, the constant $\lambda^{\prime}$ may vary for different
$k$. In order to get a uniform constant, we apply the above lemma iteratively.
First, according to (3.3) from Lemma 6,
$\bar{T}_{0}=\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})$ and
$\bar{\psi}_{0}=\chi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})$ satisfy

$\displaystyle|\bar{T}_{0}(\eta,x^{\prime})|\leq
Ce^{-\lambda_{0}\eta},\quad|\bar{\psi}_{0}(\eta,x^{\prime},\beta|\leq
Ce^{-\lambda_{0}\eta},\quad\text{for any
}x^{\prime}\in\mathbb{T}^{2},\,\beta\in\mathbb{S}^{2}.$ (3.14)

for some $C>0$ and $\lambda>0$. Therefore, with the definition of $P_{0}$ in
(2.12), we have that $P_{0}=T_{0}(0)$ is bounded, i.e. $|P_{0}|\leq C$ for
some constant $C>0$. Therefore, the assumptions of Lemma 7 with $k=1$ hold. We
take $\lambda^{\prime}=\lambda_{0}-\varepsilon_{0}$, with $\varepsilon_{0}>0$
being a sufficiently small number, in (3.5) and get that for any
$x^{\prime}\in\mathbb{T}^{2},\,\beta\in\mathbb{S}^{2}$,

$\displaystyle|\tilde{T}_{1}(\eta,x^{\prime})-\tilde{T}_{1,\infty}(x^{\prime})|\leq
Ce^{-(\lambda_{0}-\varepsilon_{0})\eta},\quad|\tilde{\psi}_{1}(\eta,x^{\prime},\beta)-\tilde{\psi}_{1,\infty}(x^{\prime},\beta)|\leq
Ce^{-(\lambda_{0}-\varepsilon_{0})\eta}.$ (3.15)

With these, $\bar{T}_{1}=\chi(\tilde{T}_{1}-\tilde{T}_{1,\infty})$ and
$\bar{\psi}_{1}=\chi(\tilde{\psi}_{1}-\tilde{\psi}_{1,\infty})$ satisfy

$\displaystyle|\bar{T}_{1}(\eta,x^{\prime})|\leq
Ce^{-(\lambda_{0}-\varepsilon_{0})\eta},\quad|\bar{\psi}_{k}(\eta,x^{\prime},\beta)|\leq
Ce^{-(\lambda_{0}-\varepsilon_{0})\eta},\quad\forall~{}(x^{\prime}\in\mathbb{T}^{2},\,\beta\in\mathbb{S}^{2}).$
(3.16)

Moreover, from Lemma 5, $|P_{1}|\leq C(1+\eta)$. Therefore, the assumption of
Lemma 7 with $k=2$ hold with $\lambda_{1}=\lambda_{0}-\varepsilon_{0}$. We
take $\lambda^{\prime}=\lambda_{1}-\varepsilon_{0}/4$ in (3.5) and get that
for any $x^{\prime}\in\mathbb{T}^{2},\,\beta\in\mathbb{S}^{2}$,

$\displaystyle|\tilde{T}_{2}(\eta,x^{\prime})-\tilde{T}_{2,\infty}(x^{\prime})|\leq
Ce^{-(\lambda_{1}-\varepsilon_{0}/4)\eta},~{}|\tilde{\psi}_{2}(\eta,x^{\prime},\beta)-\tilde{\psi}_{2,\infty}(x^{\prime},\beta)|\leq
Ce^{-(\lambda_{1}-\varepsilon_{0}/4)\eta}.$ (3.17)

These again implies
$|\bar{T}_{2}(\eta,x^{\prime})|,|\bar{\psi}_{2}(\eta,x^{\prime},\beta)|\leq
Ce^{-(\lambda_{2}-\varepsilon_{0}/8)}$ with
$\lambda_{2}=\lambda_{1}-\varepsilon_{0}/4$ if we take
$\lambda^{\prime}=\lambda_{2}-\varepsilon_{0}/9$. We can thus apply Lemma 7
iteratively with $\lambda^{\prime}=\lambda_{s-1}-\varepsilon_{0}/s^{2}$ and
$\lambda_{s}=\lambda_{s-1}-\varepsilon_{0}/(s-1)^{2}$ in the $s$-th step and
get that

$\displaystyle|\tilde{T}_{s}(\eta,x^{\prime})-\tilde{T}_{s,\infty}(x^{\prime})|\leq
Ce^{-(\lambda_{s-1}-\varepsilon_{0}/s^{2})\eta},~{}|\tilde{\psi}_{k}(\eta,x^{\prime},\beta)-\tilde{\psi}_{2,\infty}(x^{\prime},\beta)|\leq
Ce^{-(\lambda_{s}-\varepsilon_{0}/s^{2})\eta},$ (3.18)

hold for any $0\leq s\leq N$ for all $x^{\prime}\in\mathbb{T}^{2}$,
$\beta\in\mathbb{S}^{2}$. By
$\bar{T}_{s}=\chi(\tilde{T}_{s}-\tilde{T}_{s,\infty})$ and
$\bar{\psi}_{s}=\chi(\tilde{\psi}_{s}-\tilde{\psi}_{s,\infty})$, the above
inequalities imply for any $s=0,\ldots,N$,

$\displaystyle|\bar{T}_{s}(\eta,x^{\prime})|\leq
Ce^{-(\lambda_{s-1}-\varepsilon_{0}/s^{2})\eta},\quad|\bar{\psi}_{s}(\eta,x^{\prime})|\leq
Ce^{-(\lambda_{s-1}-\varepsilon_{0}/s^{2})\eta},\quad\forall~{}(x^{\prime}\in\mathbb{T}^{2},\,\beta\in\mathbb{S}^{2}).$
(3.19)

Since $\lambda_{s}=\lambda_{s-1}-\varepsilon_{0}/(s-1)^{2}$,

$\displaystyle\lambda_{N-1}-\varepsilon_{0}/N^{2}=\lambda_{N-2}-\varepsilon_{0}/(N-1)^{2}-\varepsilon_{0}/N^{2}=\cdots=\lambda_{0}-\varepsilon_{0}\left(1+\frac{1}{2^{2}}+\cdots+\frac{1}{N^{2}}\right).$
(3.20)

From the formula $\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\pi^{2}/6$, the above
equation implies

$\displaystyle\lambda_{N-1}-\varepsilon_{0}/N^{2}=\lambda_{0}-\varepsilon_{0}\sum_{n=1}^{N}\frac{1}{N^{2}}\leq\lambda_{0}-\varepsilon_{0}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\leq\lambda_{0}-\frac{\pi^{2}}{6}\varepsilon_{0}.$
(3.21)

Therefore, taking $\varepsilon_{0}=3\lambda_{0}/\pi^{2}$ and setting
$\lambda=\lambda_{0}/2$, the following lemma holds.

######  Lemma 8.

Let $\Big{\\{}\left(\tilde{T}_{s},\tilde{\psi}_{s}\right)\Big{\\}}_{0\leq
s\leq N}$ be solutions to system (2.28)-(2.29) for $s=0$ and (2.35)-(2.36) for
$s\geq 1$. Then there exist constants $\lambda>0$ and $C>0$ such that for any
$s=0,\ldots,N$,

$\displaystyle|\tilde{T}_{s}(\eta,x^{\prime})-\tilde{T}_{s,\infty}(x^{\prime})|,\,|\tilde{\psi}_{s}(\eta,x^{\prime},\beta)-\tilde{\psi}_{s,\infty}(x^{\prime},\beta)|\leq
Ce^{-\lambda\eta},$ (3.22)
$\displaystyle|\bar{T}_{s}(\eta,x^{\prime})|,\,|\bar{\psi}_{s}(\eta,x^{\prime},\beta)|\leq
Ce^{-\lambda\eta},\quad\text{for any
}(x^{\prime},\beta)\in\mathbb{T}^{2}\times\mathbb{S}^{2},$

where $C>0$ is a postive constant independent of $s$.

###  3.3. Residual Estimates

Next we estimate $\mathcal{R}_{1}(T^{a},\psi^{a})$ and
$\mathcal{R}_{2}(T^{a},\psi^{a})$ given by (2.4) and (2.4), respectively. We
will prove the following theorem.

######  Theorem 9.

Assume $\tilde{T}_{0}$ satisfies the spectral assumption (A). The composite
approximate solution $(T^{a},\psi^{a})$ constructed in section 2, satisfies
(1.20)-(1.21) with boundary conditions (1.22)-(1.23). Moreover, the error
terms
$\mathcal{R}_{1}=\mathcal{R}_{1}(T^{a},\psi^{a}),\mathcal{R}_{2}=\mathcal{R}_{2}(T^{a},\psi^{a})$
satisfy

$\displaystyle\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{2}(\Omega)},~{}\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega)},~{}\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{2}(\Omega\times\mathbb{S}^{2})},~{}\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}$
$\displaystyle\quad\leq
C\gamma_{N}\varepsilon^{N+1}+(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}},$
(3.23)

for some constant $\gamma_{N}>1$. Furthermore, for
$\delta>-\frac{4}{\lambda}(N+1)\varepsilon\log\varepsilon$, the above estimate
implies

$\displaystyle\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{2}(\Omega)},~{}\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega)},~{}\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{2}(\Omega\times\mathbb{S}^{2})},~{}\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}$
$\displaystyle\quad\leq C\varepsilon^{N+1},$ (3.24)

where $C>0$ is a positive constant independent of $\varepsilon$.

###### Proof.

We first consider $\mathcal{R}_{1}(T^{a},\psi^{a})$. From (2.4), we have

$\displaystyle\mathcal{R}_{1}(T^{a},\psi^{a})$
$\displaystyle=(\varepsilon^{N+1}\Delta T_{N-1}+\varepsilon^{N+2}\Delta
T_{N}+\varepsilon^{N+1}\Delta_{x^{\prime}}\bar{T}_{N-1}+\varepsilon^{N+2}\Delta_{x^{\prime}}\bar{T}_{N})$
$\displaystyle\quad-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)\rangle-\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle$
$\displaystyle\quad+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{0}$
$\displaystyle=:R_{11}+R_{12}+R_{13}+R_{14}.$ (3.25)

By Lemma 4, $\|\Delta T_{N-1}\|_{L^{2}(\Omega)}$, $\|\Delta
T_{N}\|_{L^{2}(\Omega)}$, $\|\Delta T_{N-1}\|_{L^{\infty}(\Omega)}$ and
$\|\Delta T_{N}\|_{L^{\infty}(\Omega)}$ are bounded. Moreover, from Lemma 8,

$\displaystyle\int_{0}^{1}\bar{T}_{N-1}^{2}dx_{1}$
$\displaystyle=\int_{0}^{1}\chi^{2}(x_{1})(\tilde{T}_{N-1}(x_{1}/\varepsilon)-\tilde{T}_{N-1,\infty})^{2}dx_{1}=\int_{0}^{1/\varepsilon}\chi^{2}(\varepsilon\eta)(\tilde{T}_{N-1}(\eta)-\tilde{T}_{N-1,\infty})^{2}d\eta$
$\displaystyle\leq\int_{0}^{\infty}(\tilde{T}_{N-1}-\tilde{T}_{N-1,\infty})^{2}d\eta\leq
C\int_{0}^{\infty}e^{-2\lambda\eta}d\eta=\frac{C}{2\lambda},$
$\displaystyle\int_{0}^{1}\bar{\psi}_{N-1}^{2}dx_{1}$
$\displaystyle=\int_{0}^{1}\chi^{2}(x_{1})(\tilde{\psi}_{N-1}(x_{1}/\varepsilon)-\tilde{\psi}_{N-1,\infty})^{2}d\eta\leq
C\int_{0}^{\infty}e^{-2\lambda\eta}d\eta=\frac{C}{2\lambda},$

The above inequalities also hold for $(\bar{T}_{N},\bar{\psi}_{N})$. It
follows from the above inequalities and (3.22) that
$\|\Delta_{x^{\prime}}\bar{T}_{N-1}\|_{L^{2}(\Omega)}$,
$\|\Delta_{x^{\prime}}\bar{T}_{N}\|_{L^{2}(\Omega)}$,
$\|\Delta_{x^{\prime}}\bar{T}_{N-1}\|_{L^{\infty}(\Omega)}$ and
$\|\Delta_{x^{\prime}}\bar{T}_{N}\|_{L^{\infty}(\Omega)}$ are bounded.
Therefore,

$\displaystyle\|R_{11}\|_{L^{2}(\Omega)},~{}\|R_{11}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{N+1}.$ (3.26)

Due to (3.22), $\|\bar{T}_{s}\|_{L^{2}(\Omega)}$,
$\|\bar{T}_{s}\|_{L^{\infty}(\Omega)}$ are bounded for any $s=0,\ldots,N$.
Therefore,

$\displaystyle\|R_{12}\|_{L^{2}(\Omega)},~{}\|R_{12}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{N+1}.$ (3.27)

Next we consider $R_{13}$. Since $\chi(x_{1})=0$ for
$x_{1}\geq\tfrac{3}{8}\delta$, $R_{13}(x_{1},x^{\prime})=0$ for
$x_{1}\geq\tfrac{3}{8}\delta$. For the region $x_{1}\leq\tfrac{3}{8}\delta$,
we rewrite it as

$\displaystyle R_{13}1_{\\{x_{1}\leq\tfrac{3}{8}\delta\\}}$
$\displaystyle=-\sum_{k=0}^{4N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle
1_{\\{x_{1}\leq\tfrac{3}{8}\delta\\}}$
$\displaystyle\quad+\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle
1_{\\{x_{1}\leq\tfrac{3}{8}\delta\\}}$ $\displaystyle=:R_{131}+R_{132}.$
(3.28)

Similarly as $R_{12}$, $R_{132}$ satisfies

$\displaystyle\|R_{132}\|_{L^{2}(\Omega)},~{}\|R_{132}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{N+1}.$ (3.29)

For $R_{132}$, from the definition (2.5) of $\mathcal{C}$,

$\displaystyle R_{131}$
$\displaystyle=-\left\langle\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}(P_{k}+\bar{T}_{k})\right)^{4}+\left(\sum_{k=0}^{N}\varepsilon^{k}P_{k}\right)^{4}\right\rangle.$
(3.30)

Using the formula $a^{4}-b^{4}=(a-b)(a+b)(a^{2}+b^{2})$ and for $a-b=c-d=f$

$\displaystyle(a^{4}-b^{4})-(c^{4}-d^{4})$
$\displaystyle=(a-b)(a+b)(a^{2}+b^{2})-(c-d)(c+d)(c^{2}+d^{2})$
$\displaystyle=f(a-c)\left(2a^{2}+2b^{2}+(c+d)(a+b+c+d)\right),$

with $a=\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})$,
$b=\sum_{k=0}^{N}\varepsilon^{k}T_{k}$,
$c=\sum_{k=0}^{N}\varepsilon^{k}(P_{k}+\bar{T}_{k})$,
$d=\sum_{k=0}^{N}\varepsilon^{k}P_{k}$,
$f=\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}$, we obtain

$\displaystyle\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}(P_{k}+\bar{T}_{k})\right)^{4}+\left(\sum_{k=0}^{N}\varepsilon^{k}P_{k}\right)^{4}$
$\displaystyle\quad=\left(\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}\right)\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}-P_{k})\right)\left(2a^{2}+2b^{2}+(c+d)(a+b+c+d)\right).$
(3.31)

Due to (3.22),

$\displaystyle\left|\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}\right|\leq
Ce^{-\lambda\eta}=Ce^{-\frac{\lambda x_{1}}{\varepsilon}}.$ (3.32)

Taylor’s formula yields

$\displaystyle
T_{k}(x_{1},x^{\prime})=\sum_{l=0}^{N-k}\frac{x_{1}^{l}}{l!}\partial_{x_{1}}^{l}T_{l}(0,x^{\prime})+\frac{\partial_{x_{1}}^{N-k+1}}{(N-k+1)!}T_{k}(\xi_{k},x^{\prime})x_{1}^{N-k+1}.$
(3.33)

Using the above formula, we get

$\displaystyle\sum_{k=0}^{N}\varepsilon^{k}(T_{k}-P_{k})$
$\displaystyle=\sum_{k=0}^{N}\varepsilon^{k}T_{k}-\sum_{k=0}^{N}\varepsilon^{k}P_{k}$
$\displaystyle=\sum_{k=0}^{N}\varepsilon^{k}\left(\sum_{l=0}^{N-k}\frac{x_{1}^{l}}{l!}\partial_{x_{1}}^{l}T_{k}(0,x^{\prime})+\frac{\partial_{x_{1}}^{N-k+1}T_{k}(\xi_{k},x^{\prime})}{(N-k+1)!}x_{1}^{N-k+1}\right)$
$\displaystyle\quad-\sum_{k=0}^{N}\varepsilon^{k}\sum_{l=0}^{k}\frac{\eta^{l}}{l!}\frac{\partial^{l}}{\partial
x_{1}^{l}}T_{k-l}(0,x^{\prime}),$ (3.34)

with $\xi_{k}\in[0,x_{1}]$. Using the formula

$\sum_{k=0}^{N}\sum_{l=0}^{k}f(l,k)=\sum_{l=0}^{N}\sum_{k=l}^{N}f(l,k)\stackrel{{\scriptstyle
s=k-l}}{{=\joinrel=}}\sum_{l=0}^{N}\sum_{s=0}^{N-l}f(l,l+s)\stackrel{{\scriptstyle
l\to k,s\to l}}{{=\joinrel=}}\sum_{k=0}^{N}\sum_{l=0}^{N-k}f(k,k+l)$

and taking
$f(k,k+l)=\varepsilon^{k}\frac{x_{1}^{l}}{l!}\partial_{x_{1}}^{l}T_{k}(0,x^{\prime})$,
we get

$f(l,k)=\varepsilon^{l}\frac{x_{1}^{k-l}}{(k-l)!}\partial_{x_{1}}^{k-l}T_{l}(0,x^{\prime})=\varepsilon^{k}\frac{\eta^{k-l}}{(k-l)!}\partial_{x_{1}}^{k-l}T_{l}(0,x^{\prime})$

and so

$\displaystyle\sum_{k=0}^{N}\sum_{l=0}^{N-k}\varepsilon^{k}\frac{x_{1}^{l}}{l!}\partial_{x_{1}}^{l}T_{k}(0,x^{\prime})=\sum_{k=0}^{N}\sum_{l=0}^{k}\varepsilon^{k}\frac{\eta^{k-l}}{(k-l)!}\partial_{x_{1}}^{k-l}T_{l}(0,x^{\prime})\stackrel{{\scriptstyle
k\to k,k-l\to
l}}{{=\joinrel=}}\sum_{k=0}^{N}\sum_{l=0}^{N}\varepsilon^{k}\frac{\eta^{l}}{l!}\partial_{x_{1}}^{l}T_{k-l}(0,x^{\prime}).$

Taking this relation into (3.3) leads to

$\displaystyle\sum_{k=0}^{N}\varepsilon^{k}(T_{k}-P_{k})=\sum_{k=0}^{N}\varepsilon^{k}\frac{\partial_{x_{1}}^{N-k+1}T_{k}(\xi_{k},x^{\prime})}{(N-k+1)!}x_{1}^{N-k+1}.$

Combining this with (3.32), (3.3) satisfies

$\displaystyle\left|\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}(P_{k}+\bar{T}_{k})\right)^{4}+\left(\sum_{k=0}^{N}\varepsilon^{k}P_{k}\right)^{4}\right|$
$\displaystyle\quad\leq Ce^{-\frac{\lambda
x_{1}}{\varepsilon}}\sum_{k=0}^{N}\varepsilon^{k}\frac{|\partial_{x_{1}}^{N-k+1}T_{k}(\xi_{k},x^{\prime})|}{(N-k+1)!}x_{1}^{N-k+1}|2a^{2}+2b^{2}+(c+d)(a+b+c+d)|$
$\displaystyle\quad\leq
C\sum_{k=0}^{N}\varepsilon^{k}\frac{1}{(N-k+1)!}x_{1}^{N-k+1}e^{-\frac{\lambda
x_{1}}{\varepsilon}},$

where the fact that $\|a,b,c,d\|_{L^{\infty}(\Omega)}\leq C$ are bounded and
$\|T_{k}\|_{C^{s}(\Omega)}\leq C$ is bounded for any $s\geq 0$. Note that the
function $h(x_{1}):=x_{1}^{N-k+1}e^{-\lambda x_{1}/\varepsilon}$ attains its
maximum at $x_{1}^{*}=(N-k+1)\varepsilon/\lambda$ with the maximum value
$h(x^{*})=(N-k+1)^{N-k+1}\varepsilon^{N-k+1}/\lambda^{N-k+1}\cdot
e^{-(N-k+1)}$. Therefore,

$\displaystyle\left|\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}\right)^{4}-\left(\sum_{k=0}^{N}\varepsilon^{k}(P_{k}+\bar{T}_{k})\right)^{4}+\left(\sum_{k=0}^{N}\varepsilon^{k}P_{k}\right)^{4}\right|$
$\displaystyle\quad\leq
C\varepsilon^{N+1}\sum_{k=0}^{N}\frac{(N-k+1)^{N-k+1}}{\lambda^{N-k+1}(N-k+1)!}e^{-(N-k+1)}=C\varepsilon^{N+1}\sum_{n=1}^{N+1}\frac{n^{n}}{\lambda^{n}n!}e^{-n}\leq
C(\gamma_{N}-1)\varepsilon^{N+1},$

where $\gamma_{N}:=\sum_{n=0}^{N+1}n^{n}/(\lambda^{n}n!)>1$ is a constant
depending on $N$. Therefore,

$\displaystyle\|R_{131}\|_{L^{2}(\Omega)}\leq
C(\gamma_{N}-1)\varepsilon^{N+1},~{}\|R_{131}\|_{L^{\infty}(\Omega)}\leq
C(\gamma_{N}-1)\varepsilon^{N+1}.$

Combining this with (3.29), we get

$\displaystyle\|R_{13}\|_{L^{2}(\Omega)}\leq
C\gamma_{N}\varepsilon^{N+1},~{}\|R_{13}\|_{L^{\infty}(\Omega)}\leq
C\gamma_{N}\varepsilon^{N+1}.$ (3.35)

Finally, we consider $R_{14}$. Recalling (2.4), we have

$\displaystyle E_{0}^{0}$
$\displaystyle=(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}+\chi\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle-\langle(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}\rangle,$
(3.36)

Since $\tilde{T}_{0}(\eta,x^{\prime})-\tilde{T}_{0,\infty}(x^{\prime})$ and
$\partial_{\eta}\tilde{T}_{0}(\eta,x^{\prime})$ exponentially decay to $0$ as
$\eta\to\infty$ and
$\partial_{\eta}\chi(\varepsilon\eta)=\varepsilon\partial_{x_{1}}\chi(x_{1})$,
$\partial_{\eta}^{2}\chi(\varepsilon\eta)=\varepsilon^{2}\partial^{2}_{x_{1}}\chi(x_{1})$
are supported on the interval $\varepsilon\eta\in(\tfrac{3}{8}\delta,\infty)$,
hence

$\displaystyle|(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}|\leq
Ce^{-\lambda\eta}(\varepsilon|\partial_{x_{1}}\chi|+\varepsilon^{2}|\partial_{x_{1}}\chi|)1_{\eta\geq\frac{3\delta}{8\varepsilon}}\leq
C\varepsilon e^{-\frac{3\lambda\delta}{8\varepsilon}}.$ (3.37)

Since $\tilde{T}_{0}(\eta,x^{\prime})-\tilde{T}_{0,\infty}(x^{\prime})$ decays
exponentially,

$\displaystyle|\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}|=|(\tilde{T}_{0}-\tilde{T}_{0,\infty}|\cdot|(\tilde{T}_{0}+\tilde{T}_{0,\infty})(\tilde{T}_{0}^{2}+\tilde{T}_{0,\infty})^{2}|\leq
Ce^{-\lambda\eta},$

and

$\displaystyle|(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}|$
$\displaystyle=\chi|(\tilde{T}_{0}-\tilde{T}_{0,\infty})|\cdot|\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty}|\cdot|(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty}))^{2}+\tilde{T}_{0,\infty}^{2}|$
$\displaystyle\leq Ce^{-\lambda\eta}.$

Note that for $\varepsilon\eta<\tfrac{1}{4}$, $\chi(\varepsilon\eta)=1$ and

$\displaystyle\langle\chi\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle-\langle(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}\rangle=\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle-\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle=0.$

Hence the left term of the above equation is supported on the interval
$\varepsilon\eta\in[\tfrac{1}{4}\delta,\infty)$. Therefore,

$\displaystyle|\langle\chi\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle-\langle(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}\rangle|\leq
Ce^{-\lambda\eta}1_{\eta\geq\tfrac{\delta}{4\varepsilon}}\leq
Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$ (3.38)

combining the above inequality with (3.37), (3.36) satisfies

$\displaystyle|E_{0}^{0}|\leq C\varepsilon
e^{-\frac{3\delta}{8\varepsilon}}+Ce^{-\frac{\lambda\delta}{4\varepsilon}}\leq
Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$ (3.39)

For $k\geq 1$, we recall (2.4):

$\displaystyle
E_{k}^{0}=((\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k})+\Big{(}\chi(\langle
4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}+4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))\rangle$
$\displaystyle\quad+\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle)-\langle\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(P,k)\rangle\Big{)}+(1-\chi\chi_{0})\Delta_{x^{\prime}}\bar{T}_{k-2}$
$\displaystyle=:E_{k1}^{0}+E_{k2}^{0}+E_{k3}^{0}.$

The term
$(\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k}$
can be estimated in the same way as (3.37) as

$\displaystyle|E_{k1}^{0}|=|(\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k}|\leq
C\varepsilon e^{-\frac{3\lambda\delta}{8\varepsilon}}.$ (3.40)

For $E_{k3}^{0}$, $(1-\chi(\varepsilon\eta)\chi_{0}(\varepsilon\eta))$ is
supported in $\varepsilon\eta\in[\tfrac{1}{4}\delta,\infty)$. By (3.22),

$\displaystyle|E_{k3}^{0}|\leq
Ce^{-\lambda\eta}1_{\eta>\tfrac{\delta}{4\varepsilon}}\leq
Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$ (3.41)

To estimate $E_{k2}^{2}$, due to (3.22),

$\displaystyle|4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}+4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))|\leq
Ce^{-\lambda\eta}+Ce^{-\lambda\eta}(1+\eta^{k})\leq
C(1+\eta^{k})e^{-\lambda\eta}.$

Since $(\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1))/\bar{T}$ is a
polynomial of $\bar{T}$, $P$ of order no bigger than $\eta^{k-1}$ and thus is
bounded, we have

$\displaystyle|\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)|=|\bar{T}|\left|\frac{\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)}{\bar{T}}\right|\leq
C(1+\eta^{k-1})e^{-\lambda\eta}.$

Similarly,

$\displaystyle|\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(P,k)|=|\bar{T}|\left|\frac{\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(C,k)}{\bar{T}}\right|\leq
C(1+\eta^{k})e^{-\lambda\eta}.$

Therefore,

$\displaystyle E_{k2}^{0}\leq C(1+\eta^{k})e^{-\lambda\eta}.$

Moreover, when $\varepsilon\eta<\tfrac{1}{4}$, $\chi(\varepsilon\eta)=1$ and
due to the formula
$\mathcal{C}(\bar{T}+P,k)-\mathcal{C}(P,k)=4(\bar{T}_{0}+P_{0})^{3}(\bar{T}_{k}+P_{k})-4P_{0}^{3}P_{k}+\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)$
as well as

$\displaystyle
4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}+4(\tilde{T}_{0}^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))$
$\displaystyle\quad=4(\bar{T}_{0}+P_{0})^{3}(\bar{T}_{k}+P_{k}(0))-4\tilde{T}_{0,\infty}^{3}P_{k}(0)+4((\bar{T}_{0}+P_{0})^{3}-\tilde{T}_{0,\infty}^{3})(P_{k}-P_{k}(0))$
$\displaystyle\quad=4(\bar{T}_{0}+P_{0})^{3}(\bar{T}_{k}+P_{k})-4P_{0}^{3}P_{k},$

where we use the fact that $P_{0}=T_{0}(0)=\tilde{T}_{0,\infty}$, we have
$E_{k2}^{0}=0$ on the interval $\varepsilon\eta\in[0,\tfrac{1}{4}\delta)$.
Thus

$\displaystyle E_{k2}^{0}\leq
C(1+\eta^{k})e^{-\lambda\eta}1_{\eta>\tfrac{\delta}{4\varepsilon}}\leq
C(1+\frac{1}{\varepsilon^{k}})e^{-\frac{\lambda\delta}{4\varepsilon}}.$

combining this with (3.40) and (3.41), we obtain

$\displaystyle|E_{k}^{0}|\leq C\varepsilon
e^{-\frac{3\delta}{8\varepsilon}}+Ce^{-\frac{\lambda\delta}{4\varepsilon}}+C(1+\frac{1}{\varepsilon^{k}})e^{-\frac{\lambda\delta}{4\varepsilon}}\leq
Ce^{-\frac{\lambda\delta}{4\varepsilon}}+C\frac{1}{\varepsilon^{k}}e^{-\frac{\lambda\delta}{4\varepsilon}}.$

combining the above estimate with (3.39), we arrive at

$\displaystyle\left|\sum_{k=0}^{N}E_{k}^{0}\right|\leq\sum_{k=0}^{N}|E_{k}^{0}|\leq
Ce^{-\frac{\lambda\delta}{4\varepsilon}}+\sum_{k=1}^{N}\varepsilon^{k}\left(1+\frac{1}{\varepsilon^{k}}\right)e^{-\frac{\lambda\delta}{4\varepsilon}}\leq(N+1)Ce^{-\frac{\lambda\delta}{4\varepsilon}}+e^{-\frac{\lambda\delta}{4\varepsilon}}\sum_{k=1}^{N}\varepsilon^{k}.$

We can take $\varepsilon<1/2$ so that
$\sum_{k=1}^{N}\varepsilon^{k}\leq\sum_{k=1}^{N}1/2^{k}\leq 1$ and get

$\displaystyle|R_{14}|=\left|\sum_{k=0}^{N}E_{k}^{0}\right|\leq(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$

By the regularity of solutions, we have

$\displaystyle\|R_{14}\|_{L^{2}(\Omega)},~{}\|R_{14}\|_{L^{\infty}(\Omega)}\leq(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$

Taking the above inequality and (3.26), (3.27) and (3.35), we obtain

$\displaystyle\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{2}(\Omega)},~{}\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega)}\leq
C\gamma_{N}\varepsilon^{N+1}+(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$
(3.42)

One can estimate $\mathcal{R}_{2}(T^{a},\psi^{a})$ given by (2.4) in the same
way. Recalling (2.4),

$\displaystyle\mathcal{R}_{2}(T^{a},\psi^{a})$
$\displaystyle=(\varepsilon^{N+1}\beta\cdot\nabla\psi_{N}+\varepsilon^{N+1}\beta^{\prime}\cdot\nabla_{x^{\prime}}\bar{\psi}_{N})-\sum_{k=N+1}^{4N}\varepsilon^{k}(\mathcal{C}(T+\bar{T},k))$
$\displaystyle\quad-\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(T+\bar{T},k)-\mathcal{C}(T,k)-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k))+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{1}$
$\displaystyle=:R_{21}+R_{22}+R_{23}+R_{24}.$

By the boundness of solutions to the interior expansion and boundary layer
corrections, $|R_{21}|=O(\varepsilon^{N+1})$, $|R_{22}|=O(\varepsilon^{N+1})$.
The term $R_{23}$ is the same as that of $R_{13}$ without integration over
$\beta\in\mathbb{S}^{2}$ and thus can be estimated in the same way,
$|R_{13}|\leq C\gamma_{N}\varepsilon^{N+1}$. Finally, $R_{24}$ can be
estimated in the same way as $R_{14}$, whereas the only difference is the term
$\mu(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})\partial_{\eta}\chi$, which is
supported on $\varepsilon\eta\in[\tfrac{1}{4}\delta,\infty)$ and thus can be
estimate by

$\displaystyle|\mu(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})\partial_{\eta}\chi|\leq
Ce^{-\lambda\eta}1_{\eta\geq\tfrac{\delta}{4\varepsilon}}\leq
Ce^{-\lambda\delta/4\varepsilon},$

and $|R_{24}|\leq(N+2)Ce^{-\frac{\lambda}{4\varepsilon}}$. Thus (3.42) also
holds for $\mathcal{R}_{2}(T^{a},\psi^{a})$, i.e.

$\displaystyle\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{2}(\Omega\times\mathbb{S}^{2})},~{}\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq
C\gamma_{N}\varepsilon^{N+1}+(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$

∎

###  3.4. An inequality

Next we prove inequality (1.29) under the spectral assumption.

######  Lemma 10.

Let $(T^{a},\psi^{a})$ be the approximate solution constructed in the previous
section. Assuming the spectral assumption (A) holds for the solution
$\tilde{T}_{0}$ of the nonlinear Milne problem (1.8)-(1.11) where
$\tilde{T}_{0}\geq a$ for some constant $a>0$. Then, for $\varepsilon>0$
sufficiently small, the following inequality holds

$\displaystyle-\int_{\Omega}4(T^{a})^{3}g\Delta
gdx=\int_{\Omega}4(T^{a})^{3}|\nabla
g|^{2}dx-\int_{\Omega}\nabla(4(T^{a})^{3})\cdot g\nabla
gdx\geq\kappa\int_{\Omega}|\nabla g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2},$ (3.43)

for any function $g$ satisfying $g(0)=0$ and for some constants $\kappa>0$,
$C>0$ depending on $M$, where $M<1$ is the constant in (1.13) of the spectral
assumption.

###### Proof.

Note that $T^{a}=\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})$ where
$\bar{T}_{k}=\chi(x_{1})(\tilde{T}_{k}-\tilde{T}_{k,\infty})$. In the domain
$x_{1}>\tfrac{3}{8}\delta$, $\chi(x_{1})=0$ and
$T^{a}=\sum_{k=0}^{N}\varepsilon^{k}T_{k}$, which only contain the interior
approximations. Since $\|T_{k}\|_{C^{s}(\Omega)}$ is bounded for any $s>0$ and
$k=1,\ldots,N$,

$\displaystyle\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}4(T^{a})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}\nabla(4(T^{a})^{3})\cdot
g\nabla gdx$
$\displaystyle\quad=\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}4(T^{a})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}2(T^{a})^{3/2}\nabla
g\cdot 6(T^{a})^{1/2}\nabla T^{a}gdx$
$\displaystyle\quad\geq\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}4(T^{a})^{3}|\nabla
g|^{2}dx-\frac{1}{2}\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}4(T^{a})^{3}|\nabla
g|^{2}dx-\frac{1}{2}\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}36T^{a}|\nabla
T^{a}|^{2}g^{2}dx$
$\displaystyle\quad\geq\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}2(T^{a})^{3}|\nabla
g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2}.$ (3.44)

In the domain $x_{1}\leq\frac{3}{8}\delta$, boundary layer effects play a
role. First we split the integral as

$\displaystyle\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}4(T^{a})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}\nabla(4(T^{a})^{3})\cdot
g\nabla gdx$
$\displaystyle\quad=\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}4(T^{a})^{3}|\nabla_{x^{\prime}}g|^{2}dx-\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}\nabla_{x^{\prime}}(4(T^{a})^{3})\cdot
g\nabla_{x^{\prime}}gdx$
$\displaystyle\qquad+\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}4(T^{a})^{3}|\partial_{x_{1}}g|^{2}dx-\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}\partial_{x_{1}}(4(T^{a})^{3})\cdot
g\partial_{x_{1}}gdx=:I_{1}+I_{2}.$

Since $\|\nabla_{x^{\prime}}(T^{a})\|_{L^{2}(\Omega)}$ is bounded, we can
estimate $I_{1}$ the same as (3.44):

$\displaystyle
I_{1}\geq\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}2(T^{a})^{3}|\nabla_{x^{\prime}}g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2}.$
(3.45)

To estimate $I_{2}$, we use the spectral assumption (A). Near the boundary,
the composite approximate solution $(T^{a},\psi^{a})$ is close to the solution
$(\tilde{T}_{0},\tilde{\psi}_{0})$ of the nonlinear Milne problem 1.8-(1.11).
Using the equation

$\displaystyle(T^{a})^{3}=\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})\right)^{3}=(T_{0}+\bar{T}_{0})^{3}+\varepsilon
G,$

where
$G=3(T_{0}+\bar{T}_{0})^{2}\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k})+6(T_{0}+\bar{T}_{0})^{2}(\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k}))^{2}+3(T_{0}+\bar{T}_{0})(\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k}))^{2}$,
we can rewrite $I_{2}$ as

$\displaystyle I_{2}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}\int_{0}^{\frac{3}{8\varepsilon}}4(T_{0}+\bar{T}_{0})^{3}+4\varepsilon
G)|\partial_{\eta}g|^{2}d\eta
dx^{\prime}-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}\int_{0}^{\frac{3}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}+4\varepsilon
G)g\partial_{\eta}gd\eta dx^{\prime}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\partial_{\eta}(4\tilde{T}_{0}^{3})g\partial_{\eta}g)d\eta+\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})|\partial_{\eta}g|^{2}d\eta$
$\displaystyle\quad-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})g\partial_{\eta}gd\eta
dx^{\prime}+\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}\varepsilon
4(G|\partial_{\eta}g|^{2}-\partial_{\eta}Gg\partial_{\eta}g)d\eta$
$\displaystyle=:I_{21}+I_{22}+I_{23}+I_{24}.$

The spectral assumption (A) implies

$\displaystyle I_{21}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\partial_{\eta}(4\tilde{T}_{0}^{3})g\partial_{\eta}g)d\eta$
$\displaystyle\geq\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\frac{1}{2}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}+36\tilde{T}_{0}|\partial_{\eta}\tilde{T}_{0}|^{2}g^{2}))d\eta$
$\displaystyle\geq\frac{1}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-36\tilde{T}_{0}|\partial_{\eta}\tilde{T}_{0}|^{2}g^{2})d\eta$
$\displaystyle\geq\frac{1-M}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta.$

For $I_{22}$, since
$\bar{T}_{0}=T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))$, it holds
that

$\displaystyle(T_{0}+\bar{T}_{0})^{3}-\tilde{T}_{0}^{3}$
$\displaystyle=(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=((T_{0}-T_{0}(0))-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=(\partial_{x_{1}}T_{0}(\xi)\varepsilon\eta-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2}).$

Since we are considering the integration over
$x_{1}=\varepsilon\eta\in[0,\tfrac{3}{8}\delta]$ and
$(1-\chi(\varepsilon\eta))$ is supported on
$[\tfrac{1}{4}\delta,\tfrac{3}{8}\delta]$,

$\displaystyle
I_{22}=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})|\partial_{\eta}g|^{2}d\eta\leq\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{\delta}{2\varepsilon}}|\partial_{\eta}g|^{2}d\eta.$

For $I_{23}$, due to

$\displaystyle\partial_{\eta}((T_{0}+\bar{T}_{0})^{3}-\tilde{T}_{0}^{3})$
$\displaystyle=\partial_{\eta}(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})\partial_{\eta}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=\varepsilon\chi^{\prime}(\varepsilon\eta)(\tilde{T}_{0}-{T}_{0}(0))(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(\chi(\varepsilon\eta)-1)\partial_{\eta}\tilde{T}_{0}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(\partial_{x_{1}}T_{0}(\xi)\varepsilon\eta-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))\partial_{\eta}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2}),$

with consideration of $\varepsilon\eta\in[0,\tfrac{3}{8}\delta]$ and and
$(1-\chi(\varepsilon\eta))$ being supported on
$[\tfrac{1}{4}\delta,\tfrac{3}{8}\delta]$, it holds that

$\displaystyle I_{23}$
$\displaystyle=-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})g\partial_{\eta}gd\eta\leq\frac{C}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(\varepsilon|g||\partial_{\eta}g|+\frac{3\delta}{8}|g||\partial_{\eta}g|)d\eta$
$\displaystyle\leq\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(g^{2}+|\partial_{\eta}g|^{2})d\eta.$

For $I_{24}$, we have

$\displaystyle
I_{24}=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}\varepsilon
4(G|\partial_{\eta}g|^{2}-\partial_{\eta}Gg\partial_{\eta}g)d\eta\leq\frac{C}{\varepsilon}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta.$

combining the above estimates gives

$\displaystyle I_{2}$
$\displaystyle\geq\frac{1-M}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta-\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta$
$\displaystyle\quad-\frac{C}{\varepsilon}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta.$

By the assumption of the lemma, $\tilde{T}_{0}\geq a$, hence
$4\tilde{T}_{0}^{3}\geq 4a^{3}$ for some constant $a>0$. We can take
sufficiently small $\varepsilon$ and $\delta$ such that
$\varepsilon<(1-M)a^{3}/C$ and $3\delta C/8\leq(1-M)/8$, and we get from the
above inequality

$\displaystyle
I_{2}\geq\frac{1-M}{4\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta.$

combining this with (3.44) and (3.45) implies

$\displaystyle-\int_{\Omega}4(T^{a})^{3}g\Delta gdx$
$\displaystyle=\int_{\Omega}4(T^{a})^{3}|\nabla
g|^{2}dx-\int_{\Omega}\nabla(4(T^{a})^{3})\cdot g\nabla gdx$
$\displaystyle\geq\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}2(T^{a})^{3}|\nabla
g|^{2}dx+\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\delta\\}}2(T^{a})^{3}|\nabla_{x^{\prime}}g|^{2}dx$
$\displaystyle\quad+\frac{1-M}{4\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta-C\|g\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\geq\kappa\|\nabla
g\|_{L^{2}(\Omega)}^{2}-C\|g\|_{L^{2}(\Omega)}^{2},$

where $\kappa=\min\\{2a^{3},(1-M)a^{3}\\}$, which finishes the proof of Lemma
10. ∎

##  4\. Diffusive Limit

In this section, we prove Theorem 1 by estimating the difference between the
solution $(T^{\varepsilon},\psi^{\varepsilon})$ to system (1.1)-(1.2) and the
constructed approximate solution $(T^{a},\psi^{a})$, which satisfies
(1.20)-(1.21). Setting
$g:=T^{\varepsilon}-T^{a},\phi:=\psi^{\varepsilon}-\psi^{a}$, functions
$(g,\phi)$ then satisfy

$\displaystyle\varepsilon^{2}\Delta
g+\langle\phi-(T^{a}+g)^{4}+(T^{a})^{4}\rangle=-\mathcal{R}_{1}(T^{a},\psi^{a}),$
(4.1)
$\displaystyle\varepsilon\beta\cdot\nabla\phi+\phi-(T^{a}+g)^{4}+(T^{a})^{4}=-\mathcal{R}_{2}(T^{a},\psi^{a}),$
(4.2)

with boundary conditions

$\displaystyle g(x)=0,\quad\text{for }x\in\partial\Omega,$
$\displaystyle\phi(x,\beta)=0,\text{ for }(x,\beta)\in\Gamma_{-}.$

In order to prove Theorem 1, we first derive suitable estimates on a
linearized system and then use Banach fixed point theorem to show the
existence of the above problem near zero solutions, leading to the convergence
of $\left(T^{\varepsilon},\psi^{\varepsilon}\right)$ to
$\left(T^{a},\psi^{a}\right)$ as $\varepsilon\to 0$.

###  4.1. Linearized system

We first consider the following linear system:

$\displaystyle\varepsilon^{2}\Delta
g+\langle\phi-4(T^{a})^{3}g\rangle=r_{1}+\langle r\rangle,$ (4.3)
$\displaystyle\varepsilon\beta\cdot\nabla\phi+\phi-4(T^{a})^{3}g=r_{2}+r,$
(4.4)

where $r_{1}=r_{1}(x),r=r(x,\beta)$, and $r_{2}=r_{2}(x,\beta)$ are given
functions and the boundary conditions are taken to be

$\displaystyle g(x)=0,\text{ for any }x\in\partial\Omega,$ (4.5)
$\displaystyle\phi(x,\beta)=0,\text{ for any }(x,\beta)\in\Gamma_{-}.$ (4.6)

The existence of solutions to the above problem and suitable estimates on the
solutions are stated in the following lemma.

######  Lemma 11.

Let $\varepsilon>0$, $(T^{a},\psi^{a})$ be the composite approximate solution
constructed in section 2. Assuming $r_{1}\in L^{2}\cap L^{\infty}(\Omega)$,
and $r,r_{2}\in L^{2}\cap L^{\infty}(\Omega\times\mathbb{S}^{2})$. Then, there
exists a unique solution $(g,\phi)\in L^{2}\cap L^{\infty}(\Omega)\times
L^{2}\cap L^{\infty}(\Omega\times\mathbb{S}^{2})$ to system (4.3)-(4.4) with
boundary conditions (4.5)-(4.6). Moreover, the solution $(g,\phi)$ satisfies
the following estimates

$\displaystyle\varepsilon\|\phi\|_{L^{2}(\Omega\times\mathbb{S})}+\varepsilon\|g\|_{H^{1}(\Omega)}+\sqrt{\varepsilon}\|\phi\|_{L^{2}(\Gamma_{+})}+\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}$
$\displaystyle\quad\leq
C\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}+\frac{C}{\varepsilon}(\|r_{1}\|_{L^{2}(\Omega)}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}).$
(4.7)

and

$\displaystyle\|\phi\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}+\|g\|_{L^{\infty}(\Omega)}$
$\displaystyle\quad\leq\frac{C}{\varepsilon^{2}}\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}+\frac{C}{\varepsilon^{3}}(\|r_{1}\|_{L^{2}(\Omega)}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})})+C\|r_{2}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}+C\|r\|_{L^{\infty}(\Omega)},$
(4.8)

where $C>0$ is a constant not depending on $\varepsilon$.

###### Proof.

Existence of the linear system (4.3)-(4.4) with homogeneous boundary
conditions (4.5)-(4.6) follows from standard theory of elliptic and transport
equations. To derive the estimates (11) and (11), we first derive the energy
estimate. Then the $L^{2}$ type estimate is derived. Finally, the $L^{\infty}$
type estimate is shown.

_Step 1: The energy estimate._ We multiply (4.3) by $4(T^{a})^{3}g$ and (4.4)
by $\phi$, and integrate over $x\in\Omega$ and $\beta\in\mathbb{S}^{2}$ to get

$\displaystyle-\int_{\Omega}\varepsilon^{2}4(T^{a})^{3}g\Delta
gdx+\iint_{\Omega\times\mathbb{S}^{2}}\varepsilon\beta\cdot\nabla\frac{\phi^{2}}{2}d\beta
dx+\iint_{\Omega\times\mathbb{S}^{2}}(\phi-4(T^{a})^{3}g)^{2}d\beta dx$
$\displaystyle\phantom{xx}{}=\iint_{\Omega\times\mathbb{S}^{2}}\phi
r_{2}d\beta
dx-\int_{\Omega}4(T^{a})^{3}gr_{1}dx+\iint_{\Omega\times\mathbb{S}^{2}}(\phi-4(T^{a})^{3}g)rd\beta
dx.$ (4.9)

The boundary condition (4.6) implies

$\displaystyle\iint_{\Omega\times\mathbb{S}^{2}}\varepsilon\beta\cdot\nabla\frac{\phi^{2}}{2}d\beta
dx$ $\displaystyle=\varepsilon\iint_{\Gamma}\beta\cdot n\phi^{2}d\beta
d\sigma_{x}=\varepsilon\iint_{\Gamma_{+}}\beta\cdot n\phi^{2}d\beta
d\sigma_{x}=\varepsilon\|\phi\|_{L^{2}(\Gamma_{+})}^{2}.$ (4.10)

By Lemma 10, inequality (3.43) gives

$\displaystyle-\int_{\Omega}4(T^{a})^{3}g\Delta gdx\geq\kappa\|\nabla
g\|_{L^{2}(\Omega)}^{2}-C\|g\|_{L^{2}(\Omega)}^{2}.$ (4.11)

Applying Young’s inequality on the last term of (4.1) gives

$\displaystyle\iint_{\Omega\times\mathbb{S}^{2}}(\phi-4(T^{a})^{3}g)rd\beta
dx\leq\frac{1}{2}\iint_{\Omega\times\mathbb{S}^{2}}(\phi-4(T^{a})^{3}g)^{2}d\beta
dx+\frac{1}{2}\iint_{\Omega\times\mathbb{S}^{2}}r^{2}d\beta dx.$

Taking the above inequality and (4.10),(4.11) into (4.1), we obtain the
following energy estimate:

$\displaystyle\varepsilon^{2}\kappa\|\nabla
g\|_{L^{2}(\Omega)}^{2}+\varepsilon\|\phi\|_{L^{2}(\Gamma_{+})}^{2}+\frac{1}{2}\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}$
$\displaystyle\quad\leq
C\varepsilon^{2}\|g\|_{L^{2}(\Omega)}^{2}+\iint_{\Omega\times\mathbb{S}^{2}}\phi
r_{2}d\beta
dx-\int_{\Omega}4(T^{a})^{3}gr_{1}dx+\frac{1}{2}\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}.$
(4.12)

_Step 2: The $L^{2}$ estimate._ First we estimate the $L^{2}$ norm of $g$.
Given $\rho=\rho(x,\beta)\in L^{2}(\Omega\times\mathbb{S}^{2})$, define the
operator $\mathcal{A}:L^{2}(\Omega)\mapsto L^{2}(\Omega)$ by

$\displaystyle\mathcal{A}h=\langle\varphi-h\rangle,\quad\text{ where
}\varphi\text{ solves
}\left\\{\begin{array}[]{cl}\varepsilon\beta\cdot\nabla\varphi+\varphi-h=\rho,&\text{
in }\Omega\times\mathbb{S}^{2},\\\ \varphi(x,\beta)=0,&\text{ for
}(x,\beta)\in\Gamma_{-},\end{array}\right.$ (4.15)

for $h=h(x)\in L^{2}(\Omega)$. Then for any function $\ell=\ell(x)\in
C^{2}(\Omega)$ satisfying $\ell=0$ on $\partial\Omega$,

$\displaystyle\int_{\Omega}\mathcal{A}h\cdot\ell dx$
$\displaystyle=\int_{\Omega}\int_{\mathbb{S}^{2}}(\varphi-h)\ell d\beta dx$
$\displaystyle=-\int_{\Omega}\int_{\mathbb{S}^{2}}\varepsilon\beta\cdot\nabla\varphi\ell
d\beta dx+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx$
$\displaystyle=\int_{\Omega}\int_{\mathbb{S}^{2}}\varepsilon\varphi\beta\cdot\nabla\ell
d\beta dx-\varepsilon\iint_{\Gamma}\beta\cdot n\varphi\ell d\beta
d\sigma_{x}+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx$
$\displaystyle=\varepsilon\int_{\Omega}\int_{\mathbb{S}^{2}}(\varphi-h)\beta\cdot\nabla\ell
d\beta dx-0+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx$
$\displaystyle=-\varepsilon\int_{\Omega}\int_{\mathbb{S}^{2}}\varepsilon\beta\cdot\nabla\varphi\beta\cdot\nabla\ell
d\beta
dx+\varepsilon\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\beta\cdot\nabla\ell
d\beta dx+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx$
$\displaystyle=\varepsilon^{2}\int_{\Omega}\int_{\mathbb{S}^{2}}(\varphi-h)(\beta\cdot\nabla)^{2}\ell
d\beta
dx+\varepsilon^{2}\int_{\Omega}\int_{\mathbb{S}^{2}}h(\beta\cdot\nabla)^{2}\ell
d\beta dx$ $\displaystyle\quad-\varepsilon^{2}\int_{\Gamma_{+}}\beta\cdot
n\varphi\beta\cdot\nabla\ell d\beta
d\sigma_{x}+\varepsilon\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\beta\cdot\nabla\ell
d\beta dx+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx$
$\displaystyle=-\varepsilon^{3}\int_{\Omega}\int_{\mathbb{S}^{2}}(\beta\cdot\nabla)^{2}\ell\beta\cdot\nabla\varphi
d\beta
dx+\varepsilon^{2}\int_{\Omega}\int_{\mathbb{S}^{2}}\rho(\beta\cdot\nabla)^{2}\ell
d\beta dx+\varepsilon^{2}\frac{4\pi}{3}\int_{\Omega}h\Delta\ell dx$
$\displaystyle\quad-\varepsilon^{2}\int_{\Gamma_{+}}\beta\cdot
n\varphi\beta\cdot\nabla\ell d\beta
d\sigma_{x}+\varepsilon\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\beta\cdot\nabla\ell
d\beta dx+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx$
$\displaystyle=\varepsilon^{3}\int_{\Omega}\int_{\mathbb{S}^{2}}(\varphi-\langle\varphi\rangle/4\pi)(\beta\cdot\nabla)^{3}\ell
d\beta dx-\varepsilon^{3}\int_{\Gamma_{+}}(\beta\cdot\nabla)^{2}\ell\beta\cdot
n\varphi d\beta d\sigma_{x}$
$\displaystyle\quad+\varepsilon^{2}\int_{\Omega}\int_{\mathbb{S}^{2}}\rho(\beta\cdot\nabla)^{2}\ell
d\beta dx+\varepsilon^{2}\frac{4\pi}{3}\int_{\Omega}h\Delta\ell
dx-\varepsilon^{2}\int_{\Gamma_{+}}\beta\cdot n\varphi\beta\cdot\nabla\ell
d\beta d\sigma_{x}$
$\displaystyle\quad+\varepsilon\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\beta\cdot\nabla\ell
d\beta dx+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx$
$\displaystyle\geq-C\varepsilon^{3}\|\varphi-\langle\varphi\rangle/(4\pi)\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\|\ell\|_{H^{3}(\Omega)}-C\varepsilon^{3}\|\nabla^{2}\ell\|_{L^{2}(\partial\Omega)}\|\varphi\|_{L^{2}(\Gamma_{+})}$
$\displaystyle\quad-C\varepsilon^{2}\|\rho\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\|\ell\|_{H^{2}(\Omega)}+\varepsilon^{2}\frac{4\pi}{3}\int_{\Omega}h\Delta\ell
dx-C\varepsilon^{2}\|\nabla\ell\|_{L^{2}(\partial\Omega)}\|\varphi\|_{L^{2}(\Gamma_{+})}$
$\displaystyle\quad-C\varepsilon\|\rho\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\|\ell\|_{H^{1}(\Omega)}+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell
d\beta dx,$

where the fact that $\langle(\beta\cdot\nabla)^{3}\ell\rangle=0$ is used,
which is due to $\ell=\ell(x)$ not depending on $\beta$. By the trace theorem
and Sobolev embeddings, the above inequality implies

$\displaystyle\int_{\Omega}\mathcal{A}h\cdot\ell dx$
$\displaystyle\geq\frac{4\pi}{3}\varepsilon^{2}\int_{\Omega}h\Delta\ell
dx-C\varepsilon^{3}\|\varphi-\langle\varphi\rangle/(4\pi)\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\|\ell\|_{H^{3}(\Omega)}-C\varepsilon^{3}\|\ell\|_{H^{3}(\Omega)}\|\varphi\|_{L^{2}(\Gamma_{+})}$
$\displaystyle\quad-C\varepsilon^{2}\|\ell\|_{H^{2}(\Omega)}\|\varphi\|_{L^{2}(\Gamma_{+})}-C\varepsilon^{2}\|\rho\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\|\ell\|_{H^{2}(\Omega)}-C\varepsilon\|\rho\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\|\ell\|_{H^{1}(\Omega)}$
$\displaystyle\quad+\int_{\Omega}\int_{\mathbb{S}^{2}}\rho\ell d\beta dx.$
(4.16)

Let $\rho=r_{2}+r,\varphi=\phi,h=4(T^{a})^{3}g$ in (4.15), equation (4.3) can
be written as

$\displaystyle\varepsilon^{2}\Delta g+\mathcal{A}(4(T^{a})^{3}g)=r_{1}+\langle
r\rangle.$

Let $\ell$ be the solution to

$\displaystyle\Delta\ell=g,\text{ in }\Omega,$ $\displaystyle\ell=0,\text{ on
}\partial\Omega.$

We multiply the previous equation by $\ell$ and integration by parts to get

$\displaystyle\varepsilon^{2}\int_{\Omega}g^{2}dx$
$\displaystyle+\int_{\Omega}\mathcal{A}(4(T^{a})^{3}g)\cdot\ell
dx=\int_{\Omega}(r_{1}+\langle r\rangle)\ell dx.$

By (4.1), and the estimates for elliptic equations
$\|\ell\|_{H^{2}(\Omega)}\leq C\|g\|_{L^{2}(\Omega)}$,
$\|\ell\|_{H^{3}(\Omega)}\leq C\|g\|_{H^{1}(\Omega)}$, we have

$\displaystyle\int_{\Omega}\mathcal{A}(4(T^{a})^{3}g)\ell dx$
$\displaystyle\quad\geq\varepsilon^{2}\frac{4\pi}{3}\int_{\Omega}4(T^{a})^{3}g^{2}dx-C\varepsilon^{3}\|\phi-\langle\phi\rangle/(4\pi)\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\|g\|_{H^{1}(\Omega)}-C\varepsilon^{3}\|g\|_{H^{1}(\Omega)}\|\phi\|_{L^{2}(\Gamma_{+})}$
$\displaystyle\qquad-C\varepsilon^{2}\|g\|_{L^{2}(\Omega)}\|\phi\|_{L^{2}(\Gamma_{+})}-C\varepsilon(\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}+\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})})\|g\|_{L^{2}(\Omega)}+\int_{\Omega}\int_{\mathbb{S}^{2}}(r_{2}+r)\ell
d\beta dx$
$\displaystyle\geq\varepsilon^{2}\frac{4\pi}{3}\int_{\Omega}4(T^{a})^{3}g^{2}dx-C\varepsilon^{3}\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}-C\varepsilon^{3}\|g\|_{H^{1}(\Omega)}^{2}-C\varepsilon^{3}\|g\|_{H^{1}(\Omega)}^{2}-C\varepsilon^{3}\|\phi\|_{L^{2}(\Gamma_{+})}^{2}$
$\displaystyle\quad-\frac{1}{4}\varepsilon^{2}\|g\|_{L^{2}(\Omega)}^{2}-C\varepsilon^{2}\|\phi\|_{L^{2}(\Gamma_{+})}^{2}-C(\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2})-\frac{1}{4}\varepsilon^{2}\|g\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\quad+\int_{\Omega}\int_{\mathbb{S}^{2}}(r_{2}+r)\ell d\beta
dx.$

Taking it into the previous equation gives

$\displaystyle\varepsilon^{2}\|g\|_{L^{2}(\Omega)}^{2}+\varepsilon^{2}\|2(T^{a})^{3/2}g\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\quad\leq
C\varepsilon^{3}\|\phi-(4T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+C\varepsilon^{3}\|g\|_{H^{1}(\Omega)}^{2}+C\varepsilon^{2}\|\phi\|_{L^{2}(\Gamma_{+})}^{2}+C(\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2})$
$\displaystyle\qquad+\int_{\Omega}r_{1}\ell
dx-\int_{\Omega}\int_{\mathbb{S}^{2}}r_{2}\ell d\beta dx$
$\displaystyle\quad\leq
C\varepsilon^{3}\|\phi-(4T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+C\varepsilon^{3}\|g\|_{H^{1}(\Omega)}^{2}+C\varepsilon^{2}\|\phi\|_{L^{2}(\Gamma_{+})}^{2}+C(\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2})$
$\displaystyle\qquad+\frac{C}{\varepsilon^{2}}(\|r_{1}\|_{L^{2}(\Omega)}^{2}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2})+\frac{1}{4}\varepsilon^{2}\|g\|_{L^{2}(\Omega)}^{2}.$

Combining this inequality with (4.1), we obtain for $\varepsilon$ sufficiently
small ($\varepsilon\leq\min\\{1,\kappa\\}/C$),

$\displaystyle\varepsilon^{2}\kappa\|\nabla
g\|_{L^{2}(\Omega)}^{2}+\varepsilon^{2}\|2(T^{a})^{3/2}g\|_{L^{2}(\Omega)}^{2}+\varepsilon\|\phi\|_{L^{2}(\Gamma_{+})}^{2}+\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}$
$\displaystyle\leq
C\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\frac{C}{\varepsilon^{2}}(\|r_{1}\|_{L^{2}(\Omega)}^{2}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2})+\iint_{\Omega\times\mathbb{S}^{2}}\phi
r_{2}d\beta
dx-\int_{\Omega}4(T^{a})^{3}gr_{1}dx+C\|r_{1}\|_{L^{2}(\Omega)}^{2}.$

Using the inequalities

$\displaystyle\int_{\Omega}\int_{\mathbb{S}^{2}}\phi r_{2}d\beta dx$
$\displaystyle=\int_{\Omega}\int_{\mathbb{S}^{2}}(\phi-4(T^{a})^{3}g)r_{2}d\beta
dx+\int_{\Omega}\int_{\mathbb{S}^{2}}4(T^{a})^{3}gr_{2}d\beta dx$
$\displaystyle\leq\frac{1}{2}\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\frac{1}{2}\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\frac{1}{4}\varepsilon^{2}\|2(T^{a})^{3/2}g\|_{L^{2}(\Omega)}^{2}+\frac{C}{\varepsilon^{2}}\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}$

and

$\displaystyle-\int_{\Omega}4(T^{a})^{3}gr_{1}dx\leq\frac{1}{4}\varepsilon^{2}\|2(T^{a})^{3/2}g\|_{L^{2}(\Omega)}^{2}+\frac{C}{\varepsilon^{2}}\|r_{1}\|_{L^{2}(\Omega)}^{2},$

we get

$\displaystyle\varepsilon^{2}\|g\|_{H^{1}(\Omega)}^{2}+\varepsilon^{2}\|2(T^{a})^{3/2}g\|_{L^{2}(\Omega)}^{2}+\varepsilon\|\phi\|_{L^{2}(\Gamma_{+})}^{2}+\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}$
$\displaystyle\quad\leq
C\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\frac{C}{\varepsilon^{2}}(\|r_{1}\|_{L^{2}(\Omega)}^{2}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}).$

Note that

$\varepsilon^{2}\|\phi\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}\leq\varepsilon^{2}\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+\varepsilon^{2}\|4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}\leq\varepsilon^{2}\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega\times\mathbb{S}^{2})}^{2}+C\varepsilon^{2}\|g\|_{L^{2}(\Omega)}^{2},$

due to $T^{a}$ being bounded. Therefore, we arrive at the estimate (11).

_Step 3: $L^{\infty}$ estimate._ We now derive the $L^{\infty}$ estimate of
$g,\phi$. First, by the maximum principle for linear transport equation, see
for example Lemma 3.1 in [15], the following estimate holds for (4.4):

$\displaystyle\|\phi\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq\|4(T^{a})^{3}g\|_{L^{\infty}(\Omega)}+\|r_{2}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}+\|r\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}.$

Since $\|T^{a}\|_{L^{\infty}(\Omega)}$ is bounded,

$\displaystyle\|\phi\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq
C\|g\|_{L^{\infty}(\Omega)}+\|r_{2}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}+\|r\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}.$
(4.17)

We now give the $L^{\infty}$ estimate of $g$. Equation (4.3) can be written as

$\displaystyle\Delta g=f,$ (4.18)

with $f=(-\langle\phi-4(T^{a})^{3}g\rangle+r_{1}+\langle
r\rangle)/\varepsilon^{2}$. According to the elliptic regularity, we have

$\displaystyle\|g\|_{L^{\infty}(\Omega)}\leq
C(\|g\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}).$

combining the above inequality with (11), we obtain

$\displaystyle\|g\|_{L^{\infty}(\Omega)}\leq
C(\|g\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)})$
$\displaystyle\quad\leq\frac{C}{\varepsilon}\|r\|_{L^{2}(\Omega\times\mathbb{S}^{2}}+\frac{C}{\varepsilon^{2}}(\|r_{1}\|_{L^{2}(\Omega)}+\|r_{2}\|_{L^{2}(\Omega\times\mathbb{S})})+\frac{C}{\varepsilon^{2}}\left(\|\phi-4(T^{a})^{3}g\|_{L^{2}(\Omega)}+\|r_{1}\|_{L^{2}(\Omega)}+\|r\|_{L^{2}(\Omega\times\mathbb{S})}\right).$

Adding the above inequality with (4.17) and using (11) on the last term of the
above inequality, we arrive to the estimate (11), which finishes the proof of
Lemma 11. ∎

######  Remark 12.

We follow a similar procedure as [15]. Here the estimate on the elliptic
equation (4.3) is needed, whereas in [15] only the estimate on the transport
equation is involved. In particular, we derive the $L^{2}$ and $L^{\infty}$
energy estimates. Here we only give the estimates on the solutions. However,
the existence of solutions can be obtained by using classical existence theory
of linear transport equations and elliptic equations.

###  4.2. Nonlinear system

We now show the existence and uniqueness of solutions to system (1.1)-(1.2)
around the constructed composite approximate solution $(T^{a},\psi^{a})$ and
finish the proof of Theorem 1.

###### Proof of Theorem 1.

The existence and uniqueness of solutions to system (1.1)-(1.2) can be
obtained by showing the existence for system (4.1)-(4.2). To achieve this, we
construct a sequence of functions $\\{g^{n},\phi^{n}\\}$ by

$\displaystyle g^{0}(x)=0,\quad\phi^{0}(x,\beta)=0,$

and for $n\geq 1$,

$\displaystyle\varepsilon^{2}\Delta
g^{n}+\langle\phi^{n}-4(T^{a})^{3}g^{n}\rangle=-\mathcal{R}_{1}+4\pi(6(T^{a})^{2}(g^{n-1})^{2}+4(T^{a})(g^{n-1})^{3}+(g^{n-1})^{4}),$
(4.19)
$\displaystyle\varepsilon\beta\cdot\nabla\phi^{n}+(\phi^{n}-4(T^{a})^{3}g^{n})=-\mathcal{R}_{2}+6(T^{a})^{2}(g^{n-1})^{2}+4(T^{a})(g^{n-1})^{3}+(g^{n-1})^{4},$
(4.20)

with boundary conditions

$\displaystyle g^{n}(x)=0,\text{ for }x\in\partial\Omega,$
$\displaystyle\phi^{n}(x,\beta)=\phi_{b}(x,\beta),\text{ for
}(x,\beta)\in\Gamma_{-}.$

The above relation defines a mapping $\mathcal{T}$ with
$(g^{n},\phi^{n})=\mathcal{T}((g^{n-1},\phi^{n-1}))$. Note that the above
system is the same with (1.24)-(1.25) in the introduction with
$T^{n}=g^{n}+T^{a}$, $\psi^{n}=\phi^{n}+\psi^{a}$.

Let $Y=L^{2}(\Omega)\cap L^{\infty}(\Omega)$ and
$W=L^{2}(\Omega\times\mathbb{S}^{2})\cap
L^{\infty}(\Omega\times\mathbb{S}^{2})$. We consider the solution in the
function space

$\displaystyle\mathcal{O}_{s}:=\\{(g,\phi)\in Y\times
W:\|g\|_{L^{2}(\Omega)}\leq C\varepsilon^{s},\|g\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{s},\|\phi\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\leq
C\varepsilon^{s},\|\phi\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq
C\varepsilon^{s}\\},$

with $s>0$ is a constant to be chosen later.

First we show $\mathcal{T}$ maps the space $\mathcal{O}_{s}$ onto itself.
Assume the residuals satisfy $\|\mathcal{R}_{1}\|_{L^{2}(\Omega)}$,
$\|\mathcal{R}_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}$,
$\|\mathcal{R}_{1}\|_{L^{\infty}(\Omega)}$,
$\|\mathcal{R}_{2}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq\varepsilon^{p}$
for some constant $p>0$. Next, we show if
$(g^{n-1},\psi^{n-1})\in\mathcal{O}_{s}$, then
$(g^{n},\psi^{n})\in\mathcal{O}_{s}$. By (11) with
$r_{1}=-\mathcal{R}_{1},r_{2}=-\mathcal{R}_{2}$ and
$r=6(T^{a})^{2}(g^{n-1})^{2}+4T^{a}(g^{n-1})^{3}+(g^{n-1})^{4}$, the following
estimate holds:

$\displaystyle\varepsilon\|\phi^{n}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}+\varepsilon\|g^{n}\|_{H^{1}(\Omega)}+\sqrt{\varepsilon}\|\phi^{n}\|_{L^{2}(\Gamma_{+})}+\|\phi^{n}-4(T^{a})^{3}g^{n}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}$
$\displaystyle\quad\leq\frac{C}{\varepsilon}\left(\|\mathcal{R}_{1}\|_{L^{2}(\Omega)}+\|\mathcal{R}_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\right)+C\left(\|6(T^{a})^{2}(g^{n-1})^{2}+4T^{a}(g^{n-1})^{3}+(g^{n-1})^{4}\|_{L^{2}(\Omega)}\right)$
$\displaystyle\quad\leq
C\varepsilon^{p-1}+C(\|g^{n-1}\|_{L^{4}}^{2}+\|g^{n-1}\|_{L^{8}}^{4})$
$\displaystyle\quad\leq
C\varepsilon^{p-1}+C(\|g^{n-1}\|_{L^{\infty}(\Omega)}\|g^{n-1}\|_{L^{2}(\Omega)}+\|g^{n-1}\|_{L^{\infty}}^{3}\|g^{n-1}\|_{L^{2}(\Omega)})$
$\displaystyle\quad\leq C\varepsilon^{p-1}+C\varepsilon^{2s}.$

Assume $p-2\geq s$ and $2s-1\geq s$, i.e. $p\geq s+2$ and $s\geq 1$, the above
inequality implies

$\displaystyle\|g^{n-1}\|_{H^{1}(\Omega)},\|\phi^{n}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\leq
C\varepsilon^{p-2}+C\varepsilon^{2s-1}\leq C\varepsilon^{s}.$

Moreover, by (11),

$\displaystyle\|\phi\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}+\|g\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq\frac{C}{\varepsilon^{2}}\left(\|6(T^{a})^{2}(g^{n-1})^{2}+4T^{a}(g^{n-1})^{3}+(g^{n-1})^{4}\|_{L^{2}(\Omega)}\right)$
$\displaystyle\quad+\frac{1}{\varepsilon^{3}}(\|\mathcal{R}_{1}\|_{L^{2}(\Omega)}+\|\mathcal{R}_{2}\|_{L^{2}(\Omega\times\mathbb{S}^{2})})+\|\mathcal{R}_{2}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}$
$\displaystyle\quad+\|6(T^{a})^{2}(g^{n-1})^{2}+4T^{a}(g^{n-1})^{3}+(g^{n-1})^{4}\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq
C\varepsilon^{2s-2}+\varepsilon^{p-3}+\varepsilon^{p}+\varepsilon^{2s}$
$\displaystyle\leq C\varepsilon^{2s-2}+\varepsilon^{p-3}.$

Assume $p-3\geq s$ and $2s-2\geq s$, i.e. $p\geq s+3$ and $s\geq 2$, the above
inequality implies

$\displaystyle\|g^{n}\|_{L^{\infty}(\Omega)},\|\phi^{n}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{2s-2}+\varepsilon^{p-3}\leq C\varepsilon^{s}.$

Thus we obtain that $(g^{n},\phi^{n})\in\mathcal{O}_{s}$ and therefore
$\mathcal{T}$ maps $\mathcal{O}_{s}$ onto itself.

Next we show the map $\mathcal{T}$ is a contraction mapping. Let
$h^{n}=g^{n}-g^{n-1},\varphi^{n}=\phi^{n}-\phi^{n-1}$, then they satisfy

$\displaystyle\varepsilon^{2}\Delta
h^{n}+\langle\varphi^{n}-4(T^{a})^{3}h^{n}\rangle=4\pi f_{n},$
$\displaystyle\varepsilon\beta\cdot\nabla\varphi^{n}+\varphi^{n}-4(T^{a})^{3}h^{n}=f_{n},$

where

$\displaystyle f_{n}$
$\displaystyle=6(T^{a})^{2}(g^{n-1})^{2}+4(T^{a})(g^{n-1})^{3}+(g^{n-1})^{4}-(6(T^{a})^{2}(g^{n-2})^{2}+4(T^{a})(g^{n-2})^{3}+(g^{n-2})^{4})$
$\displaystyle=6(T^{a})^{2}(g^{n-1}+g^{n-2})h^{n-1}+4T^{a}h^{n-1}((g^{n-1})^{2}-g^{n-1}g^{n-2}+(g^{n-2})^{2})$
$\displaystyle\quad+h^{n-1}(g^{n-1}+g^{n-2})((g^{n-1})^{2}+(g^{n-2})^{2}).$

Using (11) with $r_{1}=r_{2}=0$, $r=f_{n}$, we obtain

$\displaystyle\varepsilon\|\varphi^{n}\|_{L^{2}(\Omega\times\mathbb{S})}+\varepsilon\|h^{n}\|_{H^{1}(\Omega)}+\sqrt{\varepsilon}\|\varphi^{n}\|_{L^{2}(\Gamma_{+})}+\|\varphi^{n}-4(T^{a})^{3}h^{n}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}$
$\displaystyle\quad\leq\|f_{n}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}\leq
C\varepsilon^{s}\|h^{n-1}\|_{L^{2}(\Omega)},$

hence

$\displaystyle\|h^{n}\|_{H^{1}(\Omega)}+\|\varphi^{n}\|_{L^{2}(\Omega\times\mathbb{S})}\leq
C\varepsilon^{s-1}\|h^{n-1}\|_{L^{2}(\Omega)}.$

Using (11) with $r_{1}=r_{2}=0$, $r=f_{n}$, we obtain

$\displaystyle\|\varphi^{n}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}+\|h^{n}\|_{L^{\infty}(\Omega)}$
$\displaystyle\quad\leq\frac{C}{\varepsilon^{2}}\|f_{n}\|_{L^{2}(\Omega\times\mathbb{S}^{2})}+C\|f_{n}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{s-2}\|h^{n-1}\|_{L^{2}(\Omega)}+C\varepsilon^{s}\|h^{n-1}\|_{L^{\infty}(\Omega)}.$

Assume $s\geq 3$, then $C\varepsilon^{s-2},C\varepsilon^{s}<1$ for
$\varepsilon$ sufficiently small, the above two inequalities imply

$\displaystyle\|h^{n}\|_{Y}+\|\varphi^{n}\|_{W}\leq
C_{1}(\|h^{n-1}\|_{Y}+\|\varphi^{n-1}\|_{W})$

for some constant $0<C_{1}<1$. Therefore, for $p\geq s+3$ and $s\geq 3$,
$\mathcal{T}$ is a contraction mapping. By the Banach fixed point theorem,
there exists a unique fixed point $(g,\phi)$ such that
$(g,\phi)=\mathcal{T}((g,\phi))$. Therefore there exists a unique solution to
(4.1)-(4.2) in $\mathcal{O}_{s}$.

Taking $s=3$ and $p=6$, we can conclude that

$\displaystyle\|g\|_{Y}+\|\varphi\|_{W}\leq C\varepsilon^{3}.$

Note that in order to obain
$\mathcal{R}_{1},\mathcal{R}_{2}=O(\varepsilon^{6})$, we need to take $n=5$ in
the expansion and by Lemma 9, $\|\mathcal{R}_{1}\|_{L^{2}\cap
L^{\infty}(\Omega)},~{}\|\mathcal{R}_{2}\|_{L^{2}\cap
L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq C\varepsilon^{6}$. We have

$\displaystyle\left\|T^{\varepsilon}-\sum_{k=0}^{5}\varepsilon^{k}T_{k}-\sum_{k=0}^{5}\varepsilon^{k}\bar{T}_{k}\right\|_{L^{2}\cap
L^{\infty}(\Omega)}\leq C\varepsilon^{3},$
$\displaystyle\left\|\psi^{\varepsilon}-\sum_{k=0}^{5}\varepsilon^{k}\psi_{k}-\sum_{k=0}^{5}\varepsilon^{k}\bar{\psi}_{k}\right\|_{L^{2}\cap
L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq C\varepsilon^{3}.$

Therefore, we get

$\displaystyle\|T^{\varepsilon}-T_{0}-\bar{T}_{0}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon,$
$\displaystyle\|\psi^{\varepsilon}-T_{0}^{4}-\bar{\psi}_{0}\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq
C\varepsilon,$

which are (1.19) and finish the proof. ∎

##  Appendix A Existence of the steady state radiative transfer system

Next we prove the existence for the steady state radiative transfer system
(1.1)-(1.2) with boundary conditions (1.3)-(1.4).

######  Theorem 13.

Assume $\gamma_{1}\leq T_{b}\leq\gamma_{2}$ and
$\gamma_{1}^{4}\leq\psi_{b}\leq\gamma_{2}^{4}$ for some constants
$0\leq\gamma_{1}\leq\gamma_{2}$. There there exists a weak solution
$(T^{\varepsilon},\psi^{\varepsilon})\in L^{\infty}(\Omega)\times
L^{\infty}(\Omega\times\mathbb{S}^{2})$ to (1.1)-(1.2).

###### Proof.

We show the existence by using the fixed-point theorem. Let
$\mathcal{A}:=\\{T^{\varepsilon}\in L^{\infty}(\Omega):\gamma_{1}\leq
T^{\varepsilon}\leq\gamma_{2}\\}$. We define the operator
$\mathcal{F}:\mathcal{A}\to\mathcal{A}$ with $\theta=\mathcal{F}T$ by solving

$\displaystyle\varepsilon\beta\cdot\nabla\psi+\psi=T^{4},$ (A.1)
$\displaystyle\psi^{\varepsilon}(x,\beta)=\psi_{b}(x,\beta),\quad\text{for
}(x,\beta)\in\Gamma_{-},$

and

$\displaystyle\varepsilon^{2}\Delta\theta-4\pi^{2}\theta^{4}=-\langle\psi\rangle,$
(A.2) $\displaystyle\theta(x)=T_{b}(x),\quad\text{for }x\in\partial\Omega.$

Next we show if $\gamma_{1}\leq T\leq\gamma_{2}$, then
$\gamma_{1}\leq\theta\leq\gamma_{2}$. First the maximum principle for the
transport equation implies

$\displaystyle\gamma_{1}^{4}\leq\psi\leq\gamma_{2}^{4}.$ (A.3)

The maximum principle for equation (A.2) also implies
$\gamma_{1}\leq\theta\leq\gamma_{2}$. Suppose $\theta$ reaches its maximum at
$x_{M}\in\Omega$, then if $x_{M}\in\partial\Omega$,
$\theta(X_{M})=T_{b}\leq\gamma_{2}$ and thus
$\theta(x)\leq\theta(x_{M})\leq\gamma_{2}$ for any $x\in\Omega$. Otherwise if
$x_{M}$ is an interior point, then $\Delta\theta(x_{M})\leq 0$, and so

$\displaystyle
4\pi^{2}\theta^{4}(x_{M})\leq\langle\psi(x_{M},\cdot)\rangle\leq
4\pi^{2}\gamma_{2}^{4},$ (A.4)

hence $\theta(x)\leq\theta(x_{M})\leq\gamma_{2}$. Using a similar
contradiction argument, $\theta(x)\geq\gamma_{1}$ can be shown.

Since $\mathcal{F}$ maps $\mathcal{A}$ to itself and $\mathcal{A}$ is a convex
compact subset of the Banach space $L^{\infty}$. Hence by Schauder’s fixed
point theorem, there exists a fixed point
$(T^{\varepsilon},\psi^{\varepsilon})$ of $\mathcal{F}$. Since this fixed
point satisfies (A.1) and (A.2), hence $(T^{\varepsilon},\psi^{\varepsilon})$
is a solution to (1.1)-(1.2) with boundary conditions (1.3)-(1.4).

∎

######  Remark 14.

Unlike the time dependent case, where the uniqueness can be shown by showing
$\mathcal{F}$ is a contraction mapping (with time step small), we are not able
to show $\mathcal{F}$ in the above proof is a contraction mapping and thus
uniqueness is not obtained here.

## Acknowledgement

The work of N.M. is supported by NSF grant DMS-1716466 and by Tamkeen under
the NYU Abu Dhabi Research Institute grant of the center SITE. The work of M.G
is supported by Tamkeen under the NYU Abu Dhabi Research Institute grant of
the center SITE.

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