First Order Hyperbolic Boundary Value Problems with a Large Oscillatory Zero
Order Term

Zhaodh, Alvis

University of North Carolina at Chapel Hill

Senior Honors Thesis (Department of Mathematics)

May 4th, 2021

Approved by Williams, Mark (Thesis Advisor) Taylor, Michael (Reader) Metcalfe,
Jason (Reader)

This project was supported by the Wilson Family Honors Excellence Fund
administered by Honors Carolina.

###### Abstract.

We study the weakly stable hyperbolic boundary value problem with a large zero
order oscillatory coefficient. This problem is related to linearized problems
in the study of Mach stem and vortex sheets. We wish to establish a uniform
energy estimate with respect to $\epsilon$, which is needed in mentioned
applications and justification of geometric optics solutions, but the zero
order oscillatory term gives rise to great obstacles. In this paper we obtain
positive results in the small/medium frequency region by adapting the approach
of [Wil20a] to a more general situation than the one treated there. We also
show that it is possible to construct high order approximate solutions by the
method of geometric optics for those systems without any restriction on
frequencies.

###### Contents

  1. 1 Introduction
    1. 1.1 Singular System and Estimations
    2. 1.2 Main steps and difficulties
  2. 2 Definitions and Assumptions
  3. 3 Geometric Optics Solutions for The Hyperbolic System when ULC Holds
    1. 3.1 Introduction
    2. 3.2 Tools for constructing approximate solutions
    3. 3.3 Profile equations
    4. 3.4 Determining the solutions
    5. 3.5 Justifying WKB solutions
  4. 4 Estimations for Weakly Stable Hyperbolic System
    1. 4.1 Extensions to $\Gamma_{\delta}$ and then to $\Xi$
    2. 4.2 Tools for Estimates
    3. 4.3 Analysis of Small/Medium Frequency Region
    4. 4.4 Uniform Estimation Theorem
    5. 4.5 Application of the Main Estimate: an Existence Theorem
  5. 5 Geometric Optics Solutions for Weakly Stable Hyperbolic System
    1. 5.1 Profile Equations
    2. 5.2 Determining Solutions
    3. 5.3 Analysis of the boundary equations
  6. 6 Bibliography

##  1\. Introduction

Consider the following linear hyperbolic system on
$\Omega={\mathbb{R}}\times\\{(x_{1},x_{2}):x_{2}\geq 0\\}$:

(1.1) $\displaystyle
L(\partial)u+\mathcal{D}\left(\frac{\phi_{in}}{\epsilon}\right)u:=\partial_{t}u+B_{1}\partial_{x_{1}}u+B_{2}\partial_{x_{2}}u+\mathcal{D}\left(\frac{\phi_{in}}{\epsilon}\right)u=F(t,x,\frac{\phi_{0}}{\epsilon})\text{
in }x_{2}>0$ $\displaystyle Bu:=\epsilon
G\left(t,x_{1},\frac{\phi_{0}}{\epsilon}\right)\text{ on }x_{2}=0$
$\displaystyle u=0\text{ in }t<0$

where all $B_{j}$’s are $N\times N$ matrices, $B_{2}$ is invertible,
$\mathcal{D}(\theta_{in}),F(t,x,\theta),G(t,x,\theta)$ is of period $2\pi$ in
$\theta_{in}$, and $\theta$ receptively. We note that the oscillatory
coefficient $\mathcal{D}\left(\frac{\phi_{in}}{\epsilon}\right)$ is large,
$O(1)$, compared to the small wave length $\epsilon\in(0,\epsilon_{0}]$. The
problem is weakly stable, that is $(L(\partial),B)$ fails to satisfy the
uniform Lopatinski condition (Definition 2.6) in a specific way (Assumption
2.14). The boundary matrix $B$ is a constant $p\times N$ matrix with
appropriate rank $p$. The boundary phase is
$\phi_{0}(t,x_{1})=\beta_{l}\cdot(t,x_{1})$, where
$\beta_{l}=(\sigma_{l},\eta_{l})\in{\mathbb{R}}^{2}\setminus 0$ is a direction
where uniform Lopatinski condition fails. The interior phase:

(1.2) $\phi_{in}(t,x)=\phi_{0}(t,x_{1})+\omega_{N}(\beta_{l})x_{2}$

is incoming (Definition 2.8).

As a warm-up we first establish some techniques in the case (section 3) where
uniform Lopatinski condition defined in 2.6 holds. For simplicity, we take the
oscillatory coefficient to be $e^{i\theta_{3}}M$ in that section, where $M$ is
a constant $N\times N$ matrix. We construct an approximate WKB solutions (or
geometric optics solutions) (3.19) and use an existing estimate in [Kre70] to
justify this solution by showing that it is close to the exact solution in
$L^{\infty}$.

The main goal of this paper is to study the situation where uniform Lopatinski
condition fails in a certain way (2.14) (we say the problem is weakly stable),
the oscillatory coefficient has both positive and negative Fourier spectrum,
and there are $p\geq 2$ incoming modes ($\phi_{N}-p+1,...,\phi_{N}$) as in
definition 2.8. In this case, methods that have successfully established a
uniform estimate for other hyperbolic systems fails to a yield uniform
estimate for (1.1). Here we refer to methods developed for (a) Problems where
ULC holds [Kre70], [CGW11]; (b)weakly stable problems with non-oscillatory
coefficients [Cou04], [Cou05], [CS04]; and (c)weakly stable problems where
oscillatory coefficient is small, $O(\epsilon)$ [CGW14].

To further see the difficulties of (1.1), we note that when ULC fails, methods
established in previous papers mentioned above give an estimate to the system
$(L(\partial),B)$ (no oscillatory term) in the form:

(1.3) $|u|_{L^{2}}\leq C_{T}(|L(\partial)u|_{H^{1}}+<Bu>_{H^{1}})$

If the oscillatory term is $\epsilon\mathcal{D}$, [CG10] shows that (1.3)
still applies with a constant $C_{T}$ independent of $\epsilon\in(0,1]$ when
$L(\partial)$ is replaced by
$L(\partial)+\epsilon\mathcal{D}(\phi_{in}/\epsilon)$.In (1.1) the oscillatory
term is large (not uniformly Lipschitzean with respect to $\epsilon\in(0,1]$),
so the argument of [CG10] breaks down and fails to establish an estimate like
(1.3) when $L(\partial)$ is replaced by
$L(\partial)+\mathcal{D}(\phi_{in}/\epsilon)$. Thus, new methods are needed to
obtain estimates for (1.1).

A different method established in [Wil20a] gives an energy estimate for (1.1),
which works if either: (1) the oscillatory term has only positive Fourier
spectrum; or (2) the oscillatory term has both positive and negative Fourier
spectrum and there is only one incoming phase. In particular, theorem 2.12 in
[Wil20a] can’t be applied to the system (1.1) when there are at least two
incoming phases.

This paper mainly deals with the case when (1.1) satisfies the following
conditions: (1). The oscillatory term has both positive and negative Fourier
spectrum; (2). There are at least two incoming phases,; (3). There is only one
direction $\beta_{l}$ where ULC fails; (4). There is no resonance as defined
in definition 2.11. The first main result of this thesis is the energy
estimate of theorem 4.19 for systems of the form (1.1) satisfying these four
conditions. This estimate involves a restriction to “the small/medium
frequency region” that we clarify later. The second main result is the
construction of high order approximate solutions for a particular system like
(1.1), namely (5.1) which satisfies these four conditions. This result is
stated in Theorem 5.1. This theorem is interesting because it shows that high
order approximate solutions can be constructed even in situations where we
don’t have an unrestricted energy estimate. However, due to the lack of
uniform existence results for the system (1.1) and (5.1), we cannot justify
the geometric optics solutions.

###  1.1. Singular System and Estimations

We here establish the basic setup for studying the problem. Steps between
(1.4) and (1.11) is carried out by Williams [Wil20a]. We first rewrite (1.1)
by applying $B_{2}^{-1}$ to both sides of the first equation, for a different
$F$ we have:

(1.4)
$\displaystyle\begin{split}&D_{x_{2}}u+A_{0}D_{t}u+A_{1}D_{x_{1}}u-iB_{2}^{-1}\mathcal{D}\left(\frac{\phi_{in}}{\epsilon}\right)u=F(t,x,\frac{\phi_{0}}{\epsilon})\\\
&Bu=G(t,x_{1},\frac{\phi_{0}}{\epsilon})\text{ on }x_{2}=0\\\ &u=0\text{ in
}t<0,\end{split}$

where $D_{x_{i}}=\frac{1}{i}\partial_{x_{i}}$, $A_{0}=B_{2}^{-1}$,
$A_{1}=B_{2}^{-1}B_{1}$. We look for solutions of the form
$u(t,x)=U(t,x,\frac{\phi_{0}}{\epsilon})$ for (1.4), where $U(t,x,\theta)$ is
periodic in $\theta$. This benefits us by providing bounded sobolev norms of
the right hand side. Plugging $U(t,x,\theta)$ into (1.4) yields the
corresponding singular system:

(1.5)
$\displaystyle\begin{split}&D_{x_{2}}U+A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta})U+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta})U-iB_{2}^{-1}{\mathcal{D}}\left(\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+\theta\right)U=F(t,x,\theta)\\\
&BU=G(t,x_{1},\theta)\text{ on }x_{2}=0\\\ &U=0\text{ in }t<0.\end{split}$

We observe that a solution of the singular system immediately yields a
solution to the original system.

It is shown in [Wil20a] that we can reduce the first equation of (1.5) to:

(1.6) $\displaystyle
D_{x_{2}}U+A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta})U+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta})U-i\left(\sum_{r\in{\mathbb{Z}}\setminus
0}\alpha_{r}e^{i\left(r\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+r\theta\right)}\right)B_{2}^{-1}MU=F(t,x,\theta).$

where $M$ is any constant $N\times N$ matrix.

We want to study the Laplace-Fourier transform in $(t,x_{1},\theta)$ of the
singular system (1.5), with reduction (1.6). We can write

(1.7) $\displaystyle U(t,x,\theta)$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}{U_{k}}(t,x)e^{ik\theta},\quad
F(t,x,\theta)=\sum_{k\in{\mathbb{Z}}}{F_{k}}(t,x)e^{ik\theta},\quad
G(t,x_{1},\theta)=\sum_{k\in{\mathbb{Z}}}{G_{k}}(t,x_{1})e^{ik\theta}.$

Define $\zeta$, the dual variable of $(t,x_{1})$:

(1.8) $\displaystyle\zeta:=(\tau,\eta):=(\sigma-i\gamma,\eta),\text{ where
}(\sigma,\eta)\in{\mathbb{R}}^{2},\;\gamma\geq 0.$

Let $V_{k}(x_{2},\zeta):=\widehat{U_{k}}(\zeta,x_{2})$, the Laplace-Fourier
transform of $U_{k}$. We define:

(1.9) $\displaystyle X_{k}:=\zeta+\frac{k\beta_{l}}{\epsilon}\text{ and
}{\mathcal{A}}(\zeta)=-(A_{0}\tau+A_{1}\eta),$

Using above definitions, equations (1.5) and (1.6), we have, for each $k$, a
corersponding singular system for $V_{k}$:

(1.10)
$\displaystyle\begin{split}&D_{x_{2}}V_{k}-\mathcal{A}(X_{k})V_{k}=i\sum_{r\in{\mathbb{Z}}\setminus
0}\alpha_{r}e^{ir\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}}B_{2}^{-1}MV_{k-r}+\widehat{F_{k}}(x_{2},\zeta)\\\
&BV_{k}=\widehat{G_{k}}(\zeta)\text{ on }x_{2}=0.\end{split}$

With the help of proposition 4.11, we obtain an iteration estimate 111The name
iteration estimate goes back to [Wil20a], but we don’t iterate this estimate
here. (proposition 4.17) of the following form. For $\gamma\geq\gamma_{0}>0$,
we have:

(1.11) $||\chi V_{k}||\leq\frac{C}{\gamma}\sum_{r\in{\mathbb{Z}}\setminus
0}\sum_{t\in{\mathbb{Z}}}||\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)\chi
V_{k-r-t}||+\frac{C}{\gamma^{2}}\left|\chi\widehat{F_{k}}|X_{k}|\right|_{L^{2}}+\frac{C}{\gamma^{3/2}}\left|\chi\widehat{G_{k}}|X_{k}|\right|_{L^{2}}$

where $\chi$ is the characteristic function for $|\zeta|$ small compared to
$\epsilon^{-1}$, namely the region $|\zeta|\leq C\epsilon^{\alpha-1}$ for a
fixed $0<\alpha<1$, $||\cdot||$ is a modified $L^{2}(x_{2},\sigma,\eta)$ norm,
the constants $C$ and $\gamma_{0}$ is independent of $(\epsilon,\zeta,k)$, and
the $\alpha_{r}$ are as in (1.10). The global amplification factor,
${\mathbb{D}}$, is defined in 4.7. Estimate (1.11) is an important step toward
proving the uniform estimate of $U(t,x,\theta)$ in (1.5) (theorem 4.19):

(1.12)
$\displaystyle|\chi_{D}U^{\gamma}|_{L^{2}(t,x,\theta)}+\left|\frac{\chi_{D}U^{\gamma}(0)}{\sqrt{\gamma}}\right|_{L^{2}(t,x_{1},\theta)}\leq
K\left[\frac{1}{\gamma^{2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{F_{k}}\right|^{2}_{L^{2}(x_{2},\sigma,\eta)})^{1/2}+\frac{1}{\gamma^{3/2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{G_{k}}\right|^{2}_{L^{2}(\sigma,\eta)})^{1/2}\right]$

where $\chi_{D}$ is the Fourier multiplier corresponding to $\chi$.

######  Remark 1.1.

  1. (1)

Techniques of [CGW14] can be used to prove an estimate of the form (1.12)
where $\gamma_{0}=\gamma_{0}(\epsilon)\to\infty$ as $\epsilon\to 0$, but such
estimates can’t be used to justify WKB expansions. The main result of this
paper is a step toward obtaining an estimate with $\gamma_{0}$ independent of
$\epsilon$. The analysis also indicates some obstacles to obtaining estimates
with no restriction on $\zeta$.

  2. (2)

Since we don’t have uniform existence results for the system (1.5), estimates
(1.11), (1.12) are stated as a priori estimates.

  3. (3)

It is interesting that it is still possible to construct arbitrarily high
order WKB solutions in this situation. However, to justify those solutions
(showing the WKB solutions are close to the exact solutions), we need the
estimate (1.12) without the $\chi$ and the existence of exact solutions.

  4. (4)

We can apply estimate (1.12) to get an “restricted” existence result for a
particular singular system that is different from (1.5) by a classical duality
argument based on energy estimate for the corresponding “backward” or adjoint
boundary value problem together with the Riesz representation theorem. This is
discussed informally in section 4.5.

######  Notation 1.2.

Consider functions $f(x),g(x)$ where $x\in D$ for some domain $D$.

(1.13) $\displaystyle f\sim 1$ $\displaystyle\iff C_{1}\leq|f(x)|\leq
C_{2}\text{ on }D$ (1.14) $\displaystyle f\lesssim g$
$\displaystyle\iff|f(x)|\leq C|g(x)|\text{ on }D$

for constants $C,C_{1},\ C_{2}$ independent of $x$.

###  1.2. Main steps and difficulties

Here we briefly discuss the obstacles created by the failure of the uniform
Lopatinski condition and the presence of two incoming phases. To analyze the
singular system (1.10), we diagonalize the equation in the neighbourhood of
$\beta_{l}$ where ULC holds and write down explicitly the integral formula of
solutions ((4.20), (4.21)), which is of the form:

(1.15) $\displaystyle w^{+}_{k}(x_{2},\zeta)=\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{\infty}_{x_{2}}e^{i\xi_{+}(\epsilon,k)(x_{2}-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}[a(\epsilon,k,k-r)w^{+}_{k-r}(s,\zeta)+b(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)]ds,$
(1.16)
$\displaystyle\begin{split}&w^{-}_{k}(x_{2},\zeta)=-\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{x_{2}}_{0}e^{i\xi_{-}(\epsilon,k)(x_{2}-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}[c(\epsilon,k,k-r)w^{+}_{k-r}(s,\zeta)+d(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)]ds-\\\
&e^{i\xi_{-}(\epsilon,k)x_{2}}[Br_{-}(\epsilon,k)]^{-1}Br_{+}(\epsilon,k)\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{\infty}_{0}e^{i\xi_{+}(\epsilon,k)(-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}[a(\epsilon,k,k-r)w^{+}_{k-r}(s,\zeta)+\\\
&\qquad\qquad\qquad
b(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)]ds+e^{i\xi_{-}(\epsilon,k)x_{2}}[Br_{-}(\epsilon,k)]^{-1}\hat{G}_{k}(\zeta).\end{split}$

where $V_{k}=(w_{k}^{+}\ w_{k}^{-})$, $a,\ b,\ c,\ d$ are matrices that $\sim
1$;
$\xi_{+}(\zeta)=\text{diag}\\{{\omega_{1}(\zeta),...,\omega_{N-p}}(\zeta)\\}$,
$\xi_{-}(\zeta)=\text{diag}\\{{\omega_{N-p+1}(\zeta),...,\omega_{N}}(\zeta)\\}$,
where $\omega_{j}$ are the eigenvalues of the matrix $\mathcal{A}(\zeta)$;
$r_{+}=(r_{1},...r_{N-p}),\ r_{-}=(r_{N-p+1},...,r_{N})$ where $r_{j}$ are the
normalized eigenvector of the matrix $\mathcal{A}(\zeta)$. The indexes
$\mathcal{O}=\\{1,...,N-p\\}$, $\mathcal{I}=\\{N-p+1,...,N\\}$ corresponds to
outgoing, incoming phases respectively.

The failure of uniform Lopatinski condition as specified in assumption 2.14
implies that the Lopatinski determinant $\Delta(\beta_{l})=0$ (definition
2.13). For functions $f$ depending on $\zeta$, we use notation $f(\epsilon,k)$
to denote $f(X_{k})$ to highlight the dependence on $\epsilon,k$ (notation
4.2). Lemma 4.3 shows that:

(1.17)
$|Br_{-}(\epsilon,k)|^{-1}\lesssim|\Delta(\epsilon,k)|^{-1}\sim\frac{|X_{k}|}{|\tau-
c_{+}(\beta_{l})\eta|}$

which is of size $\geq\frac{1}{\epsilon}$ for $\zeta$ near $\beta_{l}$. This
yields an obvious obstacle when estimating the integral equations. To estimate
this term, [Wil20a] proposed the usage of $E_{i,j}(\epsilon,k,k-r)$
(definition 4.5):

(1.18)
$E_{i,j}(\epsilon,k,k-r;\beta_{l}):=\omega_{i}(\epsilon,k;\beta_{l})-\frac{r\omega_{N}(\beta_{l})}{\epsilon}-\omega_{j}(\epsilon,k-r;\beta_{l}),\text{
where }i\in\mathcal{O},j\in\mathcal{I}$

Notice that we can rewrite a component of the last line of the integral
equations (1.16) as $e^{-iE_{i,j}(\epsilon,k,k-r)s}$, provided we replace
$w_{k-r}^{-}$ by $e^{-i\xi_{-}(\epsilon,k-r)x_{2}}w_{k-r}^{-}$. An integration
by part argument using

(1.19) $e^{-iE_{i,j}s}=-\frac{1}{i}E_{i,j}^{-1}\frac{d}{ds}e^{-iE_{i,j}s}$

gives rise to the term $E_{i,j}^{-1}$. This core step allows us to control
$|\Delta(\epsilon,k)|^{-1}$ with $|E_{i,j}^{-1}|$ on regions where
$E_{i,j}(\epsilon,k,k-r)$ is large. We’ll see that the existence of more than
one incoming phases produces more difficulties when establishing a useful
lower bound on some regions. Actually we were able to prove such lower bounds
only when $|\zeta|\lesssim\epsilon^{\alpha-1}$ for some $0<\alpha<1$
(proposition 4.11).

The failure of the uniform Lopatinski condition also gives rise to more
difficulties in constructing geometric optics solutions. Since the boundary
matrix $B$ in (1.4) fails to be an isomorphism between
${\mathbb{E}}^{s}(\zeta)$ and ${\mathbb{C}}^{p}$ where
${\mathbb{E}}^{s}(\zeta)$ is the stable subspace of
$i\mathcal{A}(\zeta)$(definition 2.3), we need further analysis on the
boundary equations using the approach of [Wil20a], which is presented in
section (5.3).

##  2\. Definitions and Assumptions

In this section we establish relevant definitions, assumptions, and results
quoted directly from [Wil20a].

######  Assumption 2.1.

[Wil20a] (Strict Hyperbolicity) The $B_{j}$ are real matrices, and there exist
real valued functions $\lambda_{j}(\eta,\xi),j=1,...,N$ that are analytic on
${\mathbb{R}}^{2}\setminus 0$ and homogeneous of degree one such that

(2.1) $\det(\sigma
I+B_{1}\eta+B_{2}\xi)=\prod_{k=1}^{N}(\sigma+\lambda_{k}(\eta,\xi))\text{ for
all }(\eta,\xi)\in{\mathbb{R}}^{2}\setminus 0$

Moreover, we have

(2.2)
$\lambda_{1}(\eta,\xi)<\lambda_{2}(\eta,\xi)<...<\lambda_{N}(\eta,\xi)\text{
for all }(\eta,\xi)\in{\mathbb{R}}^{2}\setminus 0$

######  Assumption 2.2.

[Wil20a] The matrix $B_{2}$ is invertible and has $p$ positive eigenvalues,
where $1\leq p\leq N-1$. The boundary matrix $B$ is $p\times N$, real, and of
rank $p$.

We define the sets of frequencies:

(2.3)
$\displaystyle\Xi:=\\{\zeta:=(\sigma-i\gamma,\eta)\in{\mathbb{C}}\times{\mathbb{R}}\setminus(0,0):\gamma\geq
0\\}$
$\displaystyle\Xi_{0}:=\\{(\sigma,\eta)\in{\mathbb{R}}\times{\mathbb{R}}\setminus(0,0)\\}=\Xi\cap\\{\gamma=0\\}$

######  Definition 2.3.

Let $K$ be an $n\times n$ matrix with entries in ${\mathbb{C}}$. If $\lambda$
is an eigenvalue of $K$ of algebraic multiplicity $m\in{\mathbb{N}}$, the
generalized eigenspace associated to $\lambda$ is
$G_{\lambda}:=\\{x\in{\mathbb{C}}^{n}:(K-\lambda I)^{m}x=0\\}$. The stable
subspace of $K$ is the direct sum of the generalized eigenspaces associated to
eigenvalues of $K$ with real part $<0$.

######  Definition 2.4.

[Wil20a]

  1. (1)

The hyperbolic region $\mathcal{H}$ is the set of all
$(\sigma,\eta)\in\Xi_{0}$ such that the matrix $\mathcal{A}(\sigma,\eta)$ is
diagonalizable with real eigenvalues.

  2. (2)

Let $G$ denote the set of all
$(\sigma,\eta,\xi)\in{\mathbb{R}}\times{\mathbb{R}}^{2}$ such that
$(\eta,\xi)\neq 0$ and there exists an integer $k\in\\{1,...,N\\}$ satisfying:

(2.4)
$\sigma+\lambda_{k}(\eta,\xi)=\frac{\partial\lambda_{k}}{\partial\xi}(\eta,\xi)=0.$

If $\pi(G)$ denotes the projection of $G$ on the first two coordinates (that
is $\pi(\sigma,\eta,\xi)=(\sigma,\eta)$for all $(\sigma,\eta,\xi)$), the
glancing set $\mathcal{G}$ is $\mathcal{G}:=\pi(G)\subset\Xi_{0}$.

######  Proposition 2.5.

$[Kre70]$ Let the assumption 2.1 and 2.2 be satisfied. Then for all
$\zeta\in\Xi\setminus\Xi_{0}$, the matrix $i\mathcal{A}(\zeta)$ has no purely
imaginary eigenvalue and its stable subspace ${\mathbb{E}}^{s}(\zeta)$ has
dimension $p$. Furthermore, ${\mathbb{E}}^{s}$ defines an analytic vector
bundle over $\Xi\setminus\Xi_{0}$ that can be extended as a continuous vector
bundle over $\Xi$.

We define ${\mathbb{E}}^{s}(\sigma,\eta)$ to be the continuous extension of
${\mathbb{E}}^{s}(\zeta)$. It is shown in [Mét00] that

  1. (1)

The hyperbolic region $\mathcal{H}$ doesn’t contain any glancing point.

  2. (2)

Away from the glancing point, ${\mathbb{E}}^{s}(\zeta)$ depends analytically
on $\zeta$.

######  Definition 2.6.

[Kre70] Let $p$ be the number of positive eigenvalues of $B_{2}$, and let

(2.5) $L(\partial)=\partial_{t}+B_{1}\partial_{x_{1}}+B_{2}\partial_{x_{2}}$

The problem $(L(\partial),B)$ is said to be uniformly stable or to satisfy the
uniform Lopatinski condition (ULC) if

(2.6) $B:{\mathbb{E}}^{s}(\zeta)\rightarrow{\mathbb{C}}^{p}$

is an isomorphism for all $\zeta\in\Sigma$. Similarly, we say
$(L(\partial),B)$ satisfies the ULC on $\Xi$ (respectively, on a closed conic
subset $\Gamma\subset\Xi$) if (2.6) is an isomorphism on $\Sigma$
(respectively, on the subset of $\Sigma$ corresponding to $\Gamma$).

For a fixed $\beta=(\underline{\sigma},\underline{\eta})\in\mathcal{H}$, we
have the following result which follows from assumption 2.1.

######  Proposition 2.7.

[Wil20a] There exists $\delta>0$ such that on the conic neighborhood of
$\beta$:

(2.7)
$\Gamma^{+}_{\delta}(\beta)=\left\\{\zeta\in\Xi:\bigl{|}\frac{\zeta}{|\zeta|}-\frac{\beta}{|\beta|}\bigr{|}\leq\delta\right\\}$

the matrix $\mathcal{A}(\zeta)=-(A_{0}\tau+A_{1}\eta)$ has $N$ distinct
eigenvalues $\omega_{j}(\zeta)$ and corresponding eigenvectors $R_{j}(\zeta)$
such that:

(2.8) $\mathcal{A}_{j}(\zeta)R_{j}(\zeta)=\omega_{j}(\zeta)R_{j}(\zeta),\ \
j=1,...,N\text{ on }\Gamma^{+}_{\delta}(\beta)$

Functions $\omega_{j}$, $R_{j}$ map are homogeneous of degree one, analytic in
$\tau$, and $C^{\infty}$ in $\eta$. The corresponding normalized vectors

(2.9) $r_{j}(\zeta):=\frac{R_{j}(\zeta)}{|R_{j}(\zeta)|}$

are $C^{\infty}$ in $(\sigma,\gamma,\eta)$.

######  Definition 2.8.

[Wil20a] For each root $\omega_{j}(\beta)=\underline{\omega}_{j}$ there
corresponds a unique integer $k_{j}\in\\{1,...,N\\}$ such that
$\underline{\sigma}+\lambda_{k_{j}}(\underline{\eta},\underline{\omega}_{j})=0$.
We define the following real phases $\phi_{j}$ and their associated group
velocities $v_{j}$:

(2.10) $\phi_{j}(x):=\phi_{0}(t,y)+\underline{\omega}_{j}x_{2},\quad
v_{j}:=\nabla\lambda_{k_{j}}(\underline{\eta},\underline{\omega}_{j}),\quad\forall
j=1,2,...,N$

The phase $\phi_{j}$ is incoming if
$\partial_{\xi}\lambda_{k_{j}}(\underline{\eta},\underline{\omega}_{j})>0$ ,
and is outgoing if
$\partial_{\xi}\lambda_{k_{j}}(\underline{\eta},\underline{\omega}_{j})<0$. If
the phase $\phi_{j}$ is incoming (resp. outgoing), we refer to the
corresponding frequency $\underline{\omega}_{j}$ as incoming (resp. outgoing).
For $\beta\in\mathcal{H}$, there are exactly $N-p$ outgoing phases and $p$
incoming phases. We denote the corresponding set of indices by
$\mathcal{I}=\\{N-p+1,...,N\\}$ for incoming, and
$\mathcal{O}=\\{1,2,...,N-p\\}$ for outgoing.

######  Proposition 2.9.

[Wil20a] The $\omega_{j}$ are real-valued for $\gamma=0$ and can be divided
into two groups according as $j\in\mathcal{O}$ or $\mathcal{I}$. There exists
a constant $c>0$ such that for $\zeta\in\Gamma^{+}_{\delta}(\beta)$

(2.11)
$\displaystyle\begin{split}&\mathrm{Im}\;\omega_{j}(\zeta)\leq-c\gamma\text{
for }j\in\mathcal{O}\\\ &\mathrm{Im}\;\omega_{j}(\zeta)\geq c\gamma\text{ for
}j\in\mathcal{I}.\end{split}$

We define the $N\times(N-p)$ matrix $r_{+}(\zeta)$ of normalized eigenvectors
corresponding to the outgoing phases and the $N\times p$ matrix $r_{-}(\zeta)$
corresponding to the incoming phases by

(2.12) $\displaystyle
r_{+}=\begin{pmatrix}r_{1}&r_{2}&\dots&r_{N-p}\end{pmatrix},\qquad
r_{-}=\begin{pmatrix}r_{N-p+1}&\dots&r_{N}\end{pmatrix}\text{ on
}\Gamma^{+}_{\delta}(\beta).$

We similarly define $R_{\pm}(\zeta)$ using the eigenvectors $R_{j}(\zeta)$. We
also define the $N\times N$ matrix

(2.13) $S(\zeta)=(r_{+}(\zeta)\quad r_{-}(\zeta))=(r_{1},...,r_{N})\text{ on
}\Gamma_{\delta}^{+}(\beta)$

######  Proposition 2.10.

[Wil20a] For $\zeta\in\Gamma_{\delta}^{+}(\beta_{l})$, the stable subspace
${\mathbb{E}}^{s}(\beta_{l})$ admits decomposition:

(2.14) ${\mathbb{E}}^{s}(\beta_{l})=\bigoplus_{j\in\mathcal{I}}r_{j}(\zeta)$

and $r_{j}$ can be (and are) taken to be real vectors.

######  Definition 2.11.

[Wil20a] For $i\in\mathcal{O},j,N\in\mathcal{I}$ with $j\neq N$, we say that
the phases $(\phi_{j},\phi_{N},\phi_{i})$ exhibit a resonance if there exist
$p,q\in{\mathbb{Z}}\setminus 0$ such that

(2.15) $p\phi_{j}+q\phi_{N}=(p+q)\phi_{i}$

Let

(2.16)
$\Omega_{i,j}:=\frac{\omega_{i}(\beta_{l})-\omega_{N}(\beta_{l})}{\omega_{j}(\beta_{l})-\omega_{i}(\beta_{l})}\quad
i\in\mathcal{O},j\in\mathcal{I}\setminus\\{N\\}$

######  Remark 2.12.

If there exists $p,q\in{\mathbb{Z}}\setminus 0$ such that
$(\phi_{j},\phi_{N},\phi_{i})$ exhibit a resonance, then

(2.17) $p\phi_{j}+q\phi_{N}=(p+q)\phi_{i}\iff
p\omega_{j}(\beta_{l})+q\omega_{n}(\beta_{l})=(p+q)\omega_{i}(\beta_{l})\iff\frac{p}{q}=\frac{\omega_{i}(\beta_{l})-\omega_{N}(\beta_{l})}{\omega_{j}(\beta_{l})-\omega_{i}(\beta_{l})}=\Omega_{i,j}$

Above equations show that there is resonance $\iff$
$\Omega_{i,j}\in{\mathbb{Q}}$.

In this paper, we only study the case when there is no resonance. Next we
define Lopatinski determinant.

######  Definition 2.13.

[Wil20b] For $\zeta\in\Gamma^{+}_{\delta}(\beta)$, define the analytic
Lopatinski determinant

(2.18) $\Delta_{a}(\zeta)=\det BR_{-}(\zeta)$

and the normalized Lopatinski determinant:

(2.19) $\Delta(\zeta)=\det Br_{-}(\zeta)$

We next make weak stability assumption on the problem $(L(\partial),B)$ when
ULC fails. This assumption is not in effect in section 3 where we study the
case when ULC holds.

######  Assumption 2.14.

[Wil20a]

  1. (1)

For all $\zeta\in\Xi\backslash\Xi_{0},\ker
B\cap{\mathbb{E}}^{s}(\zeta)=\\{0\\}$.

  2. (2)

The set $\Upsilon_{0}:=\\{\zeta\in\Sigma_{0}:\ker
B\cap{\mathbb{E}}^{s}(\zeta)\neq\\{0\\}\\}=\\{\beta_{l},-\beta_{l}\\}$ is
included in the hyperbolic region $\mathcal{H}$.

  3. (3)

There exists a neighborhood $\Gamma^{+}_{\delta}(\beta_{l})$ as in (2.7) on
which functions $\omega_{j}$, $R_{j}$, $\Delta$ are defined and we have:

(2.20) $\Delta(\beta_{l})=0\text{ and }\partial_{\tau}\Delta(\beta_{l})\neq 0$

######  Notation 2.15.

Throughout this paper, the notation $L^{2}(t,x,\theta)$ denotes the space
$L^{2}(\Omega\times{\mathbb{T}}_{\theta})$ where
${\mathbb{T}}_{\theta}={\mathbb{R}}^{3}/2\pi{\mathbb{Z}}$. We similarly define
the space $H^{\infty}(t,x,\theta)$ etc.

##  3\. Geometric Optics Solutions for The Hyperbolic System when ULC Holds

In this section, we consider the case when uniform Lopatinski condition holds.
We construct geometric optics solutions for a $3\times 3$ system with
oscillatory term term $e^{i\frac{\phi_{3}}{\epsilon}}M$ and use an estimate
from [Kre70] to show that this solution is close to the exact solution. This
section is a warm-up to be contrasted with the more involved construction of
the final section, which deals with a problem where ULC fails.

###  3.1. Introduction

We study the following $3\times 3$ hyperbolic system on
$\Omega_{T}=(-\infty,T)\times\\{(x_{1},x_{2}):x_{2}\geq 0\\}$:

(3.1) $\displaystyle
L(\partial)u+e^{i\frac{\phi_{3}}{\epsilon}}Mu:=\partial_{t}u+B_{1}\partial_{x_{1}}u+B_{2}\partial_{x_{2}}u+e^{i\frac{\phi_{3}}{\epsilon}}Mu=0\text{
in }x_{2}>0$ $\displaystyle Bu:=\epsilon
G\left(t,x_{1},\frac{\phi_{0}}{\epsilon}\right)\text{ on }x_{2}=0$
$\displaystyle u=0\text{ in }t<0$

where all $B_{j}$’s are $3\times 3$ matrices, $B_{2}$ is invertible, $M$ is a
constant matrix, $\phi_{1}$ is outgoing, $\phi_{2},\phi_{3}$ is incoming, and
$G$ is periodic in $\theta_{0}=\frac{\phi_{0}}{\epsilon}$, where:

(3.2)
$\displaystyle\phi_{m}(t,x_{1},x_{2})=\beta_{l}\cdot(t,x_{1})+\omega_{m}(\beta_{l})x_{2}$
$\displaystyle d\phi_{m}=(\beta_{l},\omega_{m}(\beta_{l}))$

and $\beta_{l}$ being a fixed point in $\mathcal{H}$. We write
$U\left(t,x,\frac{\Phi}{\epsilon}\right)=U\left(t,x,\theta\right)|_{\theta=\frac{\Phi}{\epsilon}}$
where $\Phi=(\phi_{1},\phi_{2},\phi_{3})$ and
$\theta=(\theta_{1},\theta_{2},\theta_{3})$,
$\theta_{0}=\frac{\phi_{0}}{\epsilon}$. We set:

(3.3) $\displaystyle
L(\partial)=\partial_{t}+B_{1}\partial_{x_{1}}+B_{2}\partial_{x_{2}}\quad
L(\sigma,\eta,\xi)=\sigma I+B_{1}\eta+B_{2}\xi$
$\displaystyle\mathcal{L}(\partial_{\theta})=\sum_{m=1}^{3}L(d\phi_{m})\partial_{\theta_{m}}$

Let $\mathcal{A}(\beta_{l})$ be the matrix:

(3.4)
$\displaystyle\mathcal{A}(\beta_{l})=-(A_{0}\sigma_{l}+A_{1}\eta_{l})\text{,
where }A_{0}=B_{2}^{-1},A_{1}=B_{2}^{-1}B_{1}$

By strict hyperbolicity, the matrix $\mathcal{A}(\beta_{l})$ is diagonalizable
with eigenvalues $\omega_{m}(\beta_{l})=\omega_{m}$, $m=1,...,3$, and we
observe that the eigenspace of $\mathcal{A}(\beta_{l})$ for $\omega_{m}$ is
precisely the kernel of $L(d\phi_{m})$.

###  3.2. Tools for constructing approximate solutions

[Wil20a]

This section establishes lemmas, definitions, and assumptions for constructing
approximate solutions.

######  Lemma 3.1.

$[CG10]$ The (extended) stable subspace ${\mathbb{E}}^{s}(\beta_{l})$ admits
the decomposition

(3.5)
${\mathbb{E}}^{s}(\beta_{l})=\oplus_{m\in\mathcal{I}}\text{Ker}L(d\phi_{m})$

and each vector space in this decomposition is of real type (it admits a basis
of real vectors).

######  Lemma 3.2.

$[CG10]$ The following decomposition hold

(3.6)
${\mathbb{C}}^{3}=\oplus_{m=1}^{3}\text{Ker}L(d\phi_{m})=\oplus_{m=1}^{3}B_{2}\text{Ker}L(d\phi_{m})$

and each vector space in this decomposition is of real type.
We let $P_{m}$, respectively $Q_{m}$ ($m=1,2,3$), denote the projectors
associated with the first, respectively, second decomposition in 3.6. For each
$m$, we have Im$L(d{\phi_{m}})=$Ker$Q_{m}$.

######  Proposition 3.3.

[Wil20a] We can define operator $R_{m}$ uniquely by:

(3.7) $\displaystyle R_{m}L(d\phi_{m})=I-P_{m},\quad
L(d\phi_{m})R_{m}=I-Q_{m},\quad P_{m}R_{m}=0,\quad R_{m}Q_{m}=0.$

If we choose for each $m$ a real vector $r_{m}$ that spans
$\text{Ker}L(d\phi_{m})$, and choose a real row vector $\ell_{m}$ that satisfy
$\ell_{m}L(d\phi_{m})=0$ with the normalization
$\ell_{m}B_{2}r_{m^{\prime}}=\delta_{mm^{\prime}}$, then $P_{m}$, $Q_{m}$,
$R_{m}$ satisfies:

(3.8) $\displaystyle\forall X\in{\mathbb{C}}^{3},\quad R_{m}X=\sum_{m\neq
m^{\prime}}\frac{\ell_{m^{\prime}}X}{\omega_{m}-\omega_{m^{\prime}}}r_{m^{\prime}},\quad
P_{m}X=(\ell_{m}B_{2}X)r_{m},\quad Q_{m}X=(\ell_{m}X)B_{2}r_{m}.$

We define projection operators $E_{Q}$, $E_{P}$ on
$H^{\infty}:=H^{\infty}(\Omega_{T}\times{\mathbb{T}}_{\theta})$ and partial
inverse operators $R$ of $\mathcal{L}(\partial_{\theta})$ on a proper
subspace. We define

(3.9) $E_{P}=E_{P_{0}}+\sum_{m=1}^{3}E_{P_{m}},\quad
E_{P_{in}}=E_{P_{2}}+E_{P_{3}},\quad E_{P_{out}}=E_{P_{1}}$

where $E_{P_{0}}$ picks out the mean and $E_{P_{m}}$ picks out the pure
$\theta_{m}$ modes and then projects with $P_{m}$. For example, if we write:

(3.10)
$U(t,x,\theta)=\underline{U}(t,x)+U^{*}(t,x,\theta)=\underline{U}(t,x)+\sum_{m=1}^{3}U^{m}(t,x,\theta_{m})+U^{nc}(t,x,\theta_{1},\theta_{2},\theta_{3})$

where $\underline{U}$ is the mean of $U$, each $U^{m}$ has pure $\theta_{m}$
oscillations with mean zero, and $U^{nc}$ consists of only noncharacteristic
modes in the Fourier series of $U$, we have:

(3.11) $\displaystyle E_{P_{0}}U=\underline{U},\quad
E_{P_{m}}U=P_{m}U^{m}(t,x,\theta_{m}),\quad m=1,2,3$ (3.12)
$\displaystyle(I-E_{P})U=\sum_{m=1}^{3}(I-P_{m})U^{m}+U^{nc}$

We define $E_{Q}$ in the same way but we take the projection with $Q_{m}$
instead. Now to define the operator $R$, we define the following subspace of
$H^{\infty}$:

######  Definition 3.4.

Define $\mathcal{N}=\\{(k,l)\in{\mathbb{Z}}\setminus
0\times{\mathbb{Z}}\setminus 0\\}$. Let $\mathcal{H}^{\infty}$ be the subspace
of $H^{\infty}(t,x,\theta)$ given by222Here $\theta$ is a place holder for
$\frac{\Phi}{\epsilon}$.:

(3.13) $\mathcal{H}^{\infty}:=\\{U\in
H^{\infty}:U^{nc}=\sum_{(k,l)\in\mathcal{N}}c^{k,l}(t,x)e^{i(k\theta_{2}+l\theta_{3})}\\}$

######  Remark 3.5.

  1. (1)

We will see later the profiles $U(t,x,\theta)$ that we construct lie in
$\mathcal{H}^{\infty}$.

  2. (2)

In this section, the profiles actually lie in a subset of
$\mathcal{H}^{\infty},$:

(3.14) $\\{U\in H^{\infty}:U^{nc}=\sum_{(k,l)\in\mathcal{N},l\geq
1}c^{k,l}(t,x)e^{i(k\theta_{2}+l\theta_{3})}\\}$

since the oscillatory term in (3.1) only has positive Fourier spectrum.

We make the following small divisor assumption for both (3.1) and (5.1).

######  Assumption 3.6.

(Small divisor assumption)333This assumption is necessary for 3.17 to be well
defined.There exists a constant $C>0$ and $a\in{\mathbb{R}}$ such that
$\forall(k,l)\in\mathcal{N}$ the following holds:

(3.15) $||L(ikd\phi_{2}+ild\phi_{3})^{-1}||\geq C|(k,l)|^{a}$

For $U\in\mathcal{H}^{\infty}$ we define $R$ by:

(3.16) $\displaystyle R(\underline{U})=0$ $\displaystyle
R(U^{m})=\partial_{\theta_{m}}^{-1}R_{m}U^{m}$ $\displaystyle
R(U^{nc})=\mathcal{L}(\partial_{\theta})^{-1}U^{nc}$

where $\partial_{\theta_{m}}^{-1}$ denotes operator of finding the unique mean
zero antiderivative in $\theta_{m}$, and a calculation shows that

(3.17)
$\mathcal{L}(\partial_{\theta})^{-1}U^{nc}=\sum_{(k,l)\in\mathcal{N}}L^{-1}(ikd\phi_{2}+ild\phi_{3})c_{k,l}(t,x)e^{i(k\theta_{2}+l\theta
3)}$

The small divisor assumption 3.6, then shows
$R:\mathcal{H}^{\infty}\rightarrow\mathcal{H}^{\infty}$. The next proposition
follows directly from definition.

######  Proposition 3.7.

[Wil20a] As operators on $\mathcal{H}^{\infty}$, the operators
$\mathcal{L}(\partial_{\theta}),E_{P},E_{Q},$ and $R$ satisfy:

(3.18) $\displaystyle
E_{Q}\mathcal{L}(\partial_{\theta})=\mathcal{L}(\partial_{\theta})E_{P}=0$
$\displaystyle
R\mathcal{L}(\partial_{\theta})=I-E_{P},\quad\mathcal{L}(\partial_{\theta})R=I-E_{Q}$
$\displaystyle E_{P}R=RE_{Q}=0$

###  3.3. Profile equations

We look for approximate solutions to the system (3.1) of the form:

(3.19) $\displaystyle
u^{\epsilon}_{a}(t,x)=\sum_{k={1}}^{J}\epsilon^{k}U_{k}(t,x,\frac{\Phi}{\epsilon})$

The next proposition summaries the work of this section.

######  Proposition 3.8.

Consider the system (3.1) where we assume $Mr_{3}=0$.444The assumption
$Mr_{3}=0$ is used to decouple the problem, for example in (3.31), (3.37). If
assumption 3.6 is satisfied and there is no resonance, there exists an
approximate solution
$u^{\epsilon}_{a}(t,x)=\sum_{k={1}}^{J}\epsilon^{k}U_{k}(t,x,\frac{\Phi}{\epsilon})$
where $U_{k}\in\mathcal{H}^{\infty}(t,x,\theta)$, periodic in $\theta$, and
for any $n\in{\mathbb{N}}$, by taking $J$ large enough, one can arrange that
$u_{a}^{\epsilon}$ satisfies:

(3.20) $|u^{\epsilon}-u_{a}^{\epsilon}|_{L^{\infty}(\Omega_{T})}\leq
C\epsilon^{n}$

where $u^{\epsilon}$ is the exact solution to (3.1).

As proof of the proposition, we first construct $u_{a}^{\epsilon}$. We observe
that the first term in the sum of $u_{a}^{\epsilon}$ satisfies:

(3.21) $\displaystyle L(\partial)\left(\epsilon
U_{1}\left(t,x,\frac{\Phi}{\epsilon}\right)\right)$
$\displaystyle=\partial_{t}(\epsilon
U_{1})\left(t,x,\frac{\Phi}{\epsilon}\right)+B_{1}\partial_{x_{1}}(\epsilon
U_{1})\left(t,x,\frac{\Phi}{\epsilon}\right)+B_{2}\partial_{x_{2}}(\epsilon
U_{1})\left(t,x,\frac{\Phi}{\epsilon}\right)$
$\displaystyle+(\partial_{t}\phi_{1}+B_{1}\partial_{x_{1}}\phi_{1}+B_{2}\partial_{x_{2}}\phi_{1})\partial_{\theta_{1}}U_{1}$
$\displaystyle+(\partial_{t}\phi_{2}+B_{1}\partial_{x_{1}}\phi_{2}+B_{2}\partial_{x_{2}}\phi_{2})\partial_{\theta_{2}}U_{1}$
$\displaystyle+(\partial_{t}\phi_{3}+B_{1}\partial_{x_{1}}\phi_{3}+B_{2}\partial_{x_{2}}\phi_{3})\partial_{\theta_{3}}U_{1}$

We see that the last three lines is exactly
$\mathcal{L}(\partial_{\theta})U_{1}$. Therefore, plugging (3.19) into the
interior equation in (3.1) gives:

(3.22)
$\displaystyle\epsilon(L(\partial)U_{1})(t,x,\frac{\Phi}{\epsilon})+(\mathcal{L}(\partial_{\theta})U_{1})(t,x,\frac{\Phi}{\epsilon})+\epsilon
e^{\frac{i\phi_{3}}{\epsilon}}MU_{1}(t,x,\frac{\Phi}{\epsilon})+$
$\displaystyle\epsilon^{2}(L(\partial)U_{2})(t,x,\frac{\Phi}{\epsilon})+\epsilon(\mathcal{L}(\partial_{\theta})U_{2})(t,x,\frac{\Phi}{\epsilon})+\epsilon^{2}e^{\frac{i\phi_{3}}{\epsilon}}MU_{2}(t,x,\frac{\Phi}{\epsilon})+...=0$

The coefficients of same order $\epsilon$ gives the following interior
equations of $U_{j}(t,x,\theta)$

(3.23) $\displaystyle\mathcal{L}(\partial_{\theta})U_{1}=0$ $\displaystyle
L(\partial)U_{1}+\mathcal{L}(\partial_{\theta})U_{2}+e^{\frac{i\phi_{3}}{\epsilon}}MU_{1}=0$
$\displaystyle
L(\partial)U_{2}+\mathcal{L}(\partial_{\theta})U_{3}+e^{\frac{i\phi_{3}}{\epsilon}}MU_{2}=0$
$\displaystyle
L(\partial)U_{j}+\mathcal{L}(\partial_{\theta})U_{j+1}+e^{\frac{i\phi_{3}}{\epsilon}}MU_{j}=0\text{
for }j\geq 2$

and for boundary equations on $x_{2}=0,\ \theta_{m}=\theta_{0}$, we have:

(3.24) $\displaystyle BU_{1}=G\left(t,x_{1},\theta_{0}\right)$ $\displaystyle
BU_{j}=0\text{ for }j\geq 2.$

Now we apply the operators defined from lemma 3.2 and the discussion that
follows the lemma to above equations. Applying $E_{Q}$ to both sides of 3.23,
we reach the following interior equations:

(3.25) $\displaystyle E_{Q}[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]=0$
$\displaystyle E_{Q}[L(\partial)U_{2}+e^{i\theta_{3}}MU_{2}]=0$ $\displaystyle
E_{Q}[L(\partial)U_{j}+e^{i\theta_{3}}MU_{j}]=0$

Applying operator $R$ to 3.23 gives:

(3.26) $\displaystyle(I-E_{P})U_{1}=0$
$\displaystyle(I-E_{P})U_{2}=-R[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]$
$\displaystyle(I-E_{P})U_{3}=-R[L(\partial)U_{2}+e^{i\theta_{3}}MU_{2}]$

As for boundary equations, we first note that the space
$\mathbb{E}^{s}(\beta_{l})$ is spanned by $\\{r_{2},r_{3}\\}$. Uniform
Lopatinski condition holds implies
$B:\mathbb{E}^{s}(\beta_{l})\rightarrow{\mathbb{C}}^{2}$ is an isomorphism.
Recall that $E_{P_{in}}U_{k}\in
span\\{r_{2},r_{3}\\}=\mathbb{E}^{s}(\beta_{l})$, thus for $BU_{k}=H_{k}$ we
have the following boundary equations:

(3.27) $\displaystyle B\underline{U_{k}}=\underline{H_{k}}$ $\displaystyle
BU_{k}^{*}=H_{k}^{*}$

where $H_{k}$ is read from (3.24). We notice that using the definition of
projecting operators, the second equation of (3.27) can be written as:

(3.28) $\displaystyle
BE_{P_{in}}U_{k}=H_{k}^{*}-BE_{P_{1}}U_{k}-B[(I-E_{P})U_{k}]^{*}$

###  3.4. Determining the solutions

To equations (3.26) with the boundary condition (3.28), we will make use of
the following proposition from [Wil20a]:

######  Proposition 3.9.

[Wil20a] For $U\in\mathcal{H}^{\infty}$, let
$E_{P_{m}}U=\sigma(t,x,\theta_{m})r_{m}$ and let
$X_{\phi_{m}}=\partial_{x_{2}}-\partial_{\tau}\omega_{m}(\beta_{l})\partial_{t}-\partial_{\eta}\omega_{m}(\beta_{l})\partial_{x_{t}}$
be the transport vector field associated to the phase $\phi_{m}$. Then

(3.29) $\displaystyle
E_{Q_{m}}L(\partial)E_{P_{m}}U=Q_{m}(L(\partial)\sigma(t,x,\theta_{m})r_{m})=(X_{\phi_{m}}\sigma)B_{2}r_{m}$

First, we consider the leading profile.
Since $(I-E_{P})U_{1}=0$, for some scalar profiles $\sigma$ we write:

(3.30)
$U_{1}=\underline{U_{1}}+\sigma_{1}^{1}(t,x,\theta_{1})r_{1}+\sigma_{1}^{2}(t,x,\theta_{2})r_{2}+\sigma_{1}^{3}(t,x,\theta_{3})r_{3}$

1\. Determination of $\underline{U_{1}}$.

Taking the mean on both sides of 3.25, we have:
$E_{Q_{0}}[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]=0$. We note that the mean
$L(\partial)U_{1}$ is $L(\partial)\underline{U_{1}}$ and the mean of
$e^{i\theta_{3}}MU_{1}$ is $\underline{e^{i\theta_{3}}M\sigma_{1}^{3}r_{3}}$.
Thus, using 3.27 we have:

(3.31) $\displaystyle
L(\partial)\underline{U_{1}}+\underline{e^{i\theta_{3}}M\sigma_{1}^{3}r_{3}}=0$
$\displaystyle B\underline{U_{1}}=\underline{G}$

We note that this has the coupling involving $\sigma_{1}^{3}r_{3}$. So the
assumption that $Mr_{3}=0$ decouples this problem and will yield a solution
for $\underline{U_{1}}$.

2\. Determining the leading profile $U_{1}$

Using 3.25, 3.29, $U=E_{P}U$, and the assumption of no resonance we have:

(3.32) $\displaystyle
E_{Q_{1}}[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]=X_{\phi_{1}}\sigma_{1}^{1}B_{2}r_{1}=0$
$\displaystyle
E_{Q_{2}}[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]=X_{\phi_{2}}\sigma_{1}^{2}B_{2}r_{2}=0$
$\displaystyle
E_{Q_{3}}[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]=X_{\phi_{3}}\sigma_{1}^{3}B_{2}r_{3}+Q_{3}e^{i\theta_{3}}M\sigma_{1}^{3}r_{3}+Q_{3}e^{i\theta_{3}}M\underline{U_{1}}=0$

From the boundary equation (3.27), we have:

(3.33) $\displaystyle
B\sigma_{1}^{1}r_{1}+B\sigma_{1}^{2}r_{2}+B\sigma_{1}^{3}r_{3}=G^{*}$

To solve these equations, we note that (3.32) are homogeneous transport
equations with constant coefficients. Thus, $\sigma_{1}^{1}$ and
$\sigma_{1}^{2}$ are constant on the integral curve of $X_{\phi_{k}}$ for
$k=1,2$. Since $\phi_{1}$ is outgoing, $\sigma_{1}^{1}=0$ at $t<0$, we have
$E_{P_{1}}U_{1}=\sigma_{1}^{1}=0$.
To solve for $\sigma_{1}^{2}$, we need a boundary equation. Using
$B:\text{span}\\{r_{2},r_{3}\\}\rightarrow{\mathbb{C}}^{2}$ is an isomorphism,
the boundary equation 3.33 becomes:

(3.34) $\displaystyle
B\sigma_{1}^{2}r_{2}+B\sigma_{1}^{3}r_{3}=G^{*}=c_{1}(t,x_{1},\theta_{0})Br_{2}+c_{2}(t,x_{1},\theta_{0})Br_{3}$

which implies:

(3.35) $\displaystyle\sigma_{1}^{2}\mid_{x_{2}=0,\
\theta_{2}=\theta_{0}}=c_{1}$ $\displaystyle\sigma_{1}^{3}\mid_{x_{2}=0,\
\theta_{3}=\theta_{0}}=c_{2}$

With those boundary equations, we have uniquely determined $\sigma_{1}^{2}$.
For the equation of $\sigma_{1}^{3}$ in 3.32, using $Q_{m}X=l_{m}XB_{2}r_{m}$,
we rewrite this equation as:

(3.36)
$X_{\phi_{1}}\sigma_{1}^{3}+e^{i\theta_{3}}\sigma_{1}^{3}\ell_{m}Mr_{3}+e^{i\theta_{3}}\ell_{m}M\underline{U_{1}}=0$

and we have the following coupling equations for $\sigma_{1}^{3}$:

(3.37)
$\displaystyle\begin{cases}&X_{\phi_{1}}\sigma_{1}^{3}+e^{i\theta_{3}}\sigma_{1}^{3}\ell_{m}Mr_{3}+e^{i\theta_{3}}\ell_{m}M\underline{U_{1}}=0\\\
&\sigma_{1}^{3}\mid_{x_{2}=0,\ \theta=\theta_{3}}=c_{2}\\\ \end{cases}$
$\displaystyle\begin{cases}&L(\partial)\underline{U_{1}}+\underline{e^{i\theta_{3}}M\sigma_{1}^{3}r_{3}}=0\\\
&B\underline{U_{1}}|_{x_{2}=0}=\underline{G}\end{cases}$

To decouple the problem, we again use the assumption that $Mr_{3}=0$. Then
$\sigma_{1}^{3}$ is uniquely determined.

3\. Determining $U_{2}$

Next, we consider $U_{2}$. From (3.26), we have:

(3.38) $\displaystyle
U_{2}=E_{P}U_{2}+(I-E_{P}U_{2})=\underline{U_{2}}+\sigma_{2}^{1}r_{1}+\sigma_{2}^{2}r_{2}+\sigma_{2}^{3}r_{3}-R[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]$

We observe that $(I-E_{P})U_{2}=-R(L(\partial)U_{1}+e^{i\theta_{3}}MU_{1})$ is
determined by previous terms and is known. Plugging in this decomposition into
the equation $E_{Q}[L(\partial)U_{2}+e^{i\theta_{3}}MU_{2}]=0$, we have:

(3.39) $\displaystyle E_{Q}(L(\partial)E_{P}U_{2}+e^{i\theta_{3}}ME_{P}U_{2})$
$\displaystyle=-E_{Q}(L(\partial)(I-E_{P})U_{2}+e^{i\theta_{3}}M(I-E_{P})U_{2})$

This yields four equations for $U_{2}$:

(3.40) $\displaystyle
E_{Q_{0}}(L(\partial)E_{P}U_{2}+e^{i\theta_{3}}ME_{P}U_{2})$
$\displaystyle=-E_{Q_{0}}[L(\partial)(I-E_{P})U_{2}+e^{i\theta_{3}}M(I-E_{P})U_{2}]$
$\displaystyle=-E_{Q_{0}}[(I-E_{P})U_{2}]-E_{Q_{0}}[e^{i\theta_{3}}M(I-E_{P})U_{2}]$
(3.41) $\displaystyle X_{\phi_{1}}\sigma_{2}^{1}B_{2}r_{1}$
$\displaystyle=-E_{Q_{1}}[L(\partial)(I-E_{P})U_{2}+e^{i\theta_{3}}M(I-E_{P})U_{2}]$
$\displaystyle X_{\phi_{2}}\sigma_{2}^{2}B_{2}r_{2}$
$\displaystyle=-E_{Q_{2}}[L(\partial)(I-E_{P})U_{2}+e^{i\theta_{3}}M(I-E_{P})U_{2}]$
$\displaystyle X_{\phi_{3}}\sigma_{2}^{3}B_{2}r_{3}$
$\displaystyle+Q_{3}e^{i\theta_{3}}M\sigma_{2}^{3}r_{3}+Q_{3}e^{i\theta_{3}}M\underline{U_{2}}=$
$\displaystyle-
E_{Q_{3}}[L(\partial)(I-E_{P})U_{2}+e^{i\theta_{3}}M(I-E_{P})U_{2}]$

We observe that the first term in (3.40), $-E_{Q_{0}}[(I-E_{P})U_{2}]$ picks
out the mean of $-[(I-E_{P})U_{2}]=R(L(\partial)U_{1}+e^{i\theta_{3}}MU_{1})$.
By the definition of $R$, this term has mean zero. For the second term, we
have
$-E_{Q_{0}}[e^{i\theta_{3}}M(I-E_{P})U_{2}]=\underline{e^{i\theta_{3}}MR[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]}=\underline{e^{i\theta_{3}}MR[L(\partial)\sigma_{1}^{3}r_{3}+e^{i\theta_{3}}M(\underline{U_{1}}+\sigma_{1}^{3}r_{3})]}$

We note that all equations in (3.41) are evaluated at vector $B_{2}r_{m}$,
which makes them scalar equations. We also observe that the right hand sides
are known and we can use similar techniques as before. For the first equation
in (3.41) involving $\sigma_{2}^{1}$, the first term on the right hand side
satisfies:

(3.42)
$-E_{Q_{1}}[L(\partial)(I-E_{P})U_{2}]=-E_{Q_{1}}[L(\partial)(-R(L(\partial)U_{1}+e^{i\theta_{3}}MU_{1})]=-E_{Q_{1}}[L(\partial)(-R(L(\partial)U_{1})]$

Since $U_{1}$ has no oscillation in $\theta_{1}$, above equation equals to
zero. The second term on the right hand side of the equation (3.41) involving
$\sigma_{2}^{1}$ is $-E_{Q_{1}}[e^{i\theta_{3}}M(-R(L(\partial)U_{1})]=0$
since there is no resonance. Thus, the first equation of (3.41) becomes

(3.43) $X_{\phi_{1}}\sigma_{2}^{1}B_{2}r_{1}=0$

Since $X_{\phi_{1}}$ is outgoing and $\sigma_{2}^{1}=0$ when $t<0$, we have
$\sigma_{2}^{1}=0$.
The equation involving $\sigma_{2}^{2}$ in (3.41) is an inhomogenenous
transport equation with constant coefficients, so it has unique solutions
provided a boundary condition. Boundary condition (3.24) implies
$BE_{P}U_{2}+B(I-E_{P})U_{2}=0$. Using $B\underline{U_{2}}=0$ and isomorphism
$B:\text{span}\\{r_{2},r_{3}\\}\rightarrow{\mathbb{C}}^{2}$, we have:

(3.44)
$B\sigma_{2}^{2}r_{2}+B\sigma_{2}^{3}r_{3}=-B(I-E_{P})U_{2}=c_{2}^{2}Br_{2}+c_{2}^{3}Br_{3}$

which provides boundary equations $\sigma_{2}^{2}=c_{2}^{2}$,
$\sigma_{2}^{3}=c_{2}^{3}$ and thus gives the unique solution
$\sigma_{2}^{2}$.
Similar to the case of $U_{1}$, equations involving $\sigma_{k}^{3}$ result in
the couplings with the following equations involving $\underline{U_{k}}$:

(3.45)
$\displaystyle\begin{cases}&X_{\phi_{3}}\sigma_{2}^{3}B_{2}r_{3}+Q_{3}e^{i\theta_{3}}M\sigma_{2}^{3}r_{3}+Q_{3}e^{i\theta_{3}}M\underline{U_{2}}\\\
&=-E_{Q_{3}}[L(\partial)(I-E_{P})U_{2}+e^{i\theta_{3}}M(I-E_{P})U_{2}]\\\
&\sigma_{2}^{3}|_{x_{2}=0,\theta_{3}=\theta_{0}}=c_{2}^{3}\end{cases}$ (3.46)
$\displaystyle\begin{cases}&L(\partial)\underline{U_{2}}+\underline{e^{i\theta_{3}}M\sigma_{2}^{3}r_{3}}=\underline{e^{i\theta_{3}}MR[L(\partial)U_{1}+e^{i\theta_{3}}MU_{1}]}\\\
&B\underline{U_{2}}|_{x_{2}=0}=0\end{cases}$

which follows from (3.40). Similarly, to decouple the problem, assuming
$Mr_{3}=0$ gives $\underline{U_{2}}=0$ and thus the solution of
$\sigma_{k}^{3}$.

4\. Determining the remaining terms of the profile.

We consider $U_{k}$ for $k\geqslant 2$. First from (3.26), we have:

(3.47) $\displaystyle
U_{k}=E_{P}U_{k}+(I-E_{P}U_{k})=\underline{U_{k}}+\sigma_{k}^{1}r_{1}+\sigma_{k}^{2}r_{2}+\sigma_{k}^{3}r_{3}-R[L(\partial)U_{k-1}+e^{i\theta_{3}}MU_{k-1}]$

We again observe that
$(I-E_{P})U_{k}=-R(L(\partial)U_{k-1}+e^{i\theta_{3}}MU_{k-1})$ is determined
by previous terms and is known. Plugging this into
$E_{Q}[L(\partial)U_{k}+e^{i\theta_{3}}MU_{k}]=0$, we have:

(3.48) $\displaystyle E_{Q}(L(\partial)E_{P}U_{k}+e^{i\theta_{3}}ME_{P}U_{k})$
$\displaystyle=-E_{Q}(L(\partial)(I-E_{P})U_{k}+e^{i\theta_{3}}M(I-E_{P})U_{k})$

This gives us four equations for $U_{k}$:

(3.49) $\displaystyle
E_{Q_{0}}(L(\partial)E_{P}U_{k}+e^{i\theta_{3}}ME_{P}U_{k})$
$\displaystyle=-E_{Q_{0}}[L(\partial)(I-E_{P})U_{k}+e^{i\theta_{3}}M(I-E_{P})U_{k}]$
(3.50) $\displaystyle X_{\phi_{1}}\sigma_{k}^{1}B_{2}r_{1}$
$\displaystyle=-E_{Q_{1}}[L(\partial)(I-E_{P})U_{k}+e^{i\theta_{3}}M(I-E_{P})U_{k}]$
$\displaystyle X_{\phi_{2}}\sigma_{k}^{2}B_{2}r_{2}$
$\displaystyle=-E_{Q_{2}}[L(\partial)(I-E_{P})U_{k}+e^{i\theta_{3}}M(I-E_{P})U_{k}]$
$\displaystyle X_{\phi_{3}}\sigma_{k}^{3}B_{2}r_{3}$
$\displaystyle+Q_{3}e^{i\theta_{3}}M\sigma_{k}^{3}r_{3}+Q_{3}e^{i\theta_{3}}M\underline{U_{k}}=$
$\displaystyle-
E_{Q_{3}}[L(\partial)(I-E_{P})U_{k}+e^{i\theta_{3}}M(I-E_{P})U_{k}]$

By the same argument provided for $U_{2}$, (3.49) implies:

(3.51) $\displaystyle
L(\partial)\underline{U_{k}}+\underline{e^{i\theta_{3}}M\sigma_{k}^{3}r_{3}}$
$\displaystyle=\underline{e^{i\theta_{3}}MR[L(\partial)U_{k-1}+e^{i\theta_{3}}MU_{k-1}]}$

and the first equation in (3.50) implies:

(3.52) $\displaystyle X_{\phi_{1}}\sigma_{k}^{1}B_{2}r_{1}$ $\displaystyle=0$

Using $X_{\phi_{1}}$ is outgoing and $\sigma_{k}^{1}=0$ when $t<0$, we have
$\sigma_{k}^{1}=0$.
The equation involving $\sigma_{k}^{2}$ is also an inhomogenenous transport
equation with constant coefficients, thus has unique solutions provided
boundary conditions. Boundary condition (3.24) implies
$BE_{P}U_{k}+B(I-E_{P})U_{k}=0$. Using $B\underline{U_{k}}=0$ and isomorphism
$B:\text{span}\\{r_{2},r_{3}\\}\rightarrow{\mathbb{C}}^{2}$, we get:

(3.53)
$B\sigma_{k}^{2}r_{2}+B\sigma_{k}^{3}r_{3}=-B(I-E_{P})U_{k}=c_{k}^{2}Br_{2}+c_{k}^{3}Br_{3}$

which provides boundary equations $\sigma_{k}^{2}=c_{k}^{2}$,
$\sigma_{k}^{3}=c_{k}^{3}$ and gives the unique solution $\sigma_{k}^{2}$.
The equations involving $\sigma_{k}^{3}$ result in the couplings with the
following equations involving $\underline{U_{k}}$:

(3.54)
$\displaystyle\begin{cases}&X_{\phi_{3}}\sigma_{k}^{3}B_{2}r_{3}+Q_{3}e^{i\theta_{3}}M\sigma_{k}^{3}r_{3}+Q_{3}e^{i\theta_{3}}M\underline{U_{k}}\\\
&=-E_{Q_{3}}[L(\partial)(I-E_{P})U_{k}+e^{i\theta_{3}}M(I-E_{P})U_{k}]\\\
&\sigma_{k}^{3}|_{x_{2}=0,\theta_{3}=\theta_{0}}=c_{k}^{3}\end{cases}$ (3.55)
$\displaystyle\begin{cases}&L(\partial)\underline{U_{k}}+\underline{e^{i\theta_{3}}M\sigma_{k}^{3}r_{3}}=\underline{e^{i\theta_{3}}MR[L(\partial)U_{k-1}+e^{i\theta_{3}}MU_{k-1}]}\\\
&B\underline{U_{k}}|_{x_{2}=0}=0\end{cases}$

which follows from (3.51) and (3.24). Similarly, to decouple the problem,
assuming $Mr_{3}=0$ gives the solution of $\sigma_{k}^{3}$. Now we have
determined all unknown in 3.19 and obtain a solution of the desired form.

###  3.5. Justifying WKB solutions

We here justify the solution 3.19 we constructed above by showing that it is
close to the real solution as $\epsilon$ tends to $0$. We write

(3.56)
$u^{\epsilon}_{a}=\sum_{k=1}^{N}\epsilon^{k}U_{k}(t,x,\frac{\Phi}{\epsilon})$

as the approximate solution we constructed. Let $u$ be the exact solution to
the system. Plugging in $u-u_{a}^{\epsilon}$ to the system, we have:

(3.57) $\displaystyle
L(\partial)(u-u_{a}^{\epsilon})+e^{i\phi_{3}/\epsilon}M(u-u_{a}^{\epsilon})=-\epsilon^{N}[(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}(t,x,\theta)]|_{\theta=\Phi/\epsilon}$
$\displaystyle B(u-u_{a}^{\epsilon})=0\text{ on boundary}$

where $L(\partial):=\partial_{t}+B_{1}\partial_{x_{1}}+B_{2}\partial_{x_{2}}$.

We need the following proposition:

######  Proposition 3.10.

Let $f$ be periodic in $\theta$ and $f(t,x,\theta)\in H^{m}(t,x,\theta)$ for
$m\geq\frac{n+1}{2}$ $((t,x)\in{\mathbb{R}}^{n})$, then
$|f(t,x,\frac{\phi(t,x)}{\epsilon})|_{L^{2}(t,x)}\leq
C|f(t,x,\theta)|_{H^{1}(t,x,\theta)}$

###### Proof.

Writing

(3.58) $f(t,x,\theta)=\sum_{1}^{\infty}f_{n}(t,x)e^{i\theta/\epsilon},\quad
f(t,x,\frac{\phi(t,x)}{\epsilon})=\sum_{1}^{\infty}f_{n}(t,x)e^{i\phi(t,x)/\epsilon}$

we have

(3.59)
$\displaystyle\left|\sum_{1}^{\infty}f_{n}(t,x)e^{i\phi(t,x)/\epsilon}\right|_{L^{2}(t,x)}$
$\displaystyle\leq\sum_{1}^{\infty}|f_{n}(t,x)|_{L^{2}(t,x)}$ (3.60)
$\displaystyle\leq\left|\left(|f_{n}(t,x)|_{L^{2}(t,x)}\cdot
n\right)\right|_{\ell^{2}(n)}\left|\frac{1}{n}\right|_{\ell^{2}(n)}$ (3.61)
$\displaystyle\leq C|f(t,x,\theta)|_{H^{1}(t,x,\theta)}$

∎

We need the following estimate from [Kre70]:

######  Proposition 3.11.

[Kre70] There exists a constant $\gamma_{0}$ such that for all
$\gamma\geq\gamma_{0}$, the system

(3.62) $\displaystyle L(\partial)u=f$ (3.63) $\displaystyle Bu=g$

satisfies:

(3.64) $\gamma|e^{-\gamma t}u|_{L^{2}(t,x)}^{2}+|e^{-\gamma
t}u|_{x_{2}=0}|_{L^{2}(t,x_{1})}^{2}\leq C\left(\frac{1}{\gamma}|e^{-\gamma
t}f|^{2}_{L^{2}(t,x)}+|e^{-\gamma t}g|^{2}_{L^{2}(t,x_{1})}\right).$

######  Proposition 3.12.

The estimate (3.64) holds for the system:

(3.65) $\displaystyle L(\partial)u+\mathcal{D}u=f$ (3.66) $\displaystyle Bu=g$

where $\mathcal{D}$ is any bounded function.

###### Proof.

We can rewrite the first equation as $L(\partial)u=f-\mathcal{D}u$. Then
applying (3.64) to $(L(\partial),B)$ gives:

(3.67) $\displaystyle\gamma|e^{-\gamma t}u|_{L^{2}(t,x)}^{2}+$
$\displaystyle|e^{-\gamma t}u|_{x_{2}=0}|_{L^{2}(t,x_{1})}^{2}\leq
C\left(\frac{1}{\gamma}|e^{-\gamma
t}(f-\mathcal{D})u|^{2}_{L^{2}(t,x)}+|e^{-\gamma
t}g|^{2}_{L^{2}(t,x_{1})}\right)$ (3.68) $\displaystyle\leq
C\left(\frac{1}{\gamma}|e^{-\gamma
t}f|^{2}_{L^{2}(t,x)}+\frac{1}{\gamma}|\mathcal{D}|_{L^{\infty}}|e^{-\gamma
t}u|^{2}_{L^{2}(t,x)}+|e^{-\gamma t}g|^{2}_{L^{2}(t,x_{1})}\right).$

Taking $\gamma$ large gives the desired inequality. ∎

######  Proposition 3.13.

There exists a unique solution for the system (3.1).

###### Proof.

Consider the following system:

(3.69) $\displaystyle
L(\partial)w^{\epsilon}+e^{i\phi_{3}/\epsilon}Mw^{\epsilon}$
$\displaystyle=-\epsilon^{N}[(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}(t,x,\theta)]|_{\theta=\Phi/\epsilon}=:f$
$\displaystyle Bw^{\epsilon}$ $\displaystyle=0\text{ on boundary}$

We observe that if there exists $w^{\epsilon}$ satisfying (3.69), then
$u^{\epsilon}=u_{a}^{\epsilon}+w^{\epsilon}$ solves equation (1.1). Therefore,
it is left to prove the existence and uniqueness of $w^{\epsilon}$. Consider
the dual problem:

(3.70) $\displaystyle L^{*}(\partial)v$ $\displaystyle=f_{\sharp}$ (3.71)
$\displaystyle A_{\sharp}v$ $\displaystyle=g_{\sharp}$

where $L^{*}(\partial)$ is the adjoint operator of
$L(\partial)+e^{i\phi_{3}/\epsilon}M$, $f_{\sharp},g_{\sharp}$ are appropriate
functions and $A_{\sharp}$ satisfies:

(3.72) $B_{\sharp}^{T}B+A_{\sharp}^{T}A=I.$

Define the space

(3.73) $F=\\{V\in C^{\infty}_{0}(\overline{\Omega})\text{ such that
}A_{\sharp}V|_{x_{2}=0}=0\\}$

For all $V\in F$, define

(3.74)
$\ell[L^{*}(\partial)V]:=(f,V)_{L^{2}}+(0,B_{\sharp}V|_{x_{2}=0})_{L^{2}(t,x_{1})}=(f,V)_{L^{2}}.$

By definition $\ell$ is linear. We observe that the system (3.70) is the same
form as (1.1), and therefore satisfies the same energy estimate (3.64). So we
have:

(3.75) $\displaystyle\ell[L^{*}(\partial)V]$
$\displaystyle\leq\|f\|_{L^{2}(t,x)}\|V\|_{L^{2}(t,x)}$ (3.76)
$\displaystyle\leq C\|V\|_{L^{2}(t,x)}$ (3.77) $\displaystyle\leq
C\|L^{*}(\partial)V\|_{L^{2}(t,x)}$

This shows that $\ell$ is continuous. Now by Hahn-Banach theorem we extend
$\ell$ to $L^{2}(\Omega_{T})$. Riesz representation theorem implies that there
exists $W\in L^{2}(\Omega_{T})$ such that

(3.78)
$\ell[L^{*}(\partial)V]=(W,L^{*}(\partial)V)_{L^{2}(t,x)}=(L(\partial)W,V)_{L^{2}(t,x)}$

Thus, we have $L(\partial)W=f$. Moreover, the trace of $W$ lies in
$H^{-1/2}(t,x_{1})$, and the following Green’s formula holds:

(3.79)
$(L(\partial)W,V)_{L^{2}(t,x)}=(f,V)_{L^{2}(t,x)}+(W,V)_{H^{-1/2}(t,x_{1})H^{1/2}(t,x_{1})}\quad\forall
V\subset C_{0}^{\infty}(\overline{\Omega}).$

Comparing (3.79) and (3.74) gives:

(3.80)
$(W(t,x_{1},0),V(t,x_{1},0))_{L^{2}(t,x_{1})}=(0,B_{\sharp}V(t,x_{1},0))_{L^{2}(t,x_{1})}=0\quad\forall
V\in F$

We can therefore conclude that $W=0$ on $x_{2}=0$. This concludes the proof of
the proposition. ∎

Using proposition 3.10 and estimate (3.64), we have

(3.81) $\displaystyle\gamma|e^{-\gamma
t}(u-u_{a}^{\epsilon})|^{2}_{L^{2}(t,x)}$
$\displaystyle\leq\frac{C}{\gamma}|e^{-\gamma
t}\epsilon^{N}([L(\partial)+e^{\frac{\phi_{3}(t,x)}{\epsilon}}M)U_{N}(t,x,\theta)]|_{\theta=\Phi/\epsilon}|^{2}_{L^{2}(t,x)}$
$\displaystyle|e^{-\gamma t}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}$
$\displaystyle\leq\frac{C}{\gamma}\epsilon^{N}|e^{-\gamma
t}[(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}(t,x,\theta)]|_{\theta=\Phi/\epsilon}|_{L^{2}(t,x)}$
$\displaystyle\leq\frac{C}{\gamma}\epsilon^{N}|(L(\partial)+e^{i\theta_{3}}M)U_{N}(t,x,\theta)|_{H^{1}(t,x,\theta)}$

Since $e^{-\gamma T}|(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq|e^{-\gamma
t}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}$ for $t\in[0,T]$, we have:

(3.82) $|(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq\frac{Ce^{\gamma
T}}{\gamma}\epsilon^{N}|(L(\partial)+e^{i\theta_{3}}M)U_{N}(t,x,\theta)|_{H^{1}(t,x,\theta)}$

which gives:

(3.83) $|(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq C\epsilon^{N}$

To attain higher order derivatives, we commute the system 3.57 with tangential
derivatives $\epsilon^{|\alpha|}\partial^{\alpha}_{t,x_{1}}$ ($|\alpha|\leq
s$, $s>n/2$). We first commute the system with $\epsilon\partial_{t}$:

(3.84)
$\displaystyle\epsilon\partial_{t}L(\partial)(u-u_{a}^{\epsilon})+\epsilon\partial_{t}(e^{i\phi_{3}/\epsilon}M(u-u_{a}^{\epsilon}))=-\epsilon^{N}[\epsilon\partial_{t}(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}(t,x,\theta)]|_{\theta=\Phi/\epsilon}$
$\displaystyle\epsilon\partial_{t}B(u-u_{a}^{\epsilon})=0\text{ on boundary}$

After applying chain rule and reorganizing terms, we have
$\partial_{t}(u-u_{a}^{\epsilon})$ satisfying:

(3.85) $\displaystyle
L(\partial)(\partial_{t}(u-u_{a}^{\epsilon}))+e^{i\phi_{3}/\epsilon}M(\partial_{t}(u-u_{a}^{\epsilon}))$
$\displaystyle=-\epsilon^{N}\partial_{t}[(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}(t,x,\theta)]|_{\theta=\Phi/\epsilon}$
$\displaystyle-\frac{1}{\epsilon}ie^{i\phi_{3}/\epsilon}M\partial_{t}\phi_{3}(u-u_{a}^{\epsilon})$
$\displaystyle B(\partial_{t}(u-u_{a}^{\epsilon}))$ $\displaystyle=0\text{ on
boundary}$

where the right hand side of the first equation is
$\mathcal{O}(\epsilon^{N-1})$ by 3.83. Thus, applying the argument 3.81 to
3.83 to the system 3.85 gives:

(3.86) $|\partial_{t}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq C\epsilon^{N-1}$

We note that commuting 3.57 with $\epsilon\partial_{x_{1}}$ shows that
$\partial_{x_{1}}(u-u_{a}^{\epsilon})$ satisfies the exact system as 3.85 with
$\partial_{t}$ replaced by $\partial_{x_{1}}$, thus :

(3.87) $|\partial_{x_{1}}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq
C\epsilon^{N-1}$

We iterate the above argument and get estimate for higher derivatives:

(3.88) $|\partial^{|\alpha|}_{t,x_{1}}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq
C\epsilon^{N-|\alpha|}$

Finally, to estimate the $\partial_{x_{2}}(u-u_{a}^{\epsilon})$, using $B_{2}$
is invertible, we have from the interior equation:

(3.89)
$\displaystyle(\partial_{t}+B_{1}\partial_{x_{1}}+B_{2}\partial_{x_{2}})(u-u_{a}^{\epsilon})+e^{i\phi_{3}/\epsilon}(u-u_{a}^{\epsilon})=-\epsilon^{N}(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}$
$\displaystyle\Rightarrow$ $\displaystyle
B_{2}\partial_{x_{2}}(u-u_{a}^{\epsilon})=-\epsilon^{N}(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}-\partial_{t}(u-u_{a}^{\epsilon})-B_{1}\partial_{x_{1}}(u-u_{a}^{\epsilon})-e^{i\phi_{3}/\epsilon}(u-u_{a}^{\epsilon})$
$\displaystyle\Rightarrow$
$\displaystyle\partial_{x_{2}}(u-u_{a}^{\epsilon})=B_{2}^{-1}[-\epsilon^{N}(L(\partial)+e^{i\phi_{3}/\epsilon}M)U_{N}-\partial_{t}(u-u_{a}^{\epsilon})-B_{1}\partial_{x_{1}}(u-u_{a}^{\epsilon})-e^{i\phi_{3}/\epsilon}(u-u_{a}^{\epsilon})]$

where we already have an estimate for the all the terms that appear on the
right, which gives:

(3.90) $|\partial_{x_{2}}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq
C\epsilon^{N-1}.$

Now we iterate this process by continue applying $\partial_{x_{2}}$ to (3.89).
This gives estimates to higher order $x_{2}$ derivative:

(3.91) $|\partial^{\alpha}_{x_{2}}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq
C\epsilon^{N-|\alpha|}$

Now, for $s=|\alpha|>\frac{n}{2}$, ($n=3$ here) we have:

(3.92) $|u-u_{a}^{\epsilon}|_{L^{\infty}(t,x)}\leq
C|u-u_{a}^{\epsilon}|_{H^{s}(t,x)}=C|\partial^{\alpha}(u-u_{a}^{\epsilon})|_{L^{2}(t,x)}\leq
C\epsilon^{N-|\alpha|}$

Now we recall that from (3.56), we can compute as many terms as we want in the
expansion of $u_{a}^{\epsilon}$. Therefore, for any $n\in{\mathbb{N}}$, we can
choose the highest order of approximate solution $N\geq n+2$ to conclude that:

(3.93) $|u-u_{a}^{\epsilon}|_{L^{\infty}(t,x)}\leq C\epsilon^{n}$

##  4\. Estimations for Weakly Stable Hyperbolic System

In this section we study transformed singular system (1.10). We assume that
the uniform Lopatinski condition fails only on the set
$\Upsilon_{0}=\\{\beta_{l},-\beta_{l}\\}\subset\mathcal{H}$, and that the
oscillatory term may have both positive and negative Fourier spectrum. In the
case where there is only one incoming mode, [Wil20a] Theorem 2.12 gives the
following estimate to the system (1.10).

(4.1)
$|U^{\gamma}|_{L^{2}(t,x,\theta)}+\left|\frac{U^{\gamma}(0)}{\sqrt{\gamma}}\right|_{L^{2}(t,x_{1},\theta)}\leq
K\left[\frac{1}{\gamma^{2}}\left(\sum_{k\in{\mathbb{Z}}}\left||X_{k}|\widehat{F_{k}}\right|^{2}_{L^{2}}\right)^{1/2}+\frac{1}{\gamma^{3/2}}\left(\sum_{k\in{\mathbb{Z}}}\left||X_{k}|\widehat{G_{k}}\right|^{2}_{L^{2}(\sigma,\eta)}\right)\right]$

In the case where there are at least two incoming modes, there is so far no
analogous theorem, but we have been able to prove such a result in the
small/medium frequency region (Theorem 4.19):

(4.2)
$|\chi_{D}U^{\gamma}|_{L^{2}(t,x,\theta)}+\left|\frac{\chi_{D}U^{\gamma}(0)}{\sqrt{\gamma}}\right|_{L^{2}(t,x_{1},\theta)}\leq
K\left[\frac{1}{\gamma^{2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{F_{k}}\right|^{2}_{L^{2}(x_{2},\sigma,\eta)})^{1/2}+\frac{1}{\gamma^{3/2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{G_{k}}\right|^{2}_{L^{2}(\sigma,\eta)})^{1/2}\right]$

where $\chi_{D}$ and $\chi$ are the characteristic functions corresponds to
the small/medium frequency region $|\zeta|\leq\epsilon^{\alpha-1}$,
$0<\alpha<1$. The first step is to prove a “small/medium frequency iteration
estimate” (proposition 4.17), and that is then used to prove the main result.

###  4.1. Extensions to $\Gamma_{\delta}$ and then to $\Xi$

[Wil20a]

We here extend the variables defined in section 2 in the same way as [Wil20a].
We recall that the $\Gamma^{+}_{\delta}(\beta_{l})$ is a conic neighborhood of
$\beta_{l}$, next we define the “opposite conic”
$\Gamma^{-}_{\delta}(\beta_{l})$:

(4.3)
$\displaystyle\Gamma^{-}_{\delta}(\beta_{l}):=\\{(\sigma-i\gamma,\eta):(-\sigma-i\gamma,-\eta)\in\Gamma^{+}_{\delta}(\beta_{l})\\}.$

We extend the eigenvalues $\omega_{j}$ and $R_{j}$ to the
$\Gamma_{\delta}(\beta_{l}):=\Gamma^{+}_{\delta}(\beta_{l})\cup\Gamma^{-}_{\delta}(\beta_{l})$
by the following equations:

(4.4) $\displaystyle
R_{j}(\sigma-i\gamma,\eta)=\overline{R}_{j}(-\sigma-i\gamma,-\eta)\text{ for
}(\sigma-i\gamma,\eta)\in\Gamma^{-}_{\delta}(\beta_{l}).$ (4.5)
$\displaystyle\omega_{j}(\sigma-i\gamma,\eta)=\overline{\omega}_{j}(-\sigma-i\gamma,-\eta)\text{
for }(\sigma-i\gamma,\eta)\in\Gamma^{-}_{\delta}(\beta_{l}).$

This extension keeps $R_{j}$ analytic in $\tau$, $C^{\infty}$ in $\eta$, and
homogeneous of degree $1$ in $\Gamma_{\delta}(\beta_{l})$, and $\omega_{j}$
analytic in $\tau$. We define the new normalized
$r_{j}:=\frac{R_{j}}{|R_{j}|}$ using the extended $R_{j}$. This also gives an
extension of the matrix $S(\zeta)$ defined in (2.13).

We extend $R_{j}$ to $\Xi$ in a way such that $S(\zeta)$ is invertible and
$S^{-1}$ is uniformly bounded. We extent $\omega_{j}$ to $\Xi$ such that the
extended $\omega_{j}$ are $C^{\infty}$, analytic in $\tau$, and homogeneous of
degree $1$.

We observe that $\mathcal{A}$ is real since each $A_{j}$ are, so we have

(4.6)
$\displaystyle\mathcal{A}(\zeta)R_{j}(\zeta)=\omega_{j}(\zeta)R_{j}(\zeta)\text{
on }\Gamma_{\delta}(\beta_{l}).$

By the way of extension (4.4), the following inequality still hods on
$\Gamma_{\delta}(\beta_{l})$:

(4.7)
$\displaystyle\begin{split}&\mathrm{Im}\;\omega_{j}(\zeta)\leq-c\gamma\text{
for }j\in\mathcal{O}\\\ &\mathrm{Im}\;\omega_{j}(\zeta)\geq c\gamma\text{ for
}j\in\mathcal{I},\end{split}$

for the same constant $c$ in (2.11). This implies that on
$\Gamma_{\delta}(\beta_{l})$, $R_{j}(\zeta)\in\mathbb{E}^{s}(\zeta)$
(definition 2.3) for $j\in\mathcal{I}$. We also have $\det
BR_{-}(\pm\beta_{l})=0$ since $R_{j}$s are real. 555The extensions in this
section can be applied in the conic neighborhood of any vector and doesn’t
rely on the failure of uniform Lopatinski condition.

###  4.2. Tools for Estimates

[Wil20a]

This section quoted earlier results and establish definitions to be used in
proving a uniform estimate. This section is established by Williams [Wil20a]
but with modified definitions 4.6 and 4.7.

######  Proposition 4.1.

[Wil20a] For $\gamma>0$, $\tau=\sigma-i\gamma$ we have on $x_{2}\geq 0$:

(4.8)
$\displaystyle\begin{split}&(a)\left|\int^{x_{2}}_{0}e^{-\gamma(x_{2}-s)}f(s,\tau,\eta)ds\right|_{L^{2}(x_{2},\sigma,\eta)}\leq\frac{1}{\gamma}|f|_{L^{2}(x_{2},\sigma,\eta)}\\\
&(b)\left|\int^{\infty}_{x_{2}}e^{\gamma(x_{2}-s)}f(s,\tau,\eta)ds\right|_{L^{2}(x_{2},\sigma,\eta)}\leq\frac{1}{\gamma}|f|_{L^{2}(x_{2},\sigma,\eta)}\\\
&(c)\left|\int^{\infty}_{0}e^{-\gamma
s}f(s,\tau,\eta)ds\right|_{L^{2}(\sigma,\eta)}\leq\frac{1}{\sqrt{2\gamma}}|f|_{L^{2}(x_{2},\sigma,\eta)}\\\
&(d)\left|e^{-\gamma
x_{2}}g(\tau,\eta)\right|_{L^{2}(x_{2},\sigma,\eta)}=\frac{1}{\sqrt{2\gamma}}|g|_{L^{2}(\sigma,\eta)}.\end{split}$

######  Notation 4.2.

  1. (1)

For $\zeta=(\tau,\eta)\in\Xi$, we define
$X_{k}:=\zeta+k\frac{\beta_{l}}{\epsilon}$.

  2. (2)

For any function $f(\zeta)$, we write:

(4.9) $f(\epsilon,k)=f(\epsilon,k)(\zeta):=f(X_{k})$

  3. (3)

We define $\chi_{b}(\zeta)$ be the characteristic function of
$\Gamma_{\delta}(\beta_{l})$. Thus, $\zeta\in\text{supp}\chi_{b}(\epsilon,k)$
if and only if $X_{k}\in\Gamma_{\delta}(\beta_{l})$.

An important tool we use is the following lemma from [Wil20a], which is an
easy consequence of assumption 2.14 and relevant definitions.

######  Lemma 4.3.

[Wil20a] For $\beta_{l}\in\Upsilon^{+}_{0}$. Recall that
$X_{k}:=\zeta+k\frac{\beta_{l}}{\epsilon}$ and that
$\zeta\in\mathrm{supp}\;\chi_{b}(\epsilon,k)\Leftrightarrow
X_{k}\in\Gamma_{\delta}(\beta_{l})$. For $k\in{\mathbb{Z}}$ the following
estimates hold:

(4.10) $\displaystyle\begin{split}&(a)\;|\Delta(\epsilon,k)|\lesssim 1\text{
on }\Xi\\\
&(b)\;\left|[Br_{-}(\epsilon,k)]^{-1}\right|\lesssim|\Delta(\epsilon,k)|^{-1}\sim\frac{|X_{k}|}{|\tau-
c_{+}(\beta)\eta|}\text{ on }\mathrm{supp}\;\chi_{b}(\epsilon,k)\\\
&(c)\;\left|[Br_{-}(\epsilon,k)]^{-1}\right|\lesssim|\Delta(\epsilon,k)|^{-1}\leq
C(\delta)\text{ on }\Xi\setminus\mathrm{supp}\;\chi_{b}(\epsilon,k)\\\
&(d)\;|Br_{\pm}(\epsilon,k)|\lesssim 1\text{ on }\Xi,\\\
&(e)\;|\omega_{i}(\epsilon,k)-\omega_{j}(\epsilon,k)|\sim|X_{k}|\text{ for
}i\neq j\text{ on }\mathrm{supp}\;\chi_{b}(\epsilon,k)\\\
&(f)\;\mathrm{Im}\;\omega_{j}(\epsilon,k)\leq-c\gamma\text{ for
}j\in\mathcal{O},\;\;\;\mathrm{Im}\;\omega_{j}(\epsilon,k)\geq c\gamma\text{
for }j\in\mathcal{I},\text{ on
}\mathrm{supp}\;\chi_{b}(\epsilon,k).\end{split}$

(g) Let $r\in{\mathbb{Z}}\setminus 0$. When
$X_{k}\notin\Gamma_{\frac{\delta}{|r|}}(\beta)$ we have
$|\Delta(\epsilon,k)|^{-1}\leq C(\delta)|r|$.

We also have the following lemma:

######  Lemma 4.4.

[Wil20a] Let $X_{k}=\zeta+k\frac{\beta_{l}}{\epsilon}$,
$X_{k-r}=\zeta+(k-r)\frac{\beta_{l}}{\epsilon}$, where
$r\in{\mathbb{Z}}\setminus 0$. For $\delta\in(0,\delta_{0}]$ and
$N_{1}\in{\mathbb{N}}$ sufficiently large, assume that

(4.11) $\displaystyle
X_{k}\in\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l}),\text{ but
}X_{k-r}\notin\Gamma_{\frac{\delta}{|r|}}(\beta_{l}).$

Then $|X_{k-r}|\lesssim\frac{1}{N_{1}}|X_{k}|$.

We next define $E_{i,j}$ which is one main tool to be used to control the
amplification of $\Delta^{-1}$.

######  Definition 4.5.

[Wil20a] Let $k\in{\mathbb{Z}}$, $r\in{\mathbb{Z}}\setminus 0$. For
$\zeta=(\tau,\eta)\in\Xi$ we define the function of $\zeta$:

(4.12)
$E_{i,j}(\epsilon,k,k-r)(\zeta):=\omega_{i}(\epsilon,k)-\frac{r\omega_{N}(\beta_{l})}{\epsilon}-\omega_{j}(\epsilon,k-r),\text{
where }i\in\mathcal{O},j\in\mathcal{I}$

We separately study the following three cases for the pair
$X_{k}=\zeta+k\frac{\beta_{l}}{\epsilon}$,
$X_{k-r}=\zeta+(k-r)\frac{\beta_{l}}{\epsilon}$ for $\delta$ small and $N_{1}$
large:

(4.13)
$\displaystyle\begin{split}&(I)\;X_{k}\in\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l}),X_{k-r}\in\Gamma_{\frac{\delta}{|r|}}(\beta_{l})\\\
&(II)\;X_{k}\in\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l}),X_{k-r}\notin\Gamma_{\frac{\delta}{|r|}}(\beta_{l})\\\
&(III)\;X_{k}\in\Gamma_{\delta}(\beta_{l})\setminus\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l})\end{split}$

We proceed to define a global and local amplification factor which is used to
control the $\frac{1}{\epsilon}$ term. We’ll see later how the proof of
iteration estimate gives rise to this term.

######  Definition 4.6.

[Microlocal amplification factors] Let $C_{5}\geq 1$, $N_{1}$ be sufficiently
large constants, and let $0<\xi<1$ denote a number to be chosen (in
proposition 4.11),

For $k\in{\mathbb{Z}}$, $r\in{\mathbb{Z}}\setminus 0$, and
$(\zeta,\epsilon)\in\mathrm{supp}\;\chi_{b}(\epsilon,k)\times(0,\epsilon_{0}]$,
we define:

$\bullet$ $D(\epsilon,k,k-r)(\zeta)=C_{5}|r|$ in case $(III)$

$\bullet$ $D(\epsilon,k,k-r)(\zeta)=C_{5}|r|$ in case $(II)$.

If case $(I)$ obtains, then define

$\bullet$ $D(\epsilon,k,k-r)(\zeta)=\begin{cases}C_{5}|r|^{2+\delta},\text{
when }|r|\leq\epsilon^{-\xi}\\\ \frac{C_{5}|r|}{\epsilon\gamma}\text{
otherwise}\end{cases}$

If $\zeta\notin\mathrm{supp}\;\chi_{b}(\epsilon,k)$, define
$D(\epsilon,k,k-r)(\zeta)=0$.

######  Definition 4.7.

[Global amplification factors] For $k\in{\mathbb{Z}}$,
$r\in{\mathbb{Z}}\setminus 0$, and
$(\zeta,\epsilon)\in\mathrm{supp}\;\chi_{b}(\epsilon,k)\times(0,\epsilon_{0}]$,
we define

(4.14)
$\displaystyle\begin{split}&{\mathbb{D}}(\epsilon,k,k-r)(\zeta)=\begin{cases}\frac{C_{5}|r|}{\epsilon\gamma},\text{
if }D(\epsilon,k,k-r)(\zeta)=\frac{C_{5}|r|}{\epsilon\gamma}\\\
C_{5}|r|^{2+\delta},\;otherwise\end{cases}.\end{split}$

If $\zeta\notin\mathrm{supp}\;\chi_{b}(\epsilon,k)$ we set
${\mathbb{D}}(\epsilon,k,k-r)(\zeta)=1$.

We observe that by definition

(4.15) $\displaystyle
D(\epsilon,k,k-r)(\zeta)\leq{\mathbb{D}}(\epsilon,k,k-r)\text{ for all
}\zeta\in\Xi$

Recall that $V_{k}:=\widehat{U}_{k}(\zeta,x_{2})$, the Laplace-Fourier
transform in $(t,x_{1})$ of $U_{k}(t,x)$. Let $\chi_{g}$ be the characteristic
function of $\Gamma_{\delta}(\beta_{l})^{c}$ (the compliment of
$\Gamma_{\delta}(\beta_{l})$). We can write using notation 4.2:

(4.16)
$V_{k}(x_{2},\zeta)=\chi_{b}(\epsilon,k)V_{k}+\chi_{g}(\epsilon,k)V_{k}.$

Define $w_{k}=(w_{k}^{+},w_{k}^{-})$ for all $\zeta\in\Xi$ by

(4.17) $V_{k}(x_{2},\zeta)=S(\epsilon,k)w_{k}(x_{2},\zeta)$

where $S(\zeta)$ as in (2.13). For $\zeta\in\text{supp}\chi_{b}(\epsilon,k)$,
by definitions and obvious transforming (1.10), we see that $w_{k}$ is a
solution of the diagonalized system:

(4.18)
$\displaystyle\begin{split}&D_{x_{2}}w_{k}-\begin{pmatrix}\xi_{+}(\epsilon,k)&0\\\
0&\xi_{-}(\epsilon,k)\end{pmatrix}w_{k}=\\\ &\qquad
i\sum_{r\in{\mathbb{Z}}\setminus
0}\alpha_{r}e^{ir\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}}S^{-1}(\epsilon,k)B_{2}^{-1}MS(\epsilon,k-r)w_{k-r}+S^{-1}(\epsilon,k;\beta)\widehat{F^{\epsilon}_{k}}(x_{2},\zeta),\\\
&BS(\epsilon,k)w_{k}=\hat{G}_{k}\text{ on }x_{2}=0.\end{split}$

where

(4.19)
$\displaystyle\begin{split}&\xi_{+}(\epsilon,k)=\mathrm{diag}\;(\omega_{1}(\epsilon,k),...,\omega_{N-p}(\epsilon,k))\\\
&\xi_{-}(\epsilon,k)=\mathrm{diag}\;(\omega_{N-p+1}(\epsilon,k),...,\omega_{N}(\epsilon,k)).\end{split}$

Solutions to (4.18) for $F=0$ satisfy:

(4.20) $\displaystyle w^{+}_{k}(x_{2},\zeta)=\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{\infty}_{x_{2}}e^{i\xi_{+}(\epsilon,k)(x_{2}-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}[a(\epsilon,k,k-r)w^{+}_{k-r}(s,\zeta)+b(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)]ds,$
(4.21)
$\displaystyle\begin{split}&w^{-}_{k}(x_{2},\zeta)=-\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{x_{2}}_{0}e^{i\xi_{-}(\epsilon,k)(x_{2}-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}[c(\epsilon,k,k-r)w^{+}_{k-r}(s,\zeta)+d(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)]ds-\\\
&e^{i\xi_{-}(\epsilon,k)x_{2}}[Br_{-}(\epsilon,k)]^{-1}Br_{+}(\epsilon,k)\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{\infty}_{0}e^{i\xi_{+}(\epsilon,k)(-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}[a(\epsilon,k,k-r)w^{+}_{k-r}(s,\zeta)+\\\
&\qquad\qquad\qquad
b(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)]ds+e^{i\xi_{-}(\epsilon,k)x_{2}}[Br_{-}(\epsilon,k)]^{-1}\hat{G}_{k}(\zeta).\end{split}$

The matrices $a,b,c,d$ in (4.20), (4.21) are block matrix such that:

(4.22) $\displaystyle\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=S(\epsilon,k)^{-1}B_{2}^{-1}MS(\epsilon,k-r)$

and we have

(4.23) $\displaystyle|a(\epsilon,k,k-r)|\lesssim 1,|b(\epsilon,k,k-r)|\lesssim
1,|c(\epsilon,k,k-r)|\lesssim 1,|d(\epsilon,k,k-r)|\lesssim 1$

We also have the following form of equations for $w^{\pm}_{k}$:

(4.24) $\displaystyle w^{+}_{k}(x_{2},\zeta)=\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{\infty}_{x_{2}}e^{i\xi_{+}(\epsilon,k)(x_{2}-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}M^{+}(\epsilon,k,k-r)V_{k-r}(s,\zeta)ds$
(4.25)
$\displaystyle\begin{split}&w^{-}_{k}(x_{2},\zeta)=-\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{x_{2}}_{0}e^{i\xi_{-}(\epsilon,k)(x_{2}-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}M^{-}(\epsilon,k,k-r)V_{k-r}(s,\zeta)ds-\\\
&e^{i\xi_{-}(\epsilon,k)x_{2}}[Br_{-}(\epsilon,k)]^{-1}Br_{+}(\epsilon,k)\sum_{r\in{\mathbb{Z}}\setminus
0}\int^{\infty}_{0}e^{i\xi_{+}(\epsilon,k)(-s)+ir\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}M^{+}(\epsilon,k,k-r)V_{k-r}(s,\zeta)ds+\\\
&\qquad\qquad
e^{i\xi_{-}(\epsilon,k)x_{2}}[Br_{-}(\epsilon,k)]^{-1}\hat{G}_{k}(\zeta),\end{split}$

where the definitions of matrices $M^{\pm}$ are obvious and
$|M^{\pm}(\epsilon,k,k-r)|\lesssim 1$.

Next, we define

(4.26)
$\displaystyle\mathcal{W}_{k}(x_{2},\zeta)=(\tilde{w}^{+}_{k}(x_{2},\zeta),w^{-}_{k}(x_{2},\zeta),\frac{1}{\sqrt{\gamma}}\tilde{w}^{+}_{k}(0,\zeta),\frac{1}{\sqrt{\gamma}}w^{-}_{k}(0,\zeta)),$

where

(4.27)
$\displaystyle\tilde{w}^{+}_{k}(x_{2},\zeta)=\Delta^{-1}(\epsilon,k)w^{+}_{k}(x_{2},\zeta).$

For each $k$ we define a modified $L^{2}$ norm of $V_{k}$ by666In (4.28) the
notation $|\cdot|_{L^{2}}$ means $|\cdot|_{L^{2}(x_{2},\sigma,\eta)}$ for
components that depend on $x_{2}$ and $|\cdot|_{L^{2}(\sigma,\eta)}$ for
components that do not.

(4.28)
$\displaystyle\|V_{k}\|_{k}=|\chi_{b}(\epsilon,k)\mathcal{W}_{k}(x_{2},\zeta)|_{L^{2}}+\left|\chi_{g}(\epsilon,k)\left(V_{k}(x_{2},\zeta),\frac{1}{\sqrt{\gamma}}V_{k}(0,\zeta)\right)\right|_{L^{2}},$

We note that we ultimately want to estimate $|(\|V_{k}\|)|_{\ell^{2}(k)}$ of
$(V_{k})_{k\in{\mathbb{Z}}}$ (theorem 4.19). We usually write $\|V_{k}\|$
$\|V_{k}\|_{k}$. If $f(\zeta)$ is any function of $\zeta$ then

(4.29)
$\displaystyle\|f(\zeta)V_{k}\|_{k}:=|\chi_{b}(\epsilon,k)f(\zeta)\mathcal{W}_{k}(x_{2},\zeta)|_{L^{2}}+\left|\chi_{g}(\epsilon,k)f(\zeta)\left(V_{k}(x_{2},\zeta),\frac{1}{\sqrt{\gamma}}V_{k}(0,\zeta)\right)\right|_{L^{2}}.$

Note that for $U^{\gamma}$ as in Theorem 4.19

(4.30)
$\displaystyle|(\|V_{k}\|)|_{\ell^{2}(k)}\gtrsim|U^{\gamma}|_{L^{2}(t,x,\theta)}+\left|\frac{U^{\gamma}(0)}{\sqrt{\gamma}}\right|_{L^{2}(t,x_{1},\theta)}.$

###  4.3. Analysis of Small/Medium Frequency Region

Fix $0<\alpha<1$, we now we consider the effect of $\zeta$ small such that

(4.31) $|\zeta|\lesssim\epsilon^{\alpha-1}\quad\text{for }0<\alpha<1$

We recall (definition 2.11) that the phases $(\phi_{j},\phi_{N},\phi_{i})$
exhibit a resonance if there exist $p,q\in{\mathbb{Z}}\setminus 0$ such that:

(4.32) $p\phi_{j}+q\phi_{N}=(p+q)\phi_{i}\iff
p\omega_{j}(\beta_{l})+q\omega_{N}(\beta_{l})=(p+q)\omega_{i}(\beta_{l})\iff\frac{p}{q}=\frac{\omega_{i}(\beta_{l})-\omega_{j}(\beta_{l})}{\omega_{j}(\beta_{l})-\omega_{N}(\beta_{l})}=\Omega_{i,j}$

Proposition 4.6 in [Wil20a] showed that it is impossible to have the form of
estimates in (4.34) or (4.35) if there exists a resonance. This motivates us
to make an assumption 4.9 and study the case where $p/q$ is away from
$\Omega_{i,j}$.

######  Proposition 4.8.

  1. (1)

Thue-Siegel-Roth theorem. If $\Omega$ is an algebraic number, then there
exists a positive constant $C(x,\delta)$ such that for any
$p,q\in{\mathbb{Z}}$, $q\neq 0$, we have:

(4.33)
$\displaystyle\left|\frac{p}{q}-\Omega\right|>\frac{C(x,\delta)}{|q|^{2+\delta}}\quad\delta>0$

  2. (2)

There is a full measure set of irrational numbers such that (4.33) holds.

######  Assumption 4.9.

$\forall i\in\mathcal{O},j\in\mathcal{I}\setminus\\{N\\}$, the irrational
number
$\Omega_{i,j}=\frac{\omega_{i}(\beta_{l})-\omega_{j}(\beta_{l})}{\omega_{j}(\beta_{l})-\omega_{N}(\beta_{l})}$
(definition (2.16)) satisfies (4.33).

The following lemma is clear.

######  Lemma 4.10.

For $A,B\in{\mathbb{R}}$, there exists $\epsilon_{0}$ small such that for
$0<\epsilon<\epsilon_{0}$, we have
$A\epsilon^{-\alpha}+B\epsilon^{-\beta}\sim\epsilon^{-\beta}$ if
$0<\alpha<\beta$ and $B\neq 0$.

######  Proposition 4.11.

Suppose there is no resonance,
$X_{k}\in\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l}),X_{k-r}\in\Gamma_{\frac{\delta}{|r|}}(\beta_{l})$.
We fix arbitrary $0<\alpha<1$, and then fix arbitrary $0\leq\xi<\alpha$. There
exists $\epsilon_{0}>0$ such that for all $0<\epsilon<\epsilon_{0}$ and
$|\zeta|\lesssim\epsilon^{\alpha-1}$, there exists $C$ such that we have the
following:

  1. (1)

For all $r$ such that $|r|\leq M$ for any fixed $M>0$, we have:

(4.34) $|E_{i,j}(\epsilon,k,k-r)(\zeta)|\geq C|X_{k-r}(\zeta)|$

  2. (2)

For all $|r|\lesssim\epsilon^{-\xi}$, if $\Omega_{i,j}$ satisfies assumption
4.9, for $j\neq N$ we have:

(4.35) $|E_{i,j}(\epsilon,k,k-r)(\zeta)|\geq
C\frac{|X_{k}(\zeta)|}{|r|^{2+\delta}}$

######  Remark 4.12.

It is proved in [Wil20a] prop 3.22 that for $j=N$:

(4.36) $|E_{i,N}(\epsilon,k,k-r)(\zeta)|\geq C\frac{|X_{k}(\zeta)|}{|r|}\text{
or }|E_{i,N}(\epsilon,k,k-r;\beta)|\geq C|X_{k-r}|$

Before the proof of proposition (4.11), we first establish a set of notations
and results from [Wil20a]:

We recall that $X_{k}=\zeta+k\frac{\beta_{l}}{\epsilon}$. Define
$\tilde{X_{k}}$, the orthogonal projection of $X_{k}$ onto the line
$\mathcal{L}(\beta_{l})=\\{t\beta_{l}:t\in{\mathbb{R}}\\}$:

(4.37)
$\tilde{X}_{k}:=(X_{k}\cdot\beta_{l})\beta_{l}=s\frac{\beta_{l}}{\epsilon}+k\frac{\beta_{l}}{\epsilon}$

where $s\frac{\beta_{l}}{\epsilon}$ is the orthogonal projection of $\zeta$ on
$\beta_{l}$. Recall that

(4.38) $\displaystyle
E_{i,j}(\epsilon,k,k-r)=\omega_{i}(X_{k})-r\frac{\omega_{N}(\beta_{l})}{\epsilon}-\omega_{j}(X_{k-r}).$

We define:

(4.39)
$\displaystyle\tilde{E}_{i,j}(\epsilon,k,k-r)=\omega_{i}(\tilde{X}_{k})-r\frac{\omega_{N}(\beta_{l})}{\epsilon}-\omega_{j}(\tilde{X}_{k-r}).$

######  Lemma 4.13.

[Wil20a] For $\tilde{X}_{k}$, $\tilde{E}_{i,j}(\epsilon,k,k-r)$ defined above,
letting $t=s+k-r$, we have:

(4.40) $\displaystyle\tilde{E}_{i,j}(\epsilon,k,k-r)$
$\displaystyle=\frac{t-t_{p}}{\epsilon}(\omega_{i}(\beta_{l})-\omega_{j}(\beta_{l})):=\frac{t-t_{p}}{\epsilon}C(\beta_{l})$
(4.41) $\displaystyle=\frac{s+k-r-r\Omega_{i,j}}{\epsilon}C(\beta_{l})$

where $t_{p}(\beta_{l})=r\Omega_{i,j}(\beta_{l})$,
$\Omega_{i,j}(\beta_{l}):=\frac{\omega_{i}(\beta_{l})-\omega_{N}(\beta_{l})}{\omega_{j}(\beta_{l})-\omega_{i}(\beta_{l})}$,
and it follows that

(4.42)
$|E_{i,j}(\epsilon,k,k-r)|\geq|\tilde{E}_{i,j}(\epsilon,k,k-r)|-C(|X_{k}-\tilde{X}_{k}|+|X_{k-r}-\tilde{X}_{k-r}|).$

Now we’re ready to prove proposition 4.11:

###### Proof.

of proposition 4.11.
(1) The case where there are finitely many $r$’s ($|r|\leq M$).

Since there is no resonance, $\Omega_{i,j}$ is irrational. Picking
$k_{0},r_{0}$ such that

(4.43)
$C(\Omega_{i,j},M):=\left|\frac{k_{0}-r_{0}}{r_{0}}-\Omega_{i,j}\right|\leq\left|\frac{k-r}{r}-\Omega_{i,j}\right|\text{
for all }k\in{\mathbb{Z}},|r|\leq M$

we have:

(4.44) $\displaystyle\frac{|k-r-r\Omega_{i,j}|}{\epsilon}\geq
C(\Omega_{i,j},M)\epsilon^{-1}.$

We rewrite equation (4.40) as:

(4.45)
$\displaystyle\tilde{E}_{i,j}(\epsilon,k,k-r)(\zeta)=\left(\frac{s}{\epsilon}\right)C(\beta_{l})+\left(\frac{k-r-r\Omega_{i,j}}{\epsilon}\right)C(\beta_{l}).$

By lemma 4.10, inequality (4.44), and
$\frac{s}{\epsilon}\sim\zeta\lesssim\epsilon^{\alpha-1}$, the second term
above absorbs the first, that is

(4.46)
$\displaystyle|\tilde{E}_{i,j}(\epsilon,k,k-r)|\gtrsim\frac{|k-r-r\Omega_{i,j}|}{\epsilon}.$

Using equation (4.42):

(4.47) $\displaystyle|E_{i,.j}(\epsilon,k,k-r)|$
$\displaystyle\geq|\tilde{E}_{i,j}(\epsilon,k,k-r)|-C(|X_{k}-\tilde{X}_{k}|+|X_{k-r}-\tilde{X}_{k-r}|)$

and

(4.48)
$|\tilde{X}_{k-r}-X_{k-r}|=|\tilde{X}_{k}-X_{k}|=\left|\frac{s}{\epsilon}-\zeta\right|\sim\epsilon^{\alpha-1}$

we have by lemma 4.10

(4.49)
$\displaystyle|{E}_{i,j}(\epsilon,k,k-r)|\gtrsim\frac{|k-r-r\Omega_{i,j}|}{\epsilon}.$

If $k=0$, we have

(4.50)
$\displaystyle|X_{k-r}|\lesssim\frac{|k-r|}{\epsilon}\sim\frac{|r|}{\epsilon},\quad|{E}_{i,j}(\epsilon,k,k-r)|\gtrsim\frac{|k-r-r\Omega_{i,j}|}{\epsilon}\sim\frac{|r|}{\epsilon}$

which proves the proposition. If $k\neq 0$, we note that for every $r$, there
is exact one $k(r)\neq 0$ such that

(4.51)
$\displaystyle\left|1-\frac{r+r\Omega_{i,j}}{k(r)}\right|\leq\left|1-\frac{r+r\Omega_{i,j}}{k}\right|\quad\text{
for }k\in{\mathbb{Z}}\setminus 0.$

We define

(4.52) $C_{0}=\inf_{|r|\leq M}\left|\frac{k(r)-r-r\Omega_{i,j}}{k(r)}\right|$

so

(4.53)
$C_{0}|k|\leq\left|1-\frac{r+r\Omega_{i,j}}{k}\right||k|=|k-r-r\Omega_{i,j}|\text{
for all }k.$

Equations (4.49) then gives

(4.54)
$|{E}_{i,j}(\epsilon,k,k-r)|\gtrsim\frac{|k-r-r\Omega_{i,j}|}{\epsilon}\geq
C_{0}\frac{|k|}{\epsilon}.$

Also using

(4.55)
$\displaystyle|X_{k-r}|\lesssim\frac{|k-r|}{\epsilon}\leq\frac{|k|}{\epsilon}+\frac{M}{\epsilon}\leq(M+1)\frac{|k|}{\epsilon}$

above two equations show that (4.34) holds, which finishes the proof of part
(1) of the proposition.

(2) The case where the number of $r$’s depend on $\epsilon$, $|r|\leq
C_{0}\epsilon^{-\xi}$, for $0\leq\xi<\alpha$.

Since there is no resonance, $\Omega_{i,j}$ is irrational. If $\Omega_{i,j}$
further satisfies assumption 4.9, (4.33) implies that

(4.56)
$\displaystyle\left|\frac{k-r}{r}-\Omega_{i,j}\right|\geq\frac{C(\Omega_{i,j},\delta)}{|r|^{2+\delta}}\quad\text{
for }\delta>0.$

Using $|r|\leq C_{0}\epsilon^{-\xi}$, we have:

(4.57)
$\displaystyle\frac{|k-r-r\Omega_{i,j}|}{\epsilon}\geq\frac{C(\Omega,_{i,j}\delta)}{\epsilon|r|^{1+\delta}}\gtrsim\epsilon^{\xi+\xi\delta-1}.$

We write $\tilde{E}_{i,j}$ in the same way as before:

(4.58)
$\displaystyle\tilde{E}_{i,j}(\epsilon,k,k-r)(\zeta)=\left(\frac{s}{\epsilon}\right)C(\beta_{l})+\left(\frac{k-r-r\Omega_{i,j}}{\epsilon}\right)C(\beta_{l}).$

Since $0\leq\xi<\alpha$, $\xi+\xi\delta-1<\alpha-1$ for $\delta$ small, by
lemma 4.10, the second term absorbs the first, that is

(4.59)
$\displaystyle|\tilde{E}_{i,j}(\epsilon,k,k-r)|\gtrsim\frac{|k-r-r\Omega_{i,j}|}{\epsilon}.$

Since we have

(4.60)
$|\tilde{X}_{k-r}-X_{k-r}|=|\tilde{X}_{k}-X_{k}|=\left|\frac{s}{\epsilon}-\zeta\right|\sim\epsilon^{\alpha-1}$

and equation (4.42)

(4.61) $\displaystyle|E_{i,.j}(\epsilon,k,k-r)|$
$\displaystyle\geq|\tilde{E}_{i,j}(\epsilon,k,k-r)|-C(|X_{k}-\tilde{X}_{k}|+|X_{k-r}-\tilde{X}_{k-r}|)$

we obtain using lemma 4.10 and equations (4.59),(4.60), (4.61) :

(4.62)
$\displaystyle|{E}_{i,j}(\epsilon,k,k-r)|\gtrsim\frac{|k-r-r\Omega_{i,j}|}{\epsilon}.$

Suppose $k=0$, above equation becomes
$|{E}_{i,j}(\epsilon,k,k-r)|\gtrsim\epsilon^{\xi+\xi\delta-1}$ by (4.56), and
we have $|X_{k}|=|\zeta|\sim\epsilon^{\alpha-1}$. Thus, lemma 4.10 implies the
desired inequality.
Now for $k\neq 0$, we claim that there exists $C$ independent of
$k,r,\epsilon$ such that:

(4.63) $|k-r-r\Omega_{i,j}|\geq C\frac{|k|}{|r|^{2+\delta}}.$

Observe that by lemma 4.10 we have:

(4.64) $\displaystyle|X_{k}|$
$\displaystyle=\left|\zeta+k\frac{\beta_{l}}{\epsilon}\right|\lesssim\frac{|k|}{\epsilon}.$

Therefore, the claim and above equation directly imply the result of
proposition part (2). Now It is only left to prove the claim.

Proof of claim.

For each $r$ such that $|r|\leq C_{0}\epsilon^{-\xi}$. Let $k^{+}(r),k^{-}(r)$
be the integers such that

(4.65) $\displaystyle(a)\ k^{+}(r)-r-r\Omega_{i,j}>0,\quad
k^{-}(r)-r-r\Omega_{i,j}<0.$ (4.66) $\displaystyle(b)\ |k^{+}(r)-k^{-}(r)|=1.$
(4.67) $\displaystyle(c)\ \text{One of }k^{+}(r),k^{-}(r)\text{ satisfies
}|k^{\pm}(r)-r-r\Omega_{i,j}|=\min_{k\in{\mathbb{Z}}}|k-r-r\Omega_{i,j}|.$

Since there is no resonance and $\Omega_{i,j}$ is irrational,
$|k(r)^{\pm}-r-r\Omega_{i,j}|>0$. The proof of the claim has two steps:

Step (1). We claim that for all $r$ and for $k(r)=k(r)^{+}$ satisfying (4.65)
(the same argument shows the case when $k(r)=k(r)^{-}$), there is a constant
$C$ independent of $r$ such that:

(4.68) $C\frac{|k(r)|}{|r|^{2+\delta}}\leq|k(r)-r-r\Omega_{i,j}|.$

Using (4.56), it is sufficient to show:

(4.69)
$C\frac{|k(r)|}{|r|^{2+\delta}}\leq\frac{C(\Omega_{i,j},\delta)}{|r|^{1+\delta}}.$

If $k(r)=0$, above equation obviously holds, otherwise it holds if:

(4.70) $C\leq\frac{|r|}{|k(r)|}.$

By the definition of $k(r)^{+}$, as $r\rightarrow\infty$,
$\frac{|r|}{|k(r)|}\rightarrow\left|\frac{1}{1+\Omega_{i,j}}\right|$. Thus,
there exists a positive constant $C$ independent of $r$ such that above
equation holds, and thus (4.68) holds.

Step (2). For a fixed $r=r_{0},\ k_{0}=k(r_{0})^{+}$, we claim that for every
$k\geq k_{0}^{+}$:

(4.71) $C\frac{|k|}{|r_{0}|^{2+\delta}}\leq|k-r_{0}-r_{0}\Omega_{i,j}|.$

Let $z=k-k_{0}$, we rewrite above equation as:

(4.72)
$C\frac{|k_{0}+z|}{|r_{0}|^{2+\delta}}\leq|k_{0}+z-r_{0}-r_{0}\Omega_{i,j}|.$

We observe that the left hand side satisfies

(4.73) $C\frac{|k_{0}+z|}{|r_{0}|^{2+\delta}}\leq
C\frac{|k_{0}|}{|r_{0}|^{2+\delta}}+C\frac{|z|}{|r_{0}|^{2+\delta}}\leq|k_{0}-r_{0}-r_{0}\Omega_{i,j}|+C|z|.$

Let $x=k_{0}-r_{0}-r_{0}\Omega_{i,j}$. It is sufficient to show

(4.74) $|x|+C|z|\leq|x+z|.$

If $z=0$ it’s obviously true. Otherwise, shrink $C$ if necessary so that:

(4.75) $C\leq\frac{|x+z|-|x|}{|z|}.$

The definition of $k(r_{0})^{+}$ implies $x>0$, and thus for every $z>0$, the
right hand side of above equation equals to 1 and is thus satisfied by the
same $C$ established in proof of claim step (1) (for $C<1$). Thus equation
(4.71) holds for all $k>k_{0}^{+}$.

Using the same argument in this step, we can show that for all $k<k_{0}^{-}$,
(4.71) holds. Since $|k_{0}^{+}-k_{0}^{-}|=1$, for all integers $k$, equation
(4.71) holds. This finishes the proof of the claim, and thus the proposition.

∎

######  Remark 4.14.

Whenever we used lemma 4.10 to absorb a smaller term, the constant and
$\epsilon_{0}$ we chose only depend on the upper bound of the coefficient of
the term being absorbed. Therefore since we are only absorbing terms whose
coefficient is bounded above, the constant and $\epsilon_{0}$ chosen does not
dependent on $r,k$.

###  4.4. Uniform Estimation Theorem

Let $\chi(\sigma,\gamma,\eta)$ be the characteristic function on the set
$|\zeta|\leq\epsilon^{\alpha-1}$, that is:

(4.76)
$\displaystyle\chi(\sigma,\gamma,\eta)=\begin{cases}1\quad|\zeta|\leq\epsilon^{\alpha-1}\\\
0\quad\text{otherwise}\end{cases}$

Let $\chi_{D}$ be the Fourier multiplier associated to
$\chi(\sigma,\gamma,\eta)$, ie. $\chi_{D}$ is defined by:

(4.77)
$\widehat{(\chi_{D}U^{\gamma})}_{k}=\chi(\sigma,\gamma,\eta)V_{k}=\chi(\sigma,\gamma,\eta)\widehat{U}_{k}$

Now we are ready to prove the iteration estimate for the singular system
(1.10) on the small/medium frequency region which is:

(4.78)
$\displaystyle\begin{split}&D_{x_{2}}V_{k}-\mathcal{A}(X_{k})V_{k}=i\sum_{r\in{\mathbb{Z}}\setminus
0}\alpha_{r}e^{ir\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}}B_{2}^{-1}MV_{k-r}+\chi\widehat{F_{k}}(x_{2},\zeta)\\\
&BV_{k}=\chi\widehat{G_{k}}(\zeta)\text{ on }x_{2}=0.\end{split}$

######  Remark 4.15.

At this point, we don’t know the existence of solutions to the system (4.78).
Therefore, the following two estimates (4.79) and (4.106) will be presented as
a priori estimates.

######  Remark 4.16.

We here point out again that the assumption of more than one incoming modes
$\phi_{N-p},...\phi_{N}$ raised new difficulties in the analysis that the
theorem 2.12 in [Wil20a] didn’t treat. The major differences between the proof
of the uniform estimate in this paper and the proof in [Wil20a] are the
following, where we take advantage of the small/medium frequency cutoff
$\chi$.

  1. (1)

Using proposition 4.11, we ”control” the amplification factor for $|r|\leq
C_{0}\epsilon^{-\xi}$ in step (4) of the proof of proposition 4.17.

  2. (2)

We control the amplification factor for the remaining $r$’s using the decay of
the Fourier coefficients of $\mathcal{D}$, namely $\alpha_{r}$. This is done
in the proof of the main estimation theorem 4.19.

######  Proposition 4.17 (Iteration estimate).

Let $U^{\gamma}(t,x,\theta)\in H^{2}(\Omega\times\mathbb{T})$ for $\gamma\geq
1$ and set $V_{k}(x_{2},\zeta)=\widehat{U}_{k}(\zeta,x_{2})$. Assume that
$\Upsilon_{0}=\\{\beta_{l},-\beta_{l}\\}$, and assumptions 4.9, 2.1, 2.2, and
2.14 are satisfied. There exist positive constants $C$, $\gamma_{0}$ such that
for $\gamma\geq\gamma_{0}$, $V_{k}$ satisfies for all $k\in{\mathbb{Z}}$,

(4.79) $||\chi V_{k}||\leq\frac{C}{\gamma}\sum_{r\in{\mathbb{Z}}\setminus
0}\sum_{t\in{\mathbb{Z}}}||\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)\chi
V_{k-r-t}||+\frac{C}{\gamma^{2}}\left|\chi\widehat{F_{k}}|X_{k}|\right|_{L^{2}}+\frac{C}{\gamma^{3/2}}\left|\chi\widehat{G_{k}}|X_{k}|\right|_{L^{2}}$

with $\widehat{F_{k}}(x_{2},\zeta)$ and $\widehat{G_{k}}(\zeta)$ defined as:

(4.80)
$\displaystyle\begin{split}\widehat{F_{k}}(x_{2},\zeta):=&D_{x_{2}}V_{k}-\mathcal{A}(X_{k})V_{k}-i\sum_{r\in{\mathbb{Z}}\setminus
0}\alpha_{r}e^{ir\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}}B_{2}^{-1}MV_{k-r}\\\
\widehat{G_{k}}(\zeta):=&BV_{k}(0,\zeta)\end{split}$

Here we redefine $\alpha_{0}$ to be $1$. 777In (4.79)
$\|V_{k}\|=\|V_{k}\|_{k}$ and
$\|\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)V_{k-r-t}\|=\|\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)V_{k-r-t}\|_{k-r-t}$.

The proof of proposition 4.79 relies on a decomposition of the support of
$\chi_{b}(\epsilon,k)$. For a given $r\in{\mathbb{Z}}\setminus 0$ we write :

(4.81)
$\displaystyle\chi_{b}(\epsilon,k)=\chi^{1}_{b}(\epsilon,k,k-r)+\chi^{2}_{b}(\epsilon,k,k-r)+\chi^{3}_{b}(\epsilon,k,k-r),$

where $\chi^{i}_{b}(\epsilon,k,k-r)$, $i=1,2,3$ are respectively the
characteristic functions of

(4.82)
$\displaystyle\begin{split}&\mathcal{A}_{1}(\epsilon,k,k-r):=\\{\zeta\in\Xi:X_{k}\in\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l}),X_{k-r}\in\Gamma_{\frac{\delta}{|r|}}(\beta_{l})\\}\\\
&\mathcal{A}_{2}(\epsilon,k,k-r):=\\{\zeta\in\Xi:X_{k}\in\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l}),X_{k-r}\notin\Gamma_{\frac{\delta}{|r|}}(\beta_{l})\\}\\\
&\mathcal{A}_{3}(\epsilon,k,k-r):=\\{\zeta\in\Xi:X_{k}\in\Gamma_{\delta}(\beta_{l})\setminus\Gamma_{\frac{\delta}{N_{1}|r|}}(\beta_{l})\\}.\end{split}$

for $N_{1}\in{\mathbb{N}}$ large and $\delta$ small.

Now we summarize the related work done in [Wil20a] below:

######  Lemma 4.18.

[Wil20a] Under the same condition of proposition (4.17), the following are
true:

  1. (1)

It is sufficient to prove (4.79) in the case where $F=0$.

  2. (2)

Equation (4.79) holds on the support of $\chi_{g}$, $\chi_{b}^{2}$, and
$\chi_{b}^{3}$ that is:

(4.83) $\displaystyle||\chi_{g}(\epsilon,k)\chi V_{k}||$
$\displaystyle\leq\frac{C}{\gamma}\sum_{r\in{\mathbb{Z}}\setminus
0}\sum_{t\in{\mathbb{Z}}}||\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)\chi
V_{k-r-t}||+\frac{C}{\gamma^{2}}\left|\chi\widehat{F_{k}}|X_{k}|\right|_{L^{2}}+\frac{C}{\gamma^{3/2}}\left|\chi\widehat{G_{k}}|X_{k}|\right|_{L^{2}}$
(4.84) $\displaystyle||\chi_{b}^{2}(\epsilon,k)\chi V_{k}||$
$\displaystyle\leq\frac{C}{\gamma}\sum_{r\in{\mathbb{Z}}\setminus
0}\sum_{t\in{\mathbb{Z}}}||\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)\chi
V_{k-r-t}||+\frac{C}{\gamma^{2}}\left|\chi\widehat{F_{k}}|X_{k}|\right|_{L^{2}}+\frac{C}{\gamma^{3/2}}\left|\chi\widehat{G_{k}}|X_{k}|\right|_{L^{2}}$
(4.85) $\displaystyle||\chi_{b}^{3}(\epsilon,k)\chi V_{k}||$
$\displaystyle\leq\frac{C}{\gamma}\sum_{r\in{\mathbb{Z}}\setminus
0}\sum_{t\in{\mathbb{Z}}}||\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)\chi
V_{k-r-t}||+\frac{C}{\gamma^{2}}\left|\chi\widehat{F_{k}}|X_{k}|\right|_{L^{2}}+\frac{C}{\gamma^{3/2}}\left|\chi\widehat{G_{k}}|X_{k}|\right|_{L^{2}}$

  3. (3)

When $F=0$, we have

(4.86)
$\displaystyle\begin{split}&(a)|\chi_{b}(\epsilon,k)\tilde{w}^{+}_{k}|_{L^{2}}\leq\frac{C}{\gamma}\sum_{r\in{\mathbb{Z}}\setminus
0}\left(\left|\frac{\chi_{b}(\epsilon,k){\alpha_{r}}w^{+}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}+\left|\frac{\chi_{b}(\epsilon,k){\alpha_{r}}w^{-}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}\right),\\\
&(b)|\chi_{b}(\epsilon,k)w^{-}_{k}|_{L^{2}}\leq\frac{C}{\gamma}\sum_{r\in{\mathbb{Z}}\setminus
0}\left(\left|\frac{\chi_{b}(\epsilon,k){\alpha_{r}}w^{+}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}+\left|\frac{\chi_{b}(\epsilon,k){\alpha_{r}}w^{-}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}\right)+\frac{C}{\gamma^{3/2}}\left|\widehat{G_{k}}|X_{k}|\right|_{L^{2}(\sigma,\eta)}.\end{split}$

  4. (4)

When $F=0$, we can replace (not dominate)
$\frac{C}{\gamma}\left|\frac{\chi^{1}_{b}(\epsilon,k,k-r){\alpha_{r}}w^{+}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}$
the term on the right of (4.86) by:

(4.87)
$\displaystyle\frac{C}{\gamma}\sum_{t\in{\mathbb{Z}}}\left|{\alpha_{r}}{\alpha_{t}}D(\epsilon,k,k-r;\beta_{l})\mathcal{W}_{k-r-t}\right|_{L^{2}}.$

###### Proof.

of proposition 4.17.

We observe that lemma (2) implies that it’s sufficient to prove (4.79) on the
support of $\chi_{b}^{1}$.

Claim: We can also replace
$\frac{C}{\gamma}\left|\frac{\chi^{1}_{b}(\epsilon,k,k-r){\alpha_{r}}w^{-}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}$
on the right of (4.86) by (4.87).

Suppose the claim is true, using proposition 4.1 and the formula (4.20)
(4.21), we see that
$\frac{1}{\sqrt{\gamma}}|\tilde{w}_{k}^{+}(0,\zeta)|_{L_{2}(\sigma,\eta)},\frac{1}{\sqrt{\gamma}}|w_{k}^{-}(0,\zeta)|_{L_{2}(\sigma,\eta)}$
satisfy, respectively, the same estimates as
$\frac{1}{\sqrt{\gamma}}|\tilde{w}_{k}^{+}(x_{2},\zeta)|_{L_{2}},\frac{1}{\sqrt{\gamma}}|w_{k}^{-}(x_{2},\zeta)|_{L_{2}}$.
Therefore, (4.87) together with (4.86) gives the estimate (4.79) on the
support of $\chi_{b}^{1}$, which finishes the proof of proposition 4.17. It is
only left to prove (4.87).

Proof of claim: improving the
$\left|\frac{\chi^{1}_{b}(\epsilon,k){\alpha_{r}}w^{-}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}$
pieces of (4.86).

For any fixed $r$, the main part of
$\left|\frac{\chi^{1}_{b}(\epsilon,k){\alpha_{r}}w^{-}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}$
arises from the part of (4.21) given by:

(4.88)
$\displaystyle\begin{split}&A:=e^{i\xi_{-}(\epsilon,k)x_{2}}[Br_{-}(\epsilon,k)]^{-1}Br_{+}(\epsilon,k)\int^{\infty}_{0}e^{i\xi_{+}(\epsilon,k)(-s)+i\frac{r\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}b(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)]ds.\end{split}$

We ignore factors that are $O(1)$ and independent of $s$, and the
$p-$component of $A$ is a sum of terms of the form

(4.89) $\displaystyle
A_{p,i,j}:=e^{i\omega_{p}(\epsilon,k)x_{2}}\frac{1}{\Delta(\epsilon,k)}\int^{\infty}_{0}e^{i\omega_{i}(\epsilon,k)(-s)+i\frac{r\omega_{N}(\beta_{l})}{\epsilon}s}\alpha_{r}w^{-}_{k-r,j}(s,\zeta)ds$

where $p\in\mathcal{I}$, $i\in\mathcal{O}$ and $w^{-}_{k-r,j}$,
$j\in\mathcal{I}$ denotes a component of $w^{-}_{k-r}$. We now improve the
$\chi_{b}^{1}A_{p,i,j}$ by an integration by part. Setting:

(4.90)
$w^{*,-}_{k-r,j}(x_{2},\zeta):=e^{-i\omega_{j}(\epsilon,k-r)x_{2}}w_{k-r,j}^{-}$

we may rewrite $\chi_{b}^{1}A_{p,i,j}$ as:

(4.91)
$e^{i\omega_{p}(\epsilon,k)x_{2}}\frac{1}{\Delta(\epsilon,k)}\int^{\infty}_{0}e^{-iE_{i,j}(\epsilon,k,k-r)s}\chi_{b}^{1}\alpha_{r}w^{*,-}_{k-r}(x_{2},\zeta)ds$

where we recall that
$E_{i,j}(\epsilon,k,k-r):=\omega_{i}(\epsilon,k)-\frac{r\omega_{N}(\beta_{l})}{\epsilon}-\omega_{j}(\epsilon,k-r)$.
From (4.18), we have:

(4.92)
$\partial_{x_{2}}w^{*,-}_{k-r,j}=e^{-i\omega_{j}(\epsilon,k-r)x_{2}}ih_{k-r,j}$

where $h_{k-r,j}$ is the $j$th component of the right hand side of
$\eqref{a3a}_{k-r}$. We also have:

(4.93) $-\frac{1}{i}E_{i,j}^{-1}\frac{d}{ds}e^{-iE_{i,j}s}=e^{-iE_{i,j}s}$

Using (4.92) (4.93), an integration by part on (4.91) gives:

(4.94) $\displaystyle e^{i\omega_{p}x_{2}}$
$\displaystyle\frac{1}{i\Delta(\epsilon,k)}\chi_{b}^{1}E_{i,j}^{-1}(\epsilon,k,k-r)\left[\alpha_{r}w_{k-r,j}^{-}(0,\zeta)+\int_{0}^{\infty}e^{-iE_{i,j}(\epsilon,k,k-r)s}e^{-i\omega_{j}(\epsilon,k-r)s}{\alpha_{r}}ih_{k-r}(s,\zeta)\right]ds$

Now by proposition 4.11, in particular (4.35), on the support of
$\chi_{b}^{1}$, for $|r|\leq\epsilon^{-\xi}$, when $j\neq N$ we have:

(4.95)
$\left|\frac{1}{\Delta(\epsilon,k)E_{1,j}}\right|\leq\frac{|X_{k}|}{\gamma}\frac{|r|^{2+\delta}}{C|X_{k}|}\sim\frac{|r|^{2+\delta}}{\gamma}$

and for $j=N$, using (4.36), we have the following three cases: (I). The first
alternative in (4.36) holds; (II). The second alternative holds and
$|X_{k-r}|\geq\frac{|X_{k}|}{N_{2}}$; (III). The second alternative holds and
$|X_{k-r}|<\frac{|X_{k}|}{N_{2}}$ for $N_{2}\in{\mathbb{Z}}$ chosen below.

We have, in case (I):

(4.96)
$\left|\frac{1}{\Delta(\epsilon,k)E_{i,N}}\right|\leq\frac{|X_{k}|}{\gamma}\frac{|r|}{C|X_{k}|}\sim\frac{|r|}{\gamma},$

in case (II):

(4.97)
$\left|\frac{1}{\Delta(\epsilon,k)E_{i,N}}\right|\leq\frac{|X_{k}|}{\gamma}\frac{1}{C|X_{k-r}|}\sim\frac{N_{2}}{\gamma}$

in case (III): the third term in $E_{i,j}$ is dominated by the first two
terms, more precisely,

(4.98)
$|X_{k-r}|=|X_{k}-\frac{r\beta_{l}}{\epsilon}|\leq\frac{1}{N_{2}}|X_{k}|,$

so taking $N_{2}$ large, we have888Use (4.98) and (4.10)(e) for the last
$\sim$ of (4.99).:

(4.99)
$|\omega_{N}(X_{k-r})|\lesssim|X_{k-r}|\leq\frac{1}{N_{2}}|X_{k}|\sim|\omega_{i}(X_{k})-\omega_{N}(\frac{r\beta_{l}}{\epsilon})|$

This implies that $E_{i,N}$ satisfies

(4.100) $|E_{i,N}^{-1}|\leq\frac{C(N_{2})}{|X_{k}|}\text{ and hence
}\left|\frac{1}{\Delta(\epsilon,k)E_{i,N}}\right|\leq\frac{C(N_{2})}{\gamma}$

provided $N_{2}$ is large enough.

Equation (4.94), (4.95), (4.96), (4.97), (4.100), proposition 4.1, and
definition (4.6) yield the desired inequality on the support of $\chi$:

(4.101) $\displaystyle|\chi_{b}^{1}(\epsilon,k,k-r)\chi
A_{p,i,j}|_{L^{2}}\lesssim$ (4.102)
$\displaystyle\quad\frac{C}{\gamma^{3/2}}\left[|{\alpha_{r}}D(\epsilon,k,k-r)\chi
w_{k-r,j}^{-}(0,\zeta)|_{L^{2}(\sigma,\eta)}+\frac{C}{\sqrt{\gamma}}\sum_{t\in{\mathbb{Z}}\setminus
0}|{\alpha_{r}}\alpha_{t}D(\epsilon,k,k-r)\chi w_{k-r-t}|_{L^{2}}\right]$

We observe that when $|r|\geq\epsilon^{-\xi}$, above inequality holds by the
definition of amplification factor.

The piece of the second term on the right of (4.86)(a) given by
$\left|\frac{\chi^{1}_{b}(\epsilon,k){\alpha_{r}}w^{-}_{k-r}}{\Delta(\epsilon,k)}\right|_{L^{2}}$
arises from the following part of (4.20):

(4.103)
$\frac{\chi_{b}^{1}(\epsilon,k,k-r)}{\Delta(\epsilon,k)}\int_{x_{2}}^{\infty}e^{i\xi_{+}(\epsilon,k)(x_{2}-s)+i\frac{\omega_{N}(\beta_{l})}{\epsilon}s}{\alpha_{r}}b(\epsilon,k,k-r)w^{-}_{k-r}(s,\zeta)ds$

A same integration by part argument would give the estimate of the form
(4.101).

Now as we finished the proof of claim, the proof of proposition 4.17 is
completed. ∎

The next theorem gives an a priori estimate in the small/medium frequency
region for the singular system:

(4.104)
$\displaystyle\begin{split}&D_{x_{2}}U+A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta})U+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta})U-iB_{2}^{-1}{\mathcal{D}}\left(\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+\theta\right)U=F(t,x,\theta)\\\
&BU=G(t,x_{1},\theta)\text{ on }x_{2}=0\\\ \end{split}$

######  Theorem 4.19.

Let $U^{\gamma}(t,x,\theta)\in H^{2}(\Omega\times\mathbb{T})$ for $\gamma\geq
1$. Assume that the singular system (4.104) has $p$ incoming phases
$\phi_{N-p},...,\phi_{N}$. Let:

(4.105)
$\displaystyle\begin{split}&F(t,x,\theta):=D_{x_{2}}U+A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta})U+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta})U-iB_{2}^{-1}{\mathcal{D}}\left(\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+\theta\right)U\\\
&G(t,x_{1},\theta):=BU(t,x_{2},0,\theta)\\\ \end{split}$

Suppose the coefficients $\widehat{\mathcal{D}}(r)$ satisfy
$\widehat{\mathcal{D}}(0)=0$ and
$\widehat{\mathcal{D}}(r)\lesssim|r|^{-(M+N)}$ for some $M\geq 4$ and
$N\geq\xi^{-1}$ where $\xi$ is as in proposition 4.11. Under the usual
structural assumptions 2.1, 2.14, and 2.2, we assume further that
$\Upsilon_{0}=\\{\beta_{l},-\beta_{l}\\}$ and assumption 4.9 holds. Then there
exist positive constants $\epsilon_{0},\ \gamma_{0},\ K$ such that for
$0<\epsilon\leq\epsilon_{0}$ and $\gamma\geq\gamma_{0}$ we have:

(4.106)
$\displaystyle|\chi_{D}U^{\gamma}|_{L^{2}(t,x,\theta)}+\left|\frac{\chi_{D}U^{\gamma}(0)}{\sqrt{\gamma}}\right|_{L^{2}(t,x_{1},\theta)}\leq
K\left[\frac{1}{\gamma^{2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{F_{k}}\right|^{2}_{L^{2}(x_{2},\sigma,\eta)})^{1/2}+\frac{1}{\gamma^{3/2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{G_{k}}\right|^{2}_{L^{2}(\sigma,\eta)})^{1/2}\right]$

###### Proof.

We note that by equation (4.30), the proof is reduced to the proof of the
following equation:

(4.107) $(||\chi V_{k}||)_{\ell^{2}(k)}\leq
K\left[\frac{1}{\gamma^{2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{F_{k}}\right|^{2}_{L^{2}})^{1/2}+\frac{1}{\gamma^{3/2}}(\sum_{k\in{\mathbb{Z}}}\left|\chi|X_{k}|\widehat{G_{k}}\right|^{2}_{L^{2}(\sigma,\eta)})^{1/2}\right]$

Define

(4.108) $\beta_{r}:=\begin{cases}\alpha_{r}C|r|^{2+\delta}\text{ for
}r\leq\epsilon^{-\xi}\\\ \frac{\alpha_{r}C|r|}{\epsilon}\text{ otherwise
}\end{cases}$

By definition of ${\mathbb{D}}(\epsilon,k,k-r)(\zeta)$, we see that
$\beta_{r}\geq\alpha_{r}{\mathbb{D}}$. Using the fact that when
$|r|\gtrsim\epsilon^{-\xi}$, the corresponding Fourier coefficients of
$\mathcal{D}$, namely $\alpha_{r}$, satisfy $|\alpha_{r}|\leq
C_{M,N}<r>^{-(M+N)}\leq C_{M,N}\epsilon^{N\xi}|r|^{-M}$, we have:

(4.109) $\displaystyle|\beta_{r}|\leq|r|^{-3/2}\text{ for
}r\leq\epsilon^{-\xi}$ (4.110) $\displaystyle|\beta_{r}|\leq
C\epsilon^{N\xi-1}|r|^{-(M-1)}\leq C|r|^{1-M}\text{ otherwise }$

The above inequalities ensure that $|(\beta_{r})|_{\ell^{1}}$ is finite. From
this point we can repeat the proof of Proposition 4.7 in [Wil20a] to obtain
the desired estimate. That is999We note that the definition of $\mathbb{D}$
and $\beta_{r}$ are different from [Wil20a].:

(4.111) $\displaystyle\begin{split}&\sum_{r\in{\mathbb{Z}}\setminus
0}\sum_{t\in{\mathbb{Z}}}\|\alpha_{r}\alpha_{t}{\mathbb{D}}(\epsilon,k,k-r)\chi
V_{k-r-t}\|\leq\sum_{r\in{\mathbb{Z}}\setminus
0}\sum_{t\in{\mathbb{Z}}}\|\beta_{r}\alpha_{t}\chi V_{k-r-t}\|=\\\
&\qquad\qquad\sum_{s}\left(\sum_{r+t=s}|\beta_{r}\alpha_{t}|\right)\|\chi
V_{k-s}\|:=\sum_{s}\gamma_{s}\|\chi V_{k-s}\|.\end{split}$

Young’s convolution inequality gives

(4.112) $\displaystyle\left|\left(\sum_{s}\gamma_{s}\|\chi
V_{k-s}\|\right)\right|_{\ell^{2}(k)}\leq|(\|\chi
V_{k}\|)|_{\ell^{2}}|(\gamma_{s})|_{\ell^{1}}.$

Since $\gamma_{s}=\sum_{r}|\beta_{r}||\alpha_{s-r}|$, using Young’s
convolution inequality again we have

(4.113)
$\displaystyle|(\gamma_{s})|_{\ell^{1}}\leq|(\beta_{r})|_{\ell^{1}}|(\alpha_{t})|_{\ell^{1}}:=K_{1}.$

Thus, by propsition 4.17 with remark 4.4, we have

(4.114) $||\chi V_{k}||_{\ell^{2}}\lesssim\frac{K_{1}}{\gamma}|(\|\chi
V_{k}\|)|_{\ell^{2}}+|(|\chi|X_{k}|\widehat{F_{k}}|_{L^{2}(x_{2},\sigma,\eta)})|_{\ell^{2}}+|(|\chi|X_{k}|\widehat{G_{k}}|_{L^{2}(\sigma,\eta)})|_{\ell^{2}}$

and the result follows by taking $\gamma_{0}$ large enough. ∎

###  4.5. Application of the Main Estimate: an Existence Theorem

In this section, we informally discuss an application of theorem 4.19 of
getting a “restricted” existence result for a forward problem different from
(1.5). The idea is to use a classical duality argument based on an energy
estimate for the corresponding “backward” or adjoint boundary value problem
together with the Riesz representation theorem.

We first explain why the argument fails to yield an existence result for
(1.5). In (1.5), we recall that $\omega_{N}$ is incoming, which is used to
prove the main energy estimate (4.106). If we want to prove an existence
result for (1.5) by duality, we need an estimate for the corresponding
backward problem. In that backward problem, $\omega_{N}$ still appears in the
oscillatory term but now it is outgoing, and thus the proof of the energy
estimate breaks down for this backward problem. In the proof of proposition
4.17 (and lemma 4.18), we generally rely on a lower bound for terms like
$E_{i,j}$ to control the term $\Delta^{-1}(\epsilon,k)$. More precisely, to
prove the second equation in (4.83) of lemma 4.18, we need a lower bound for
$E_{i}$:

(4.115)
$E_{i}(\epsilon,k,k-r)=\left(\omega_{i}(X_{k})-r\omega_{N}(\frac{\beta_{l}}{\epsilon})\right)I_{N}-{\mathcal{A}}(X_{k-r}).$

Now since $\omega_{i}$ is outgoing, we may have $i=N$. In this case, for any
fixed $k=r\neq 0$, we have:

(4.116)
$E_{i}(\epsilon,k,k-r)=\left(\omega_{N}(\zeta+r\frac{\beta_{l}}{\epsilon})-\omega_{N}(\frac{r\beta_{l}}{\epsilon})\right)I_{N}-{\mathcal{A}}(\zeta).$

which can be small, for example, for $\zeta$ near $0$. It is impossible to
establish an lower bound for $E_{i}$, so we can’t prove even a restricted
existence result for (1.5).

Given above observation, in this section, we are considering the following
singular problem that is different from (1.5) for which $\omega_{N}$ is
outgoing. This means that the new corresponding backward problem $\omega_{N}$
is incoming, so we will have a restricted estimate for this backward problem.
We study the forward singular problem:

(4.117)
$\displaystyle\begin{split}&D_{x_{2}}U+A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta})U+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta})U-iB_{2}^{-1}{\mathcal{D}}\left(\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+\theta\right)U=F(t,x,\theta)\\\
&BU=G(t,x_{1},\theta)\text{ on }x_{2}=0\\\ &U=0\text{ in }t<0.\end{split}$

where we now assume that the $B_{j}$ used to define the $A_{j}$ are now
symmetric (recall that $A_{0}=B_{2}^{-1}$, $A_{1}=B_{2}^{-1}B_{1}$).

Suppose that the $\omega_{j}$ are incoming for $j\in\\{1,...,N-p\\}$ and
outgoing for $j\in\\{N-p+1,...,N\\}$. We denote the operator on the left hand
side by
$L:=D_{x_{2}}+A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta})+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta})-iB_{2}^{-1}{\mathcal{D}}\left(\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+\theta\right)$.
Commuting (4.117) with $e^{-\gamma t}$, we are led to find solutions in
$L^{2}(t,x,\theta)$ of the system:

(4.118) $\displaystyle L^{\gamma}\tilde{U}=F^{\gamma}$ (4.119) $\displaystyle
B\tilde{U}=G^{\gamma}$

where $L^{\gamma}:=-i\gamma A_{0}+L$, $F^{\gamma}=e^{-\gamma t}F$,
$G^{\gamma}=e^{-\gamma t}G$. We define the formal adjoint $L^{*}$ of $L$:

(4.120) $\displaystyle L^{*}:=$ $\displaystyle
D_{x_{2}}+A_{0}^{T}(D_{t}+\frac{\sigma_{l}}{\epsilon})D_{\theta}+A_{1}^{T}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon})D_{\theta}+\left(iB_{2}^{-1}{\mathcal{D}}\left(\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+\theta\right)\right)^{T}$
(4.121) $\displaystyle(L^{\gamma})^{*}:=$ $\displaystyle i\gamma
A_{0}^{T}+L^{*}$

with respect to the $L^{2}$ inner product:

(4.122)
$\displaystyle(U,V)_{L^{2}(\Omega\times{\mathbb{T}})}=\int_{\Omega\times{\mathbb{T}}}U(t,x,\theta)\overline{V(t,x,\theta)}$

Then, the dual problem (or backward problem) of (4.118) is:

(4.123) $\displaystyle(L^{\gamma})^{*}V=f_{\sharp}$ (4.124) $\displaystyle
M_{\sharp}V|_{x_{2}=0}=g_{\sharp}$

for suitable $f_{\sharp},g_{\sharp}$, and $M_{\sharp}$ as a $(n-p)\times n$
matrix of maximal rank satisfying:

(4.125) $B^{T}_{\sharp}B+M^{T}_{\sharp}M=I$

for suitable matrices $B_{\sharp}$ and $M$.

We observe that the backward problem (4.123) is of the same form as (4.104).
In particular, in (4.123) $\omega_{j}$ is outgoing for $j\in\\{1,...,N-p\\}$
and is incoming for $j\in\\{N-p+1,...,N\\}$101010Here we use the assumption
that the $B_{j}$’s are symmetric.. We can apply theorem 4.19 to (4.123) to get
an estimate of the form:

(4.126) $\|\chi_{D}V\|_{0}+\frac{1}{\sqrt{\gamma}}|\chi_{D}V(0)|_{0}\leq
C\left[\frac{1}{\gamma^{2}}\|f_{1}\|_{1}+\frac{1}{\gamma^{3/2}}|g_{1}|_{1}\right]$

for appropriate $f_{1},g_{1}$. Here $V(0):=V|_{x_{2}=0}$, $\|V\|_{0}$ denotes
the $L^{2}(\Omega\times{\mathbb{T}})$ norm, and $|V|_{s}$ denotes the
$L^{2}({\mathbb{R}}^{2}\times{\mathbb{T}}^{3})$ norm, and $\|f\|_{1},|g|_{1}$
are the norms equivalent to the norm appeared on the right hand side of
(4.106). We can obtain an estimate of the following form by commuting a
singular operator with symbol $\langle X_{k}\rangle^{-1}$ through the problem:

(4.127) $\|\chi_{D}V\|_{-1}+\frac{1}{\sqrt{\gamma}}|\chi_{D}V(0)|_{-1}\leq
C\left[\frac{1}{\gamma^{2}}\|f_{2}\|_{0}+\frac{1}{\gamma^{3/2}}|g_{2}|_{0}\right]$

for appropriate $f_{2},g_{2}$.

With the help of above estimates, we may apply an argument similar to that in
[Cou05] to get the existence of solutions to (4.117). We sketch this argument
below. Define a space of test functions:

(4.128) $Q:=\\{V\in C_{0}^{\infty}(\overline{\Omega\times{\mathbb{T}}})\text{
s.t. }M_{\sharp}V=0\\}$

We may define a linear functional $\ell$ on $(L^{\gamma})^{*}Q$ by:

(4.129)
$\ell[(L^{\gamma})^{*}V]:=(F^{\gamma},\chi_{D}V)_{L^{2}(\Omega\times{\mathbb{T}})}+(G^{\gamma},B_{\sharp}\chi_{D}V(0))_{L^{2}({\mathbb{R}}^{2}\times{\mathbb{T}})},\quad\forall
V\in Q$

and we have estimate:

(4.130) $\displaystyle|\ell[(L^{\gamma})^{*}V]|$
$\displaystyle\leq\|F^{\gamma}\|_{1}\|\chi_{D}V\|_{-1}+C|G^{\gamma}|_{1}|\chi_{D}V(0)|_{-1}$
(4.131) $\displaystyle\leq
C\left(\|\chi_{D}V\|_{-1}+|\chi_{D}V(0)|_{-1}\right)$ (4.132)
$\displaystyle\leq\frac{C}{\gamma^{3/2}}\|(L^{\gamma})^{*}V\|_{0}$

which shows that $\ell$ is continuous. Now by Hahn-Banach theorem, we can
extend $\ell$ from the subspace $(L^{\gamma})^{*}Q$ to a continuous linear
functional on the whole space of $L^{2}(\Omega\times{\mathbb{T}})$. Riesz
representation theorem gives a function $\tilde{U}\in
L^{2}(\Omega\times{\mathbb{T}})$ such that:

(4.133)
$\ell[(L^{\gamma})^{*}V]=(U,(L^{\gamma})^{*}V)_{L^{2}(\Omega\times{\mathbb{T}})}.$

In particular, we have $L^{\gamma}\tilde{U}=\chi_{D}F^{\gamma}$ in the sense
of distributions.

As for the boundary, the trace $\tilde{U}$ belongs to
$H^{-1/2}({\mathbb{R}}^{2}\times{\mathbb{T}})$, and the following Green’s
formula holds:

(4.134)
$(\tilde{U},(L^{\gamma})^{*}V)_{L^{2}(\Omega)}=(\chi_{D}F^{\gamma},V)_{L^{2}(\Omega)}+(\tilde{U}(0),V(0))_{H^{-1/2}({\mathbb{R}}^{2}\times{\mathbb{T}}),H^{1/2}({\mathbb{R}}^{2}\times{\mathbb{T}})},\quad\forall
V\in C_{0}^{\infty}(\overline{\Omega\times{\mathbb{T}}}).$

We observe that (4.125) gives:

(4.135)
$\displaystyle(\tilde{U}(0),V(0))_{H^{-1/2}({\mathbb{R}}^{2}\times{\mathbb{T}}),H^{1/2}({\mathbb{R}}^{2}\times{\mathbb{T}})}$
$\displaystyle=(B^{T}_{\sharp}B+M^{T}_{\sharp}M\tilde{U}(0),V(0))_{H^{-1/2}({\mathbb{R}}^{2}\times{\mathbb{T}}),H^{1/2}({\mathbb{R}}^{2}\times{\mathbb{T}})}$
(4.136)
$\displaystyle=(B\tilde{U}(0),B_{\sharp}V(0))+(M\tilde{U}(0),M_{\sharp}V(0))$

Combining (4.129), (4.134), and (4.135) gives:

(4.137)
$(\chi_{D}G^{\gamma}-B\tilde{U}(0),B_{\sharp}V(0))_{H^{-1/2}({\mathbb{R}}^{2}\times{\mathbb{T}}^{3}),H^{1/2}({\mathbb{R}}^{2}\times{\mathbb{T}})}=0$

Using (4.125), we observe that:

(4.138) $\begin{pmatrix}B_{\sharp}\\\ M_{\sharp}\end{pmatrix}\in
W^{2,\infty}({\mathbb{R}}^{2}\times{\mathbb{T}})$

is invertible. We therefore conclude that $B\tilde{U}(0)=\chi_{D}G^{\gamma}$,
which completed showing that $\tilde{U}$ is a solution to (4.117) on
$\chi_{D}$.

We conclude that now we have shown the existence of solutions $U\in e^{\gamma
t}L^{2}(\Omega\times{\mathbb{T}}_{\theta})$ to the system:

(4.139) $\displaystyle
D_{x_{2}}U+A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta})U+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta})U-iB_{2}^{-1}{\mathcal{D}}\left(\frac{\omega_{N}(\beta_{l})}{\epsilon}x_{2}+\theta\right)U=\chi_{D}F(t,x,\theta)$
(4.140) $\displaystyle BU=\chi_{D}G(t,x_{1},\theta)\text{ on }x_{2}=0$

##  5\. Geometric Optics Solutions for Weakly Stable Hyperbolic System

In this section, we construct an approximate solution to system (5.1).
Although we use the tools established in section 3.2, there are some new
difficulties in the construction due to the failure of the uniform Lopatinski
condition at $\beta_{l}$.

We study the following $3\times 3$ system on
$\Omega=(-\infty,\infty)\times\\{(x_{1},x_{2}):x_{2}\geq 0\\}$:

(5.1) $\displaystyle
L(\partial)u+\left(e^{-i\frac{\phi_{3}}{\epsilon}}+e^{i\frac{\phi_{3}}{\epsilon}}\right)Mu=0\text{
in }x_{2}>0$ $\displaystyle Bu=\epsilon
G\left(t,x_{1},\frac{\phi_{0}}{\epsilon}\right)\text{ on }x_{2}=0$
$\displaystyle u=0\text{ in }t<0$

where
$L(\partial)u:=\partial_{t}u+B_{1}\partial_{x_{1}}u+B_{2}\partial_{x_{2}}u$,
and $G$ is a periodic function in $\theta_{0}$. Here all $B_{j}$’s are
$3\times 3$ matrices and $B_{2}$ is invertible. We take $\phi_{1}$ to be the
outgoing phase, $\phi_{2}$, $\phi_{3}$ to be the incoming phase, and there is
no resonance. The boundary phase is taken to be
$\phi_{0}(t,x_{1})=\beta_{l}\cdot(t,x_{1})$, where
$\beta_{l}=(\sigma_{l},\eta_{l})\in{\mathbb{R}}^{2}\setminus{0}$ is the only
direction where the uniform Lopatinski condition fails. Also,

(5.2)
$\displaystyle\phi_{m}(t,x_{1},x_{2})=\beta_{l}\cdot(t,x_{1})+\omega_{m}(\beta_{l})x_{2}$
$\displaystyle d\phi_{m}=(\beta_{l},\omega_{m})$

######  Theorem 5.1.

Consider the system (5.1) where we assume $Mr_{3}=0$.111111The assumption
$Mr_{3}=0$ is used to decouple some of the profile equations below. If the
small divisor assumption 3.6 is satisfied and there is no resonance, there
exists an approximate solution
$u^{\epsilon}_{a}(t,x)=\sum_{k={1}}^{J}\epsilon^{k}U_{k}(t,x,\frac{\Phi}{\epsilon})$
where $U_{k}\in\mathcal{H}^{\infty}(t,x,\theta)$, periodic in $\theta$.

(5.3)
$\begin{split}D_{x_{2}}U^{\epsilon}_{a}+&A_{0}(D_{t}+\frac{\sigma_{l}}{\epsilon}D_{\theta_{0}})U^{\epsilon}_{a}+A_{1}(D_{x_{1}}+\frac{\eta_{l}}{\epsilon}D_{\theta_{0}})U^{\epsilon}_{a}\\\
-i&\left(e^{i\left(\frac{\omega_{3}(\beta_{l})}{\epsilon}x_{2}+\theta_{0}\right)}+e^{-i\left(\frac{\omega_{3}(\beta_{l})}{\epsilon}x_{2}+\theta_{0}\right)}\right)B_{2}^{-1}MU^{\epsilon}_{a}\\\
&=\epsilon^{J}\left(\mathcal{L}(\partial_{\theta})U_{J}+L(\partial)U_{J-1}+\left(e^{i\left(\frac{\omega_{3}(\beta_{l})}{\epsilon}x_{2}+\theta_{0}\right)}+e^{-i\left(\frac{\omega_{3}(\beta_{l})}{\epsilon}x_{2}+\theta_{0}\right)}\right)MU_{J-1}\right)\\\
BU^{\epsilon}_{a}&=G(t,x_{1},\theta_{0})\text{ on }x_{2}=0\\\
U^{\epsilon}_{a}&=0\text{ for }t<0\end{split}$

As proof of the theorem, we construct approximate solutions.

###  5.1. Profile Equations

We use the same set of tools defined in section 3.2, and apply similar method
as before. However, since $B$ fails to be an isomorphism between
${\mathbb{E}}^{s}(\zeta)$ and ${\mathbb{C}}^{p}$, a new discussion on boundary
equation is needed, which is discussed in section 5.3. First we construct
approximate solutions of the form

(5.4)
$u_{a}^{\epsilon}(t,x)=\sum_{k=0}^{J}\epsilon^{k}U_{k}\left(t,x,\frac{\Phi}{\epsilon}\right)$

Here all $U_{k}\in\mathcal{H}^{\infty}$. Let
$f(\theta_{3}):=e^{-i\theta_{3}}+e^{i\theta_{3}}$. Using the same notation as
before, we plug in the approximate solution to our system of equations and
get:

(5.5) $\displaystyle\mathcal{L}(\partial_{\theta})U_{0}=0$
$\displaystyle\mathcal{L}(\partial_{\theta})U_{1}+L(\partial)U_{0}+f(\theta_{3})MU_{0}=0$
$\displaystyle\mathcal{L}(\partial_{\theta})U_{2}+L(\partial)U_{1}+f(\theta_{3})MU_{1}=0$
$\displaystyle\mathcal{L}(\partial_{\theta})U_{j}+L(\partial)U_{j-1}+f(\theta_{3})MU_{j-1}=0\text{
for }j\geq 1$

and the boundary equations at
$x_{2}=0,\theta_{1}=\theta_{2}=\theta_{3}=\theta_{0}$:

(5.6) $\displaystyle BU_{0}=0$ $\displaystyle BU_{1}=G(t,x_{1},\theta_{0})$
$\displaystyle BU_{j}=0\text{ for }j\geq 2$

Applying $E_{Q}$ to (5.5), we have:

(5.7) $\displaystyle E_{Q}[L(\partial)U_{0}+f(\theta_{3})MU_{0}]=0$
$\displaystyle E_{Q}[L(\partial)U_{1}+f(\theta_{3})MU_{1}]=0$ $\displaystyle
E_{Q}[L(\partial)U_{j}+f(\theta_{3})MU_{j}]=0\text{ for }j\geq 0$

And applying $R$ to (5.5), by (3.18) we have:

(5.8) $\displaystyle(I-E_{P})U_{0}=0$
$\displaystyle(I-E_{P})U_{1}=-R[L(\partial)U_{0}+f(\theta_{3})MU_{0}]$
$\displaystyle(I-E_{P})U_{2}=-R[L(\partial)U_{1}+f(\theta_{3})MU_{1}]$
$\displaystyle(I-E_{P})U_{j}=-R[L(\partial)U_{j-1}+f(\theta_{3})MU_{j-1}]\text{
for }j\geq 1$

We now proceed to define $e,\check{e},\check{E}(\beta_{l})$ as in [Wil20a]. By
proposition 2.10, the vector space $E^{s}(\beta_{l})$ is spanned by
$\\{r_{2},r_{3}\\}$. $B$ fails to be an isomorphism as in assumption 2.14
implies that the subspace $\ker B\cap E^{s}(\beta_{l})$ is one-dimensional and
is thus we can define $e=e_{2}+e_{3}$ such that $\ker B\cap
E^{s}(\beta_{l})=\text{span}\\{e\\}$ for $e_{m}\in$ span$\\{r_{m}\\}$. We know
that the vector space $BE^{s}(\beta_{l})$ is one-dimensional and of real type,
so we have:

(5.9) $BE^{s}(\beta_{l})=\\{X\in{\mathbb{C}}^{2}:b\cdot X=0\\}$

for an appropriate nonzero vector $b$. We next choose $\check{E}(\beta_{l})$,
such that $E^{s}(\beta_{l})=\check{E}(\beta_{l})\oplus\text{span}\\{e\\}$. We
pick $\check{e}$ such that $\check{E}(\beta_{l})=\text{span}\\{\check{e}\\}$,
where $\check{e}=\check{e}_{2}+\check{e}_{3}$,
$\check{e}_{m}\in\text{span}\;r_{m}$. Thus, by definition we have an
isomorphism:

(5.10) $B:\check{E}^{s}(\beta_{l})\rightarrow BE^{s}(\beta_{l})$

To analyze an equation of the form $BU_{k}=H(t,x,\theta_{0})$, using notation
3.10, we take the mean on both sides: $B\underline{U_{k}}=\underline{H}$, and
the mean zero oscillation:

(5.11)
$(BU_{k})^{*}=H^{*}\Rightarrow(B{E_{P}}{U_{k}}+B(I-E_{P})U_{k})^{*}=H^{*}\Rightarrow
B{E_{P_{in}}}{U_{k}}=H^{*}-E_{P_{1}}U_{k}-B[I-E_{P})U_{k}]^{*}.$

Since $B{E_{P_{in}}}U_{k}=B\check{U}_{k}$, we have:

(5.12) $\displaystyle B\underline{U_{k}}=\underline{H}$ $\displaystyle
B\check{U}_{k}=H^{*}-BEP_{1}U_{k}-B[(I-E_{P})U_{k}]^{*}.$

Using this along with (LABEL:eqn:bdy1) and (5.8) we have:

(5.13) $\displaystyle B\underline{U_{0}}=0\quad
B\underline{U_{1}}=\underline{G}$ $\displaystyle B\underline{U_{2}}=0\quad
B\underline{U_{j}}=0\text{ for }j\geq 2$ (5.14) $\displaystyle B\check{U}_{0}$
$\displaystyle=-BE_{P_{1}}U_{0}$ $\displaystyle B\check{U}_{1}$
$\displaystyle=G^{*}-BE_{P_{1}}U_{1}+BR[L(\partial)U_{0}+f(\theta_{3})MU_{0}]^{*}$
$\displaystyle B\check{U}_{2}$
$\displaystyle=-BE_{P_{1}}U_{2}+BR[L(\partial)U_{1}+f(\theta_{3})MU_{1}]^{*}$
$\displaystyle B\check{U}_{j}$
$\displaystyle=-BE_{P_{1}}U_{j}+BR[L(\partial)U_{j-1}+f(\theta_{3})MU_{j-1}]^{*}\text{
for }j\geq 2.$

Using (5.9), we have:

(5.15) $\displaystyle
b\cdot[G^{*}-BE_{P_{1}}U_{1}+BR(L(\partial)U_{0}+f(\theta_{3})MU_{0})^{*}]=0$
$\displaystyle
b\cdot[-BE_{P_{1}}U_{2}+BR(L(\partial)U_{1}+f(\theta_{3})MU_{1})^{*}]=0$
$\displaystyle
b\cdot[-BE_{P_{1}}U_{j}+BR(L(\partial)U_{j-1}+f(\theta_{3})MU_{j-1})^{*}]=0\text{
for }j\geq 2$

###  5.2. Determining Solutions

We write as before:

(5.16)
$U_{k}=\underline{U_{k}}+\sigma_{k}^{1}r_{1}+\sigma_{k}^{2}r_{2}+\sigma_{k}^{3}r_{3}+(I-E_{P})U_{k}.$

1\. Determining the leading profile $U_{0}$

From (5.8), we have:

(5.17)
$U_{0}=\underline{U_{0}}+\sigma_{0}^{1}(t,x,\theta_{1})r_{1}+\sigma_{0}^{2}(t,x,\theta_{2})r_{2}+\sigma_{0}^{3}(t,x,\theta_{3})r_{3}$

where $\sigma_{0}^{m}(t,x,\theta_{m})$ are scalars.
The first equation (5.7) gives rise to four equations. We first consider the
mean of both sides:

(5.18) $\displaystyle E_{Q_{0}}[L(\partial)U_{0}+f(\theta_{3})MU_{0}]=0$
$\displaystyle\Leftrightarrow$ $\displaystyle
L(\partial)\underline{U_{0}}+\underline{f(\theta_{3})M\sigma_{0}^{3}(t,x,\theta_{3})r_{3}}=0.$

Using the boundary condition given by (5.13) and the fact $Mr_{3}=0$, we have:

(5.19) $\displaystyle L(\partial)\underline{U_{0}}=0$ $\displaystyle
B\underline{U_{0}}=0.$

This gives $\underline{U_{0}}=0$.

We then take the pure $\theta_{1}$ mode on both sides of the first equation of
(5.7):

(5.20) $\displaystyle E_{Q_{1}}[L(\partial)U_{0}+f(\theta_{3})MU_{0}]=0.$

Notice that the only oscillation of $L(\partial)U_{0}+f(\theta_{3})MU_{0}$ in
$\theta_{1}$ is $\sigma_{0}^{1}$ since there is no resonance, thus we have:

(5.21) $E_{Q_{1}}L(\partial)E_{P_{1}}U_{0}=0.$

Using proposition 3.9, we have $X_{\phi_{1}}\sigma_{0}^{1}B_{2}r_{1}=0$. Since
$\phi_{1}$ is outgoing, and $\sigma_{0}^{1}=0$ when $t<0$, we have
$\sigma_{0}^{1}=0$.

Now we take a moment to discuss implications of $\sigma_{0}^{1}=0$. We claim
that $\sigma_{0}^{1}=0$ implies $E_{P_{1}}U_{j}=0$ for all $j$. First, we
consider $U_{1}$:

(5.22)
$U_{1}=\underline{U_{1}}+\sigma_{1}^{1}r_{1}+\sigma_{1}^{2}r_{2}+\sigma_{1}^{3}r_{3}+(I-E_{P})U_{1}$

where $(I-E_{P})U_{1}=-R[L(\partial)U_{0}+f(\theta_{3})MU_{0}]$. Plugging this
equation in (5.7), we have:

(5.23) $\displaystyle 0$
$\displaystyle=E_{Q}[L(\partial)U_{1}+f(\theta_{3})MU_{1}]=E_{Q}[L(\partial)(E_{P}U_{1}+(I-E_{P})U_{1})+f(\theta_{3})M(E_{P}U_{1}+(I-E_{P})U_{1})]$
$\displaystyle\Leftrightarrow
E_{Q}[L(\partial)(E_{P}U_{1})+f(\theta_{3})ME_{P}U_{1}]=-E_{Q}[L(\partial)(I-E_{P})U_{1}+f(\theta_{3})M(I-E_{P})U_{1}].$

We take the pure $\theta_{1}$ oscillation of (5.23). Since there is no
resonance, all the terms involving $f(\theta_{3})$ is zero and on the right
hand side only the terms involving $U_{0}$ is left. We also know $U_{0}$ has
no $\theta_{1}$ mode, so we obtain:

(5.24)
$E_{Q_{1}}L(\partial)E_{P_{1}}U_{1}=X_{\phi_{1}}\sigma_{1}^{1}B_{2}r_{1}=0.$

Therefore, $X_{\phi_{1}}$ is outgoing and $\sigma_{1}^{1}=0$ in $t<0$
$\Rightarrow\sigma_{1}^{1}=0$.

We do the above calculation for other profiles, same calculation shows that
$U_{j}$ has no $\theta_{1}$ mode if $U_{j-1}$ has no $\theta_{1}$ mode.
Therefore, $E_{P_{1}}U_{j}=0$ for all $j$. We will use this observation to
continue our determining the solutions.

We continue to solve the rest of $U_{0}$. The other two equations given by
(5.7) is :

(5.25) $\displaystyle E_{Q_{2}}[L(\partial)U_{0}+f(\theta_{3})MU_{0}]=0$
$\displaystyle E_{Q_{3}}[L(\partial)U_{0}+f(\theta_{3})MU_{0}]=0.$

Using proposition 3.9 and no resonance, we have:

(5.26) $\displaystyle X_{\phi_{2}}\sigma_{0}^{2}B_{2}r_{2}=0$ $\displaystyle
X_{\phi_{3}}\sigma_{0}^{3}B_{2}r_{3}+Q^{3}[f(\theta_{3})M\underline{U_{0}}+f(\theta_{3})M\sigma_{0}^{3}r_{3}]=0.$

Knowing $\underline{U_{0}}=0$ and $Mr_{3}=0$, the second equation become
$X_{\phi_{3}}\sigma_{0}^{3}B_{2}r_{3}=0$. To solve these transport equations,
we need a boundary condition for $\sigma_{0}^{2},\sigma_{0}^{3}$. To
summarize, we now have $U_{0}=\sigma_{0}^{2}r_{2}+\sigma_{0}^{3}r_{3}$.

On the boundary, we write:

(5.27)
$U_{0}=\sigma_{0}^{2}(t,x_{1},0,\theta_{0})r_{2}+\sigma_{0}^{3}(t,x_{1},0,\theta_{0})r_{3}=a_{0}(t,x_{1},\theta_{0})e+\check{a}_{0}(t,x_{1},\theta_{0})\check{e}$

where $e=e_{2}+e_{3}$,and $\check{e}=\check{e}_{2}+\check{e}_{3}$ for
$e_{m},\check{e}_{m}\in$span $r_{m}$. From (5.14), we have: $B\check{U}=0$,
thus $\check{a}_{0}=0$.

It is left to to determine $a_{0}$. We apply $\partial_{\theta_{0}}$ to both
sides of the first equation in (5.15). We already know that $EP_{1}U_{1}=0$.
We have121212Whenever the boundary equation involves the operator $R$ like
this, the evaluation at boundary is taken after applying $R$.:

(5.28) $\displaystyle\partial_{\theta_{0}}b\cdot
G^{*}+\partial_{\theta_{0}}b\cdot BR[L(\partial)U_{0}+f(\theta_{3})MU_{0}]$
$\displaystyle=0$ $\displaystyle\Leftrightarrow\ \partial_{\theta_{0}}b\cdot
BR[L(\partial)(a_{0}(t,x_{1},\theta_{2})e_{2}+a_{0}(t,x_{1},\theta_{3})e_{3}))$
$\displaystyle\hskip
72.26999pt+f(\theta_{3})M(a_{0}(t,x_{1},\theta_{2})e_{2}+a_{0}(t,x_{1},\theta_{3})e_{3}))]$
$\displaystyle=\partial_{\theta_{0}}b\cdot G^{*}.$

Knowing the right hand side, we can solve this equation by the discussion in
section 5.3, which gives the unique solution $a_{0}(t,x_{1},\theta_{0})$. Now
we know $\sigma_{0}^{2}(t,x_{1},0,\theta_{0}),$ and
$\sigma_{0}^{3}(t,x_{1},0,\theta_{0})$. Putting these boundary conditions with
interior equations (5.26), we have

(5.29) $\displaystyle\begin{cases}&X_{\phi_{2}}\sigma_{0}^{2}B_{2}r_{2}=0\\\
&\sigma_{0}^{2}(t,x_{1},0,\theta_{0})=a_{0}(t,x_{1},\theta_{0})e_{2}+\check{a}_{0}(t,x_{1},\theta_{0})\check{e}_{2}=a_{0}(t,x_{1},\theta_{0})e_{2}\end{cases}$
$\displaystyle\begin{cases}&X_{\phi_{3}}\sigma_{0}^{3}B_{2}r_{3}=0\\\
&\sigma_{0}^{3}(t,x_{1},0,\theta_{0})=a_{0}(t,x_{1},\theta_{0})e_{3}+\check{a}_{0}(t,x_{1},\theta_{0})\check{e}_{3}=a_{0}(t,x_{1},\theta_{0})e_{3}.\end{cases}$

These equations uniquely determined $\sigma_{0}^{2},\sigma_{0}^{3}$, thus we
now have $U_{0}$.

2\. Determining $U_{1}$:

We write

(5.30)
$U_{1}=\underline{U_{1}}+\sigma_{1}^{1}r_{1}+\sigma_{1}^{2}r_{2}+\sigma_{1}^{3}r_{3}+(I-E_{P})U_{1}$

where $(I-E_{P})U_{1}=-R[L(\partial)U_{0}+f(\theta_{3})MU_{0}]$.

We plug this equation in (5.7), we have:

(5.31) $\displaystyle 0$
$\displaystyle=E_{Q}[L(\partial)U_{1}+f(\theta_{3})MU_{1}]=E_{Q}[L(\partial)(E_{P}U_{1}+(I-E_{P})U_{1})+f(\theta_{3})M(E_{P}U_{1}+(I-E_{P})U_{1})]$
$\displaystyle\Leftrightarrow
E_{Q}[L(\partial)(E_{P}U_{1})+f(\theta_{3})ME_{P}U_{1}]=-E_{Q}[L(\partial)(I-E_{P})U_{1}+f(\theta_{3})M(I-E_{P})U_{1}].$

Taking the mean of this equation together with the boundary condition gives
$U_{1}$:

(5.32) $\displaystyle L(\partial)\underline{U_{1}}=0$ $\displaystyle
B\underline{U_{1}}=\underline{G}.$

This gives $\underline{U_{1}}$.

Taking the $\theta_{2},\theta_{3}$ mode of (5.31) yields equations:

(5.33) $\displaystyle
E_{Q_{2}}L(\partial)(E_{P_{2}}U_{1})=-E_{Q_{2}}[L(\partial)(I-E_{P})U_{1}+f(\theta_{3})M(I-E_{P})U_{1}]$
$\displaystyle
E_{Q_{3}}[L(\partial)(E_{P_{3}}U_{1})+f(\theta_{3})M(\underline{U_{1}}+\sigma_{3}^{3}r_{3})]=-E_{Q_{3}}[L(\partial)(I-E_{P})U_{1}+f(\theta_{3})M(I-E_{P})U_{1}].$

Using $Mr_{3}=0$ and moving the term involving $\underline{U_{1}}$ to the
right hand side, we have:

(5.34) $\displaystyle
X_{\phi_{2}}\sigma_{1}^{2}B_{2}r_{2}=-E_{Q_{2}}[L(\partial)(I-E_{P})U_{1}+f(\theta_{3})M(I-E_{P})U_{1}]$
$\displaystyle
X_{\phi_{3}}\sigma_{1}^{3}B_{2}r_{3}=-E_{Q_{3}}[L(\partial)(I-E_{P})U_{1}+f(\theta_{3})M(I-E_{P})U_{1}-f(\theta_{3})M\underline{U_{1}}].$

These are inhomogeneous transport equations. Knowing
$\sigma_{1}^{2},\sigma_{1}^{3}=0$ in $t<0$, we only need a boundary condition.
We have on the boundary:

(5.35)
$E_{P_{in}}U_{1}=\sigma_{1}^{2}(t,x_{1},0,\theta_{0})r_{2}+\sigma_{1}^{3}(t,x_{1},0,\theta_{0})r_{3}=a_{1}(t,x_{1},\theta_{0})e+\check{a}_{1}(t,x_{1},\theta_{0})\check{e}.$

The equation from (5.14)
$B\check{U}_{1}=G^{*}-BE_{P_{1}}U_{1}+BR[L(\partial)U_{0}+f(\theta_{3})MU_{0}]^{*}$
yields $\check{U}_{1}$ since the right hand side is known. It is left to
determine $a_{1}$. We apply $\partial_{\theta_{0}}$ to both sides of (5.15),
we have:

(5.36)
$\displaystyle\partial_{\theta_{0}}b\cdot[-BE_{P_{1}}U_{2}+BR(L(\partial)U_{1}+f(\theta_{3})MU_{1})^{*}]$
$\displaystyle=0$
$\displaystyle\Leftrightarrow\partial_{\theta_{0}}b\cdot[BR(L(\partial)(a_{1}(t,x_{1},\theta_{2})e_{2}+a_{0}(t,x_{1},\theta_{3})e_{3})+$
$\displaystyle\hskip
72.26999ptf(\theta_{3})M(a_{1}(t,x_{1},\theta_{2})e_{2}+a_{0}(t,x_{1},\theta_{3})e_{3}))^{*}]$
$\displaystyle=h_{1}$

where the second line is obtained by using $BE_{P_{1}}U_{2}=0$, moving
$\check{U}_{1},\underline{U_{1}},(I-E_{P})U_{1}$ terms to the right hand side,
and letting $h_{1}$ denotes the known right hand side. This equation yields
$a_{1}$ by the discussion in section 5.3. Thus we have now determined
$\sigma_{1}^{2},\sigma_{1}^{3}$ from the following equations:

(5.37)
$\displaystyle\begin{cases}&X_{\phi_{2}}\sigma_{1}^{2}B_{2}r_{2}=H_{1}^{2}\\\
&\sigma_{1}^{2}(t,x_{1},0,\theta_{0})=a_{1}(t,x_{1},\theta_{0})e_{2}+\check{a}_{1}(t,x_{1},\theta_{0})\check{e}_{2}\end{cases}$
$\displaystyle\begin{cases}&X_{\phi_{3}}\sigma_{1}^{3}B_{2}r_{3}=H_{1}^{3}\\\