# Formation and construction of a shock wave for one dimensional $n\times n$
strictly hyperbolic conservation laws with small smooth initial data 111The
research of Ding Min was supported in part by the National Natural Science
Foundation of China under Grant No.11701435, and in part by the Natural
Science Foundation of Hubei Province (No.202111j0209). The research of Yin
Huicheng was supported by the National Natural Science Foundation of China
(No.11731007) and the National key research and development program
(No.2020YFA0713803).

Ding Min1, Yin Huicheng2 222Email addresses: Ding Min (minding@whut.edu.cn),
Yin Huicheng<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Under the genuinely nonlinear assumptions for one dimensional $n\times n$
strictly hyperbolic conservation laws, we investigate the geometric blowup of
smooth solutions and the development of singularities when the small initial
data fulfill the generic nondegenerate condition. At first, near the unique
blowup point we give a precise description on the space-time blowup rate of
the smooth solution and meanwhile derive the cusp singularity structure of
characteristic envelope. These results are established through extending the
smooth solution of the completely nonlinear blowup system across the blowup
time. Subsequently, by utilizing a new form on the resulting 1-D strictly
hyperbolic system with $(n-1)$ good components and one bad component, together
with the choice of an efficient iterative scheme and some involved analyses, a
weak entropy shock wave starting from the blowup point is constructed.

1\. Department of Mathematics, College of Science, Wuhan University of

Technology, Wuhan, 430070, China.

2\. School of Mathematical Sciences and Institute of Mathematical Sciences,

Nanjing Normal University, Nanjing, 210023, China.

Key words: Geometric blowup, blowup system, cusp, genuinely nonlinear,

generic nondegenerate condition, shock wave

AMS Subject Classifications. 35L65, 35L67, 35L72

###### Contents

  1. 1 Introduction
  2. 2 Geometric blowup of the hyperbolic system
    1. 2.1 Reformulation
    2. 2.2 Boundary conditions on the $i$-th characteristics
    3. 2.3 Solvability of the blowup system
  3. 3 Analysis on the cusp singularity near the blowup point
    1. 3.1 The singular behavior of solutions in cusp domain
    2. 3.2 Estimates of the first approximation of shock solutions
  4. 4 Approximate shock solutions
    1. 4.1 Reformulated problem
    2. 4.2 Iteration schemes for the initial-value problem and initial-boundary value problem
    3. 4.3 The proof of Lemma $4.1$
  5. 5 Construction of a shock solution
  6. 6 Appendix

##  1 Introduction

As is well known, no matter how smooth and how small the initial data are, the
classical solutions will generally form singularities in finite time for one
dimensional strictly quasilinear hyperbolic conservation laws with genuinely
nonlinear structures (see [1], [14], [17] and [23]). Therefore, it is
important to understand the physical process of singularity development from
the smooth solutions and the evolution of singularities starting from the
blowup points.

Consider the following Cauchy problem for one dimensional $n\times n$ strictly
hyperbolic conservation law

$\begin{cases}\partial_{t}u+\partial_{x}f(u)=0,\\\ u(x,0)=\epsilon
u_{0}(x),\end{cases}$ (1.1)

where $t\geq 0$, $x\in\mathbb{R}$, $\epsilon>0$ is a sufficiently small
parameter, $u=(u_{1},\cdots,u_{n})^{\top}\in\mathbb{R}^{n}$,
$f(u)=(f_{1}(u),\cdots,f_{n}(u))^{\top}\in C^{\infty}$, and

$u_{0}(x)=(u^{1}_{0}(x),\cdots,u^{n}_{0}(x))^{\top}\in
C^{\infty}_{0}(\mathbb{R}),\quad\text{supp }u_{0}(x)\subset[a,b],$ (1.2)

here $a$ and $b$ with $a<b$ are constants.

For $C^{1}$ solution $u$, the system in $\eqref{eq:1.1}_{1}$ can be rewritten
as

$\displaystyle\partial_{t}u+F(u)\partial_{x}u=0,$ (1.3)

where $F(u)=\nabla_{u}f(u)$. By the strict hyperbolicity of (1.3),
$\det(\lambda\mathbb{I}_{n}-F(u))=0$ has $n$ distinct real eigenvalues, which
are denoted by

$\lambda_{1}(u)<\lambda_{2}(u)<\cdots<\lambda_{n}(u).$ (1.4)

The corresponding left and right eigenvectors are $l_{1}(u)$, $\cdots$,
$l_{n}(u)$, and $r_{1}(u)$, $\cdots$, $r_{n}(u)$, respectively. If

$\nabla_{u}\lambda_{j}(u)\cdot r_{j}(u)\neq 0\quad\text{ for all }u,$

then (1.3) is called genuinely nonlinear with respect to $\lambda_{j}$.
Otherwise, if

$\nabla_{u}\lambda_{j}(u)\cdot r_{j}(u)\equiv 0\quad\text{ for all }u,$

then (1.3) is called linearly degenerate for $\lambda_{j}$.

When $\eqref{Y-1}$ is genuinely nonlinear for all eigenvalues $\lambda_{j}(u)$
($1\leq j\leq n$), and the initial data $u(x,0)=\psi(x)\in C^{2}$ satisfy that
$\text{supp}\psi\subseteq[a,b]$ holds and
$(b-a)\sup_{x\in\mathbb{R}}|\psi^{\prime\prime}(x)|$ is sufficiently small
(but $\psi\not\equiv 0$), F. John [15] introduced the wave decomposition
method to prove that the first order derivatives of $u$ blow up in finite
time. T.P. Liu [20] generalized the result of [15] to the system $\eqref{Y-1}$
with the features that some eigenvalues are genuinely nonlinear and others are
linearly degenerate. L. Hörmander [13] improved the results in [15] and [20]
such that a sharp estimate for the lifespan $T_{\epsilon}$ of smooth solutions
is established. In addition, employing the method of geometric optics, S.
Alinhac [2] reconsidered the result in [13], and gave a more precise
description on $T_{\epsilon}$ through the asymptotic expansion form of
$\epsilon$. As shown in [13, 14], the lifespan $T_{\epsilon}$ of
$\eqref{eq:1.1}$ satisfies

$\lim_{\epsilon\to 0^{+}}\epsilon
T_{\epsilon}=-\displaystyle\frac{1}{\displaystyle\min_{1\leq j\leq n}M_{j}},$
(1.5)

where $M_{j}=\min_{x}h_{j}(x)$ and
$h_{j}(x)=\displaystyle\frac{\nabla_{u}\lambda_{j}(0)r_{j}(0)}{l_{j}(0)r_{j}(0)}l_{j}(0)u_{0}^{\prime}(x)$.

Recently, motivated by [4], through taking the efficient decomposition of $u$
along the different characteristic directions, the involved analysis on the
characteristics with large variations, the suitable introduction of the
modulated coordinates together with the global weighted energy estimates and
the characteristics method, Li-Xu-Yin [19] have established the geometric
blowup mechanism of smooth large solutions to (1.3) for a class of initial
data $u(x,0)$. The geometric blowup by the terminology in [1] means that up to
the blowup time $T_{*}$, the solution $u\in C([0,T_{*})\times\mathbb{R})\cap
L^{\infty}([0,T_{*}]\times\mathbb{R})$ but
$\|\nabla_{x,t}u(\cdot,t)\|_{L^{\infty}}\to\infty$ holds as $t\to T_{*}-$. In
the present paper, we wonder to know under which condition, a shock can be
generated and how it develops from the blowup time. For the 1-D or M-D scalar
conservation laws, the authors in [25] and [26] have solved the problem of
shock formation and construction under the various assumptions of initial
data. For the $p$-system of gases dynamics, under the assumptions that one
Riemann invariant is a constant and the initial data satisfy the related
generic nondegenerate condition, M.P. Lebaud [18] constructed a shock solution
from the blowup point. This result was extended to the more general case of
$p$-system in [5] and [16], where both the Riemann invariants are not
constants and it is additionally assumed that only one family of
characteristics is squeezed, while another family of the characteristics does
not squeeze at the same point. Here it is pointed out that the existence of
Riemann invariants plays a crucial role in the analysis of [5, 16, 18] since
the $2\times 2$ $p$-system can be diagonalized in this situation. For the
$3\times 3$ case of (1.1), by introducing a suitable invertible transformation
$u=u(w)$ to find the new unknown function $w=(w_{1},w_{2},w_{3})$ with two
good components $(w_{1},w_{3})$ and one bad component $w_{2}$, a 2-shock
starting from the blowup point is constructed in [6] under the generic
nondegenerate condition of initial data and by the extension result of smooth
solutions to the resulting blowup system across the blowup time in [2]. For
the three dimensional full compressible Euler equations with spherical
symmetric structure, a symmetric shock solution after the blowup time is
constructed in [24] (also see the independent work of [9]). In addition, the
authors in [3] study the simultaneous development and cusps for the 2D
compressible Euler system with azimuthal symmetry from smooth data.

Benefiting from the analysis of the blowup mechanism of smooth solutions for
the $3\times 3$ strictly hyperbolic system (1.3) in [2], and the construction
of a formed shock for the $3\times 3$ hyperbolic conservation law (1.1) in
[6], we shall study the shock formation and construction of (1.1) for the
general case of $n\geq 3$. For this purpose, at first, we will derive the
geometric blowup mechanism of (1.1) and extend the smooth solution of the
corresponding blowup system across the lifespan $T_{\epsilon}$. Note that for
treating the geometric blowup mechanism of $3\times 3$ case in (1.1), S.
Alinhac [2] utilized the geometric optics method and the special properties of
$3\times 3$ system that the solution $u$ is 1-simple and 3-simple on the left
side of 2-characteristics through $(a,0)$ and on the right side of
2-characteristics through $(b,0)$, respectively. In the paper, we will
directly deal with the blowup system of (1.1) for all $n\geq 3$ by solving the
related initial-boundary value problem through some basic observations on the
properties of the corresponding boundary values and on the nonlinear structure
of blowup system, where the geometric optics method and the applications on
the exclusive properties of $3\times 3$ system in [2] are avoided. We point
out that it is difficult for us to directly use the methods in [2] to solve
the blowup mechanism of the general hyperbolic system (1.3) with $n\geq 4$,
some resulting difficulties require to be overcome by new ingredients and
careful observations. Based on the extension property of smooth solution to
the blowup system across $T_{\epsilon}$ and the cusp property of
characteristic envelope, through choosing a new form of (1.1) such that its
solution is more singular along one direction than other left directions, and
taking the corresponding iterative scheme, the weak entropy shock solution
issuing from the blowup point of (1.1) can be constructed, meanwhile, the
detailed descriptions on the location of the shock as well as the estimates of
the solution near the blowup point are also given. Although the main argument
procedures for the uniform boundedness and convergence of the iterative scheme
of approximate shock solutions are analogous to those in [6] and [24], we
still give all the details due to the general forms of $n\times n$ cases
together with more precise and complete computations.

The paper is organized as follows. In §$2$, we study the geometric blowup
mechanism and extend the smooth solution of the blowup system across
$T_{\epsilon}$ for problem (1.1). In order to solve the blowup system, some
suitable boundary conditions and boundary values are derived by basic
observations. In addition, the precise descriptions on the formation and
construction of a shock wave are given. In §$3$, close to the blowup point,
the crucial cusp properties and estimates on the solutions of the blowup
system are derived. In §$4$, by introducing a transformation to fix the free
shock curve and taking a suitable iterative scheme to construct the
approximate shock wave solutions which satisfy the Rankine-Hugoniot conditions
and the Lax’s geometric entropy conditions, we can trace the location of the
approximate shock and obtain the estimates of approximate solutions. In §$5$,
the convergence of the approximate shock solutions is shown. Subsequently, the
main conclusions are established.

##  2 Geometric blowup of the hyperbolic system

To study the geometric blowup of the hyperbolic system (1.3), motivated by [6]
for the $3\times 3$ case, we now give a generalized simplification for the
general $n\times n$ case ($n\geq 3$) as follows.

######  Lemma 2.1.

Assume that (1.4) holds and (1.3) is genuinely nonlinear with respect to the
eigenvalue $\lambda_{i}(u)$ (fixed number $i$ with $1\leq i\leq n$). Then
there exists an invertible transformation in the neighbourhood of the origin:
$u\longrightarrow w(u)$ with $w(0)=0$ such that (1.3) can be equivalently
reduced into

$\begin{cases}\partial_{t}w+A(w)\partial_{x}w=0,\\\\[5.0pt] w(x,0)=\epsilon
w_{0}(x)+\epsilon^{2}w_{1}(x,\epsilon),\end{cases}$ (2.1)

where $w_{0}(x)=(w^{1}_{0}(x),\cdots,w^{n}_{0}(x))^{\top}$ and
$w_{1}(x,\epsilon)=(w_{1}^{1}(x,\epsilon),\cdots,w_{n}^{1}(x,\epsilon))$ are
$C^{\infty}$ about their arguments and compactly supported in $[a,b]$ for the
variable $x$, the $n\times n$ matrix $\displaystyle
A(w)=(\partial_{u}w)F(u(w))(\partial_{u}w)^{-1}:=(a_{ij})_{n\times n}$
satisfies that

  * $(1)$

the eigenvalues of $A(w)$ are $\lambda_{1}(u(w)),\cdots,\lambda_{n}(u(w))$;

  * $(2)$

for $j\neq i$, $a_{ji}=0$, and $a_{ii}=\lambda_{i}(u(w))$;

  * $(3)$

the $i$-th right eigenvector of $A(w)$ is
$r_{i}(w):=(\partial_{u}w)r_{i}(u(w))$, which is parallel to the unit vector
$(0,\cdots,1,0,\cdots,0)^{\top}$;

  * $(4)$

$A(0)=\text{diag}\Big{(}\lambda_{1}(0),\lambda_{2}(0),\cdots,\lambda_{n}(0)\Big{)}$.

###### Proof.

By Theorem $8.1$ in [11], we know that there exist $(n-1)$ Riemann invariants
$q_{j}(u)$ $(j\neq i)$ whose gradients are linearly independent and satisfy
for $|u|\ll 1$

$\nabla_{u}q_{j}(u)\cdot r_{i}(u)=0,\qquad j\neq i.$ (2.2)

Let $\\{\zeta_{j}\\}_{j\neq i}$ be $(n-1)$ linearly independent column
constant vectors orthogonal to $r_{i}(0)$. Inspired by (2.2), set

$q_{j}(u)=\zeta_{j}\cdot u+\tilde{q}_{j}(u),$ (2.3)

where $\\{\tilde{q}_{j}(u)\\}_{j\neq i}$ satisfy

$\begin{cases}\nabla_{u}\tilde{q}_{j}(u)\cdot
r_{i}(u)=-\zeta_{j}\cdot(r_{i}(u)-r_{i}(0)),\\\ \tilde{q}_{j}(0)=0,\quad j\neq
i.\end{cases}$ (2.4)

From the standard theory of the first order scalar quasilinear partial
differential equations, problem (2.4) is uniquely solved for $|u|\ll 1$ and
$\tilde{q}_{j}(u)=O(|u|^{2})$ holds.

Introduce a transformation: $u\to\tilde{u}$ as

$\tilde{u}_{j}=q_{j}(u)\quad\text{for $j\neq
i$},\qquad\tilde{u}_{i}=r^{\top}_{i}(0)\cdot u.$ (2.5)

Then

$\det(\partial_{u}\tilde{u})|_{u=\vec{0}}=\det(\zeta_{1},\cdots,\zeta_{i-1},\
r_{i}(0),\ \zeta_{i+1},\cdots,\zeta_{n})^{\top}\neq 0,$

and the mapping $u\to\tilde{u}(u)$ is invertible when $|u|\ll 1$.

Under the transformation (2.5), the system (1.3) can be reduced into

$\partial_{t}\tilde{u}+\tilde{A}(\tilde{u})\partial_{x}\tilde{u}=0,$ (2.6)

where
$\tilde{A}(\tilde{u})=(\partial_{u}\tilde{u})F(u)(\partial_{u}\tilde{u})^{-1}=\Big{(}\tilde{a}_{ij}(\tilde{u})\Big{)}_{n\times
n}$. By direct calculations, it is easy to know that $\tilde{A}(\tilde{u})$
has $n$ distinct eigenvalues $\\{\lambda_{k}(\tilde{u})\\}^{n}_{k=1}$ with
$\lambda_{k}(\tilde{u})=\lambda_{k}(u(\tilde{u}))$ and the corresponding right
eigenvectors are
$\Big{\\{}(\partial_{u}\tilde{u})r_{k}(u)\Big{\\}}^{n}_{k=1}$. Moreover, it
holds that

$(\partial_{u}\tilde{u})r_{i}(u)=r^{\top}_{i}(0)\cdot
r_{i}(u(\tilde{u}))\bm{e_{i}}\neq\vec{0},$

which implies that

$\tilde{a}_{ji}(\tilde{u})=0\;\;\text{for $j\neq
i$},\qquad\tilde{a}_{ii}(\tilde{u})=\lambda_{i}(\tilde{u}).$

Then the matrix
$\tilde{A}_{n-1}(\tilde{u})=(\tilde{a}_{ij})_{(n-1)\times(n-1)}$ has $(n-1)$
distinct eigenvalues

$\lambda_{i}(\tilde{u})<\cdots<\lambda_{i-1}(\tilde{u})<\lambda_{i+1}(\tilde{u})<\cdots<\lambda_{n}(\tilde{u}).$

Therefore, there exists a $(n-1)\times(n-1)$ invertible constant matrix
$B_{n-1}=(b_{ij})_{(n-1)\times(n-1)}$ such that

$B_{n-1}\tilde{A}_{n-1}(0)B^{-1}_{n-1}=\text{diag}(\lambda_{1}(0),\cdots,\lambda_{i-1}(0),\lambda_{i+1}(0),\cdots,\lambda_{n}(0)).$

Let

$\left(\begin{array}[]{c}w_{1}\\\\[3.0pt] \vdots\\\ \hat{w}_{i}\\\\[3.0pt]
\vdots\\\
w_{n}\end{array}\right)=B_{n-1}\left(\begin{array}[]{c}\tilde{u}_{1}\\\\[3.0pt]
\vdots\\\ \hat{\tilde{u}}_{i}\\\\[3.0pt] \vdots\\\
\tilde{u}_{n}\end{array}\right),$ (2.7)

where $\hat{w}_{i}$ and $\hat{\tilde{u}}_{i}$ represent that the related
$(n-1)$ dimensional vectors do not contain the components $w_{i}$ and
$\tilde{u}_{i}$ respectively. It follows from (2.6) and (2.7) that

$\partial_{t}\left(\begin{array}[]{c}w_{1}\\\\[3.0pt] \vdots\\\
\hat{w}_{i}\\\\[3.0pt] \vdots\\\
w_{n}\end{array}\right)+\bar{B}_{n-1}(w_{1},\cdots,\tilde{u}_{i},\cdots,w_{n})\partial_{x}\left(\begin{array}[]{c}w_{1}\\\\[3.0pt]
\vdots\\\ \hat{w}_{i}\\\\[3.0pt] \vdots\\\ w_{n}\end{array}\right)=0,$ (2.8)

where

$\bar{B}_{n-1}(w_{1},\cdots,\tilde{u}_{i},\cdots,w_{n})=B_{n-1}\tilde{A}_{n-1}(w)B^{-1}_{n-1}$

and

$\bar{B}_{n-1}(0)=\text{diag}(\lambda_{1}(0),\cdots,\lambda_{i-1}(0),\lambda_{i+1}(0),\cdots,\lambda_{n}(0)).$

Let

$\tilde{u}_{i}=w_{i}+\sum_{j\neq i}k_{j}w_{j},$ (2.9)

where $k_{j}$ are some constants determined later. In this case, equation
$\eqref{Eq:2.106}_{i}$ can be rewritten as

$\begin{split}\partial_{t}w_{i}+\tilde{a}_{ii}\partial_{x}w_{i}+\sum_{j\neq
i}\Big{(}\tilde{a}_{ij}+k_{j}(\tilde{a}_{ii}-\tilde{a}_{jj})-\sum_{l\neq
i,j}k_{l}\tilde{a}_{lj}\Big{)}\partial_{x}w_{j}=0.\end{split}$

By $\tilde{a}_{ii}(0)=\lambda_{i}(0)$, we set the following equalities

$k_{j}(\lambda_{i}(0)-\tilde{a}_{jj}(0))-\sum_{l\neq
i,j}k_{l}\tilde{a}_{lj}(0)=-\tilde{a}_{ij}(0)\qquad\text{for $j\neq i$}.$
(2.10)

Due to

$\det\left(\begin{matrix}\lambda_{i}(0)-\tilde{a}_{11}(0)&-\tilde{a}_{21}(0)&\cdots&-\tilde{a}_{(i-1)1}(0)&-\tilde{a}_{(i+1)1}(0)&\cdots&-\tilde{a}_{n1}(0)\\\\[5.0pt]
-\tilde{a}_{12}(0)&\lambda_{i}(0)-\tilde{a}_{22}(0)&\cdots&-\tilde{a}_{(i-1)2}(0)&-\tilde{a}_{(i+1)2}(0)&\cdots&-\tilde{a}_{n2}(0)\\\\[5.0pt]
\vdots\\\\[5.0pt]
-\tilde{a}_{1n}(0)&-\tilde{a}_{2n}(0)&\cdots&-\tilde{a}_{(i-1)1}(0)&-\tilde{a}_{(i+1)1}(0)&\cdots&\lambda_{i}(0)-\tilde{a}_{nn}(0)\end{matrix}\right)_{n-1}\neq
0,$

then $k_{j}$ ($j\not=i$) can be uniquely solved from (2.10).

From (2.3), (2.5), (2.7) and (2.9), $A(w)$ with properties (1)-(4) can be
obtained. Then Lemma 2.1 is proved. ∎

Under Lemma 2.1, in terms of (1.5), the lifespan $T_{\epsilon}$ of (1.1) can
be expressed as

$\lim_{\epsilon\to 0^{+}}\epsilon
T_{\epsilon}=-\displaystyle\frac{1}{\displaystyle\min_{1\leq j\leq
n}N_{j}}>0,$ (2.11)

where $N_{j}=\min_{x}\mathcal{H}_{j}(x)$ and
$\mathcal{H}_{j}(x)=\partial_{u_{j}}\lambda_{j}(0)(w_{0}^{j})^{\prime}(x)$.
Without loss of generality,

$N_{i}<\displaystyle\min_{j\not=i}N_{j}$ (2.12)

is assumed, meanwhile the following generic nondegenerate condition is posed:

There exists a unique point $x_{0}$ such that

$\mathcal{H}_{i}(x_{0})=N_{i}=\min\mathcal{H}_{i}(x),\quad\mathcal{H}^{\prime}_{i}(x_{0})=0,\quad\mathcal{H}^{\prime\prime}_{i}(x_{0})>0.$
(2.13)

Denote $x=\varphi_{j}(y,t)$ by the $j$-th characteristics of (2.1) passing
through the point $(y,0)$. Set $D_{j,t_{0}}=\\{(x,t):\varphi_{j}(a,t)\leq
x\leq\varphi_{j}(b,t),t_{0}\leq t\leq T_{\epsilon}\\}$ for some large fixed
$t_{0}>0$ (see Figure $1$). In this case, the domains $D_{j,t_{0}}$ for
different $j$ ($1\leq j\leq n$) are disjoint. On the other hand, it follows
from Chapter 4 of [14] or [15] that the blowup points at the blowup time only
appear in $D_{i,t_{0}}$ under the assumption (2.12).

For simplicity, we denote $x=\varphi(y,t)$ as the $i$-th characteristics of
(2.1) passing through the point $(y,t_{0})$, which means

$\begin{cases}\partial_{t}\varphi(y,t)=\lambda_{i}(w(\varphi(y,t),t)),\\\
\varphi(y,t_{0})=y.\end{cases}$ (2.14)

Define $a_{0}=\varphi_{i}(a,t_{0})$ and $b_{0}=\varphi_{i}(b,t_{0})$.

$t=0$$(a,0)$$(b,0)$$D_{i,t_{0}}$$D_{j,t_{0}}$$D_{l,t_{0}}$$D_{k,t_{0}}$$(a_{0},t_{0})$$(b_{0},t_{0})$$t=t_{0}$$t=T_{\epsilon}$Figure
$1$. Domain $D_{j,t_{0}}$

Set $v(y,t)=w(\varphi(y,t),t)$. Then it follows from (2.1) and (2.14) together
with direct computation that

$\begin{cases}\partial_{t}\varphi(y,t)=\lambda_{i}(v),\\\\[3.0pt]
l_{i}(v)\partial_{t}v=0,\\\\[5.0pt]
l_{j}(v)\Big{(}\partial_{t}v\partial_{y}\varphi(y,t)+(\lambda_{j}-\lambda_{i})(v(y,t))\partial_{y}v\Big{)}=0,\qquad
j\neq i,\end{cases}$ (2.15)

which is called the blowup system corresponding to the $i$-th eigenvalue
$\lambda_{i}(w)$ by the terminology in [1]-[2]. Note that (2.15) is a
completely nonlinear evolution system of $(\varphi,v)$, which is degenerate at
the points satisfying $\partial_{y}\varphi(y,t)=0$.

Next, we illustrate the result on the extension of smooth solution
$(\varphi,v)$ of (2.15) across $T_{\epsilon}$ when the initial data are given
by

$\varphi(y,t_{0})=y,\quad v(y,t_{0})=w(y,t_{0}).$ (2.16)

######  Theorem 2.2.

Assume that (1.4) holds and (1.3) is genuinely nonlinear with respect to
$\lambda_{i}(u)$. Under assumption (2.13), there exist a small constant
$\delta_{0}$ and a unique smooth solution $(\varphi,v)(y,t)$ to (2.15)-(2.16)
in the domain $D=\\{(y,t):a_{0}\leq y\leq b_{0},t_{0}\leq t\leq
T_{\epsilon}+\delta_{0}\\}$. Moreover, the following estimates hold that for
$|\alpha|\geq 0$ and $(y,t)\in D$,

$|\partial_{y,t}^{\alpha}\varphi(y,t)|\leq
C_{\alpha},\quad|\partial^{\alpha}_{y,t}v(y,t)|\leq C_{\alpha}\epsilon,$
(2.17)

where $C_{\alpha}$ stands for the generic positive constant independent of
$\epsilon$.

######  Remark 2.3.

By Chapter 4 of [14] or [15], we know that the smooth solution of (1.1) exists
in $\\{(y,t):y\in\mathbb{R},0\leq t\leq T_{\epsilon}+\delta_{0}\\}\setminus
D$. Moreover, when $(y,t)\in D\cap\\{(y,t):y\in\mathbb{R},t_{0}\leq t\leq
T_{\epsilon}\\}$, $\partial_{x,t}^{\alpha}v_{j}(x,t)=O(\epsilon^{2})$ hold for
$j\not=i$ and $|\alpha|>0$, while $v(x,t)=O(\epsilon)$. In addition, without
loss of generality, we will assume $\delta_{0}=1$ in Theorem 2.2 from now on.

######  Remark 2.4.

If there exist two numbers $N_{i_{0}}$ and $N_{j_{0}}$ with
$N_{i_{0}}=N_{j_{0}}$ ($i_{0}\not=j_{0}$, $1\leq i_{0},j_{0}\leq n$) in
(2.11), and the first blowup point appears in $D_{i_{0},t_{0}}$, then under
the condition (2.13) for $\mathcal{H}_{i_{0}}(x)$, Theorem 2.2 holds
analogously by the property of $D_{i_{0},t_{0}}\cap
D_{j_{0},t_{0}}=\emptyset$.

######  Remark 2.5.

In this paper, assume $n\geq 3$ and $2\leq i\leq n-1$. In fact, for $i=1$ or
$i=n$, it is much simpler to show Theorem 2.2 since the solution $u$ is
$1$-simple on the left side of the $2$-nd characteristics, and is $n$-simple
on the right side of the $(n-1)$-th characteristics.

######  Remark 2.6.

In recent years, the studies on the shock formation of smooth solutions to the
multidimensional hyperbolic conservation laws or the second order potential
equations of polytropic gases have made much progress (see [4], [7], [8],
[10], [12] and [22]), which illustrate that the formation of the
multidimensional shock is due to the compression of the characteristic
surfaces. However, the related constructions of a multidimensional shock wave
after the blowup of smooth solutions are not obtained.

###  2.1 Reformulation

Introduce a quantity in the domain $D=\\{(y,t):a_{0}\leq y\leq b_{0},t_{0}\leq
t\leq T_{\epsilon}+1\\}$ as

$v(y,t)=\epsilon D_{\epsilon}\omega$

with the matrix $D_{\epsilon}=\text{diag}(\epsilon,\cdots,1,\cdots,\epsilon)$
(the number $1$ is at the $(i,i)-$position).

We shall investigate the blowup system (2.15) reformulated by $\omega$. It
follows from Remark 2.3 that one can only expect the uniform boundedness of
$D_{\epsilon}\omega$ (rather than $\omega$) and
$\partial_{x,t}^{\alpha}\omega$ with $|\alpha|>0$ if $n\geq 4$, which is
different from the case of $n=3$ in [2] (where $\partial_{x,t}^{\alpha}\omega$
for all $|\alpha|\geq 0$ are uniformly bounded).

In addition, set

$\tilde{l}_{j}(\epsilon D_{\epsilon}\omega)=l_{j}(\epsilon
D_{\epsilon}\omega)D_{\epsilon},\quad\tilde{r}_{j}=D^{-1}_{\epsilon}r_{j}(\epsilon
D_{\epsilon}\omega),\quad j=1,\cdots,n.$

Note that $\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega)=\epsilon(l_{j1},...,0,...,l_{jn})$ for $j\neq i$ includes
the small factor $\epsilon$ due to the simplification in Lemma 2.1.

Then the blowup system (2.15) can be reduced into

$\begin{cases}\partial_{t}\varphi(y,t)=\lambda_{i}(\epsilon
D_{\epsilon}\omega),\\\\[3.0pt] \tilde{l}_{i}(\epsilon
D_{\epsilon}\omega)\partial_{t}\omega=0,\\\\[5.0pt] \tilde{l}_{j}(\epsilon
D_{\epsilon}\omega)\Big{(}K\partial_{t}\omega+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega)\partial_{y}\omega\Big{)}=0,\qquad j\neq i,\end{cases}$
(2.18)

where $K:=\partial_{y}\varphi(y,t)$.

Motivated by [2], set

$h_{i}=\tilde{l}_{i}\partial_{y}\omega,\quad
h_{j}=\tilde{l}_{j}\partial_{t}\omega\;\;\text{for $j\neq i$}.$ (2.19)

This leads to

$\partial_{t}\omega=\sum_{j\neq
i}h_{j}\tilde{r}_{j},\quad\partial_{y}\omega=-\sum_{j\neq
i}\frac{Kh_{j}}{\lambda_{j}-\lambda_{i}}\tilde{r}_{j}+h_{i}\tilde{r}_{i}.$
(2.20)

Due to $\eqref{eq:2.2}_{1}$, one has

$\partial_{t}K=\epsilon\nabla\lambda_{i}D_{\epsilon}\partial_{y}\omega=\epsilon
L^{(1)}(\epsilon D_{\epsilon}\omega)\tilde{h}K+\epsilon\nabla\lambda_{i}\cdot
r_{i}h_{i},$ (2.21)

where and below $L^{(k)}(\epsilon D_{\epsilon}\omega)$ represents the row
vector depending on $\epsilon D_{\epsilon}\omega$, and
$\tilde{h}=(h_{1},\cdots,\hat{h}_{i},\cdots,h_{n})^{\top}$ stands for the
vector that replaces the component $h_{i}$ with $0$.

Next, we derive the equation of $h_{j}$ for $j=1,\cdots,n$. Differentiating
$\eqref{eq:2.2}_{2}$ with respect to $y$ yields

$\epsilon\Big{(}\nabla\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega)D_{\epsilon}\partial_{y}\omega\Big{)}^{\top}\partial_{t}\omega+\tilde{l}_{i}\partial^{2}_{ty}\omega=0,$
(2.22)

where

$\tilde{l}_{i}\partial^{2}_{ty}\omega=\partial_{t}h_{i}-\epsilon(\nabla\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega)D_{\epsilon}\partial_{t}\omega)^{\top}\partial_{y}\omega.$
(2.23)

By (2.20) and (2.22)-(2.23), we arrive at

$\begin{split}\partial_{t}h_{i}=&\epsilon K\sum_{j\neq i,k\neq
i}h_{j}h_{k}\frac{\lambda_{k}-\lambda_{j}}{(\lambda_{j}-\lambda_{i})(\lambda_{k}-\lambda_{i})}(\nabla\tilde{l}_{i}r_{j})^{\top}\tilde{r}_{k}+\epsilon
h_{i}\sum_{j\neq
i}h_{j}\Big{(}(\nabla\tilde{l}_{i}r_{j})^{\top}\tilde{r}_{i}-(\nabla\tilde{l}_{i}r_{i})^{\top}\tilde{r}_{j}\Big{)}\\\
=&\epsilon\tilde{h}^{\top}Q^{(1)}(\epsilon
D_{\epsilon}\omega)\tilde{h}K+\epsilon L^{(2)}(\epsilon
D_{\epsilon}\omega)\tilde{h}h_{i},\end{split}$

where and below $Q^{(k)}(\epsilon D_{\epsilon}\omega)$ represents a symmetric
$n\times n$ matrix.

On the other hand, taking the first order derivative of $\eqref{eq:2.2}_{3}$
with respect to $t$, one has

$\begin{split}\tilde{l}_{j}\Big{(}K\partial_{t}+(\lambda_{j}-&\lambda_{i})\partial_{y}\Big{)}\partial_{t}\omega+\tilde{l}_{j}\Big{(}\partial_{t}K\partial_{t}\omega+\epsilon\nabla(\lambda_{j}-\lambda_{i})D_{\epsilon}\partial_{t}\omega\partial_{y}\omega\Big{)}\\\
&+\epsilon(\nabla\tilde{l}_{j}D_{\epsilon}\partial_{t}\omega)^{\top}(K\partial_{t}\omega+(\lambda_{j}-\lambda_{i})\partial_{y}\omega)=0,\end{split}$

which drives

$\begin{split}&\Big{(}K\partial_{t}+(\lambda_{j}-\lambda_{i})\partial_{y}\Big{)}h_{j}+\epsilon
K\Big{(}\tilde{h}^{\top}Q^{(2)}(\epsilon
D_{\epsilon}\omega)\tilde{h}+L^{(3)}(\epsilon
D_{\epsilon}\omega)\tilde{h}h_{j}\Big{)}\\\ &\qquad\quad+\epsilon
h_{i}\Big{(}r^{\top}_{i}Q^{(3)}(\epsilon
D_{\epsilon}\omega)\tilde{h}+\nabla\lambda_{i}(\epsilon
D_{\epsilon}\omega)r_{i}(\epsilon
D_{\epsilon}\omega)h_{j}\Big{)}=0.\end{split}$

Therefore, the blowup system (2.18) in domain $D$ can be reformulated as

$\begin{cases}\partial_{t}K=\epsilon L^{(1)}(\epsilon
D_{\epsilon}\omega)\tilde{h}K+\epsilon\nabla\lambda_{i}\cdot r_{i}(\epsilon
D_{\epsilon}\omega)h_{i},\\\\[5.0pt]
\partial_{t}h_{i}=\epsilon\tilde{h}^{\top}Q^{(1)}(\epsilon
D_{\epsilon}\omega)\tilde{h}K+\epsilon L^{(2)}(\epsilon
D_{\epsilon}\omega)\tilde{h}h_{i},\\\\[5.0pt]
\Big{(}K\partial_{t}+(\lambda_{j}-\lambda_{i})\partial_{y}\Big{)}h_{j}+\epsilon
K\Big{(}\tilde{h}^{\top}Q^{(2)}(\epsilon
D_{\epsilon}\omega)\tilde{h}+L^{(3)}(\epsilon
D_{\epsilon}\omega)\tilde{h}h_{j}\Big{)}\\\\[3.0pt] \qquad\qquad\quad+\epsilon
h_{i}\Big{(}r^{\top}_{i}Q^{(3)}(\epsilon
D_{\epsilon}\omega)\tilde{h}+\nabla\lambda_{i}(\epsilon
D_{\epsilon}\omega)r_{i}(\epsilon
D_{\epsilon}\omega)h_{j}\Big{)}=0,\\\\[5.0pt] \tilde{l}_{i}(\epsilon
D_{\epsilon}\omega)\partial_{t}\omega=0,\\\\[5.0pt] \tilde{l}_{j}(\epsilon
D_{\epsilon}\omega)\Big{(}K\partial_{t}\omega+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega)\partial_{y}\omega\Big{)}=0,\qquad j\neq i.\end{cases}$
(2.24)

From (2.24), it is known that $K$ and $h_{i}$ can be solved by the direct
integration on $t$ through their initial data, while $h_{j}$ ($j\not=i$) are
determined by their initial data and suitable boundary values on $y=a_{0}$ or
$y=b_{0}$ (the signs of $K$ and $\lambda_{j}-\lambda_{i}$ on the boundaries
play a key role). Additionally, we have to overcome the difficulties arising
from the boundedness of $D_{\epsilon}\omega$ rather than $\omega$ since the
uniform bounds or smallness orders of $(K,h,D_{\epsilon}\omega)$ and their
derivatives should be derived. In this process, the basic observations on the
tangent derivatives
$\partial_{t}h_{i}|_{y=a_{0}}=O(\epsilon),\partial_{t}h_{i}|_{y=b_{0}}=O(\epsilon)$
and $\partial_{t}K=O(\epsilon)$ are very necessary.

###  2.2 Boundary conditions on the $i$-th characteristics

In this subsection, we will mainly study the estimates of $h_{k}$ and
$\tilde{l}_{k}w$ on the boundaries $y=a_{0}$ or $y=b_{0}$ of the domain $D$
for the integer $k\neq i$.

By $\text{supp }u_{0}(x)\subset[a,b]$, for the solution $u$ of problem (1.1)
one has

$\text{supp }u(x,t)\subset\Big{\\{}(x,t):a+\lambda_{1}(0)t\leq x\leq
b+\lambda_{n}(0)t,t\geq 0\Big{\\}}.$

The $i$-th characteristics $\Gamma_{i}:x=\varphi(y,t)$ is defined in (2.14).
Set

$a(t)=\varphi(a_{0},t),\quad b(t)=\varphi(b_{0},t),$

and the domain $R_{i,t_{0}}$ (see Figure $2$) can be described as

$R_{i,t_{0}}:=\\{(x,t):a(t)\leq x\leq b(t),t_{0}\leq t\leq t_{1}\\},$

where and below $t_{1}=:T_{\epsilon}+1$ is defined.

$x$$t=t_{0}$$(a_{0},t_{0})$$(b_{0},t_{0})$$R_{i,t_{0}}$$\Gamma_{i,-}:x=a(t)$$\Gamma_{i,+}:x=b(t)$Figure
$2$. Domain $R_{i,t_{0}}$

Since we will investigate the well-posedness of solution of (2.24) in the
coordinate $(x,t)$, especially the behavior of solution located in the domain
$R_{i,t_{0}}$, it is required to put forward the appropriate boundary
conditions of $h_{k}$ and $\tilde{l}_{k}\omega_{k}$ on $x=a(t)$ and $x=b(t)$
for $k\neq i$.

Under the transformation $w=\epsilon D_{\epsilon}\omega$,
$\eqref{eq:2.88}_{1}$ can be reduced into
$\partial_{x}\omega=-D^{-1}_{\epsilon}A^{-1}D_{\epsilon}\partial_{t}\omega$.
In addition, without loss of generality, it is assumed $\lambda_{k}\not=0$ for
all $1\leq k\leq n$ (otherwise, one can achieve this by a simple space
transformation). By direct calculations, we have

$\begin{split}L_{j}h_{j}:=&\partial_{t}h_{j}+\lambda_{j}\partial_{x}h_{j}\\\
=&\epsilon(\nabla\tilde{l}_{j}D_{\epsilon}(\partial_{t}\omega+\lambda_{j}\partial_{x}\omega))^{\top}\partial_{t}\omega+\tilde{l}_{j}\partial^{2}_{t}\omega+\lambda_{j}\tilde{l}_{j}\partial^{2}_{xt}\omega\\\
=&\epsilon(\nabla\tilde{l}_{j}D_{\epsilon}(\partial_{t}\omega+\lambda_{j}\partial_{x}\omega))^{\top}\partial_{t}\omega-\epsilon(\nabla\tilde{l}_{j}D_{\epsilon}\partial_{t}\omega)^{\top}(\partial_{t}\omega+\lambda_{j}\partial_{x}\omega)-\epsilon(\nabla\lambda_{j}D_{\epsilon}\partial_{t}\omega)\tilde{l}_{j}\partial_{x}\omega\\\
=&\epsilon\sum_{k\neq i,l\neq
i}h_{k}h_{l}(\lambda_{j}\lambda^{-1}_{l}-\lambda_{j}\lambda^{-1}_{k})(\nabla\tilde{l}_{j}r_{k})^{\top}\tilde{r}_{l}+\epsilon\nabla\lambda_{j}\sum_{k\neq
i,l\neq i}h_{k}h_{l}\lambda^{-1}_{l}\delta_{jl}r_{k}\\\
=&\epsilon(Q_{0}h^{2}_{j}+Q_{1}h_{j}+Q_{2})=\epsilon\sum_{k\neq i,l\neq
i}\gamma_{jkl}h_{k}h_{l},\\\\[5.0pt]
L_{j}p_{j}:=&\partial_{t}(\tilde{l}_{j}\omega)+\lambda_{j}\partial_{x}(\tilde{l}_{j}\omega)\\\
=&\epsilon\sum_{k\neq i,1\leq l\leq
n}(1-\lambda_{j}\lambda^{-1}_{k})(\nabla\tilde{l}_{j}r_{k})^{\top}h_{k}p_{l}\tilde{r}_{l}=\epsilon\sum_{k\neq
i,1\leq l\leq n}\zeta_{jkl}h_{k}p_{l},\end{split}$ (2.25)

where $p_{j}=\tilde{l}_{j}\omega$, and

$\begin{split}Q_{0}:=&\gamma_{jjj}=\frac{1}{\lambda_{j}}\nabla\lambda_{j}\cdot
r_{j},\quad Q_{1}:=\sum_{k\neq i,j}\gamma_{jjk},\quad Q_{2}:=\sum_{k\neq
i,j\atop l\neq i,j}\gamma_{jkl}h_{k}h_{l},\quad\gamma_{jkk}|_{k\neq j}=0,\\\
\gamma_{jkl}=&\lambda_{j}(\lambda^{-1}_{l}-\lambda^{-1}_{k})(\nabla\tilde{l}_{j}r_{k})^{\top}\tilde{r}_{l}+\delta_{jl}\lambda^{-1}_{l}\nabla\lambda_{j}r_{k},\;\;\;\zeta_{jkl}=(1-\lambda_{j}\lambda^{-1}_{k})(\nabla\tilde{l}_{j}r_{k})^{\top}\tilde{r}_{l}.\end{split}$

Note that

$R_{i,t_{0}}\cap R_{j,t_{0}}=\emptyset\quad\text{for $i\neq j$}.$

We claim that there exist two positive constants $A_{0}$ and $A_{1}$
independent of $\epsilon$ such that for sufficiently small $\epsilon>0$,
$(x,t)\notin R_{j,t_{0}}$ and $t_{0}\leq t\leq t_{1}$, then

$|h_{j}(x,t)|<A_{0}\epsilon,\quad|p_{j}(x,t)|<A_{1}.$ (2.26)

To prove this claim, first of all, we will show

$\int_{\Gamma_{j}\cap\\{t_{0}\leq t\leq t_{1}\\}}|h_{k}|\text{d}s\leq
C\epsilon,\quad k\neq j,$ (2.27)

where and below $C>0$ is a generic constant independent of $\epsilon$.

Following Chapter 4 of [14], let

$\begin{split}L(t)&=\sum_{j\neq
i}\int^{+\infty}_{-\infty}|h_{j}(x,t)|\text{d}x,\quad
Q(t)=\epsilon\sum_{j,k\neq
i,j>k}\iint_{x<z}|h_{j}(x,t)h_{k}(z,t)|\text{d}x\text{d}z,\\\
R(t)&=\sum_{j>k}\int^{+\infty}_{-\infty}|h_{j}(x,t)h_{k}(x,t)|\text{d}x.\end{split}$

Then we have the following lemma.

######  Lemma 2.7.

There exist some positive constants $C_{0}$ and $C_{1}$ such that if
$L(t_{0})\leq\frac{C_{0}}{3C_{1}}$, then it holds that

$L(t)\leq
2L(t_{0}),\;\;\int^{t_{1}}_{t_{0}}R(t)\text{d}t\leq\frac{3\epsilon}{2C_{0}}L^{2}(t_{0}).$

###### Proof.

Note that

$\begin{split}&\text{d}(h_{j}(\text{d}x-\lambda_{j}(w)\text{d}t))\\\
=&\Big{(}\partial_{t}h_{j}+\lambda_{j}\partial_{x}h_{j}-\epsilon\nabla\lambda_{j}\sum_{k\neq
i}h_{j}h_{k}\lambda^{-1}_{k}r_{k}\Big{)}\text{d}t\wedge\text{d}x\\\
=&\epsilon\sum_{k\neq i,l\neq
i}\Gamma_{jkl}h_{k}h_{l}\text{d}t\wedge\text{d}x,\end{split}$

where
$\Gamma_{jkl}:=\gamma_{jkl}-\frac{1}{2}\nabla\lambda_{j}(\lambda^{-1}_{k}\delta_{jl}r_{k}+\lambda^{-1}_{l}\delta_{jk}r_{l})$,
$\Gamma_{jkk}=0$ for any $j,k\neq j$, and

$\delta_{jk}=\begin{cases}1,\;k=j,\\\ 0,\;k\neq
j,\end{cases}\;\;\delta_{jl}=\begin{cases}1,\;l=j,\\\ 0,\;l\neq j.\end{cases}$

On the other hand,

$\begin{split}\text{d}(|h_{j}|(\text{d}x-\lambda_{j}(w)\text{d}t))=&\epsilon\text{sgn
}h_{j}\sum_{k,l\neq i}\Gamma_{jkl}h_{k}h_{l}\text{d}t\wedge\text{d}x\\\
=&\Big{(}\partial_{t}|h_{j}|+\partial_{x}(|h_{j}|\lambda_{j}(w))\Big{)}\text{d}t\wedge\text{d}x.\end{split}$
(2.28)

Then applying Stokes’ formula to (2.28) in the interval
$(-\infty,x]\times[t_{0},t_{1}]$ yields

$\begin{split}&\int^{x}_{-\infty}|h_{j}(x,t_{1})|\text{d}x-\int^{x}_{-\infty}|h_{j}(x,t_{0})|\text{d}x+\int^{t_{1}}_{t_{0}}\lambda_{j}(w(x,t))|h_{j}(x,t)|\text{d}t\\\
=&\int^{x}_{-\infty}\int^{t_{1}}_{t_{0}}\Big{(}\partial_{t}|h_{j}(x,t)|+\partial_{x}(\lambda_{j}(w(x,t))|h_{j}(x,t)|)\Big{)}\text{d}t\text{d}x\\\
=&\epsilon\int^{x}_{-\infty}\int^{t_{1}}_{t_{0}}\text{sgn}h_{j}\sum_{k,l\neq
i}\Gamma_{jkl}h_{k}h_{l}\text{d}t\text{d}x.\end{split}$

Therefore,

$\int^{x}_{-\infty}\partial_{t}|h_{j}(x,t)|\text{d}x=-\lambda_{j}(w(x,t))|h_{j}(x,t)|+\epsilon\int^{x}_{-\infty}\text{sgn}h_{j}\sum_{k,j\neq
i}\Gamma_{jkl}h_{k}h_{l}(x,t)\text{d}x.$ (2.29)

Similarly,

$\int^{+\infty}_{x}\partial_{t}|h_{j}(x,t)|\text{d}x=\lambda_{j}(w(x,t))|h_{j}(x,t)|+\epsilon\int^{+\infty}_{x}\text{sgn}h_{j}\sum_{k,j\neq
i}\Gamma_{jkl}h_{k}h_{l}(x,t)\text{d}x.$ (2.30)

Based on (2.29) and (2.30), then there exist two positive constant $C_{0}$ and
$C_{1}$ such that

$\begin{split}Q^{\prime}(t)=&\epsilon\sum_{j>k}\Big{(}\int^{\infty}_{-\infty}\text{d}z\int^{z}_{-\infty}\partial_{t}|h_{j}(x,t)||h_{k}(z,t)|\text{d}x+\int^{\infty}_{-\infty}\text{d}x\int^{\infty}_{x}\partial_{t}|h_{k}(z,t)||h_{j}(x,t)|\text{d}z\Big{)}\\\
=&\epsilon\sum_{j>k}\int^{\infty}_{-\infty}(\lambda_{k}(w)-\lambda_{j}(w))|h_{j}(x,t)||h_{k}(x,t)|\text{d}x\\\
&+\epsilon^{2}\sum_{j>k}\int^{\infty}_{-\infty}\int^{z}_{-\infty}\text{sgn}h_{j}\sum_{\mu,\nu\neq
i}\Gamma_{j\mu\nu}h_{\mu}h_{\nu}(x,t)|h_{k}(z,t)|\text{d}x\text{d}z\\\
&+\epsilon^{2}\sum_{j>k}\int^{\infty}_{-\infty}\int^{\infty}_{x}\text{sgn}h_{k}\sum_{\mu,\nu\neq
i}\Gamma_{k\mu\nu}h_{\mu}h_{\nu}(z,t)|h_{j}(x,t)|\text{d}z\text{d}x\\\
\leq&-C_{0}\epsilon R(t)+C_{1}\epsilon^{2}R(t)L(t),\end{split}$ (2.31)

where $\lambda_{j}(u)-\lambda_{k}(u)\geq C_{0}$ for $j>k$. It follows from
(2.28) that

$L^{\prime}(t)=\epsilon\sum_{j\neq i,\;k\neq i\atop l\neq i}\int\text{sgn
}h_{j}\Gamma_{jkl}h_{k}h_{l}\text{d}x\leq C_{1}\epsilon R(t).$ (2.32)

Assume that $L(t)\leq\frac{2C_{0}}{3C_{1}}$ holds for $t_{0}\leq t\leq t_{1}$.
By continuous induction, we next show that for $t<t_{1}$, there exists a
positive constant $M_{0}$ with $M_{0}<\frac{2C_{0}}{3C_{1}}$ such that

$L(t)<M_{0}.$

Along with (2.31) and (2.32), one has

$\begin{split}\Big{(}3C_{1}Q(t)+C_{0}L(t)\Big{)}^{\prime}=&3C_{1}Q^{\prime}(t)+C_{0}L^{\prime}(t)\leq
3\epsilon C_{1}R(t)\Big{(}-C_{0}+C_{1}\epsilon L(t)\Big{)}+\epsilon
C_{1}C_{0}R(t)\\\ =&\epsilon(3C_{1}\epsilon L(t)-2C_{0})C_{1}R(t)\leq
0.\end{split}$

Therefore, it yields that

$3C_{1}Q(t)+C_{0}L(t)\leq
3C_{1}Q(t_{0})+C_{0}L(t_{0})\leq\frac{3C_{1}}{2}\epsilon
L^{2}(t_{0})+C_{0}L(t_{0})\leq\frac{3C_{0}}{2}L(t_{0})$

and

$L(t)\leq\frac{3}{2}L(t_{0})<2L(t_{0})<\frac{2C_{0}}{3C_{1}}.$

Meanwhile, we can calculate directly that

$Q^{\prime}(t)\leq\epsilon R(t)(C_{1}\epsilon L(t)-C_{0})\leq\epsilon
R(t)(2C_{1}\epsilon L(t_{0})-C_{0})\leq-\frac{C_{0}}{3}R(t).$

Then

$Q(t)-Q(t_{0})\leq-\frac{C_{0}}{3}\int^{t_{1}}_{t_{0}}R(s)\text{d}s$

and

$\int^{t_{1}}_{t_{0}}R(s)\text{d}s\leq\frac{3}{C_{0}}Q(t_{0})\leq\frac{3}{2C_{0}}\epsilon
L^{2}(t_{0}).$

Thus, we complete the proof of Lemma 2.7. ∎

Based on Lemma 2.7, we will estimate the integral of $h_{j}(j\neq i)$ along
the different families of characteristics, which is crucial for evaluating the
boundedness of $h_{j}$ on the $i$-th characteristics.

######  Lemma 2.8.

Under the assumption in Lemma 2.7, it holds that

$\int_{\Gamma_{k}\cap\\{t_{0}\leq t\leq
t_{1}\\}}|h_{j}|\text{d}s\leq\frac{4\epsilon}{3C_{1}},\quad k\neq j,$

where the positive constant $C_{1}$ has been given in Lemma 2.7.

###### Proof.

Denote by $\mathcal{D}$ the domain bounded by the $k$-th characteristics
$\Gamma_{k}$, the $j$-th characteristics $\Gamma_{j}$, the straight lines
$t=t_{0}$ and $t=t_{1}$ (see Figure $3$). Applying Stokes’ formula to (2.28)
yields

$\int_{\partial\mathcal{D}}|h_{j}|(\text{d}x-\lambda_{j}(w)\text{d}t)=\epsilon\iint_{\mathcal{D}}\text{sgn
}h_{j}\sum_{k,l\neq i}\Gamma_{jkl}h_{k}h_{l}\text{d}t\text{d}x.$ (2.33)

$t=t_{1}$$t=t_{0}$$\mathcal{D}$$\Gamma_{j}$$\Gamma_{k}$Figure $3$. Applying
Stokes’ formula in the domain $\mathcal{D}$

Along $\Gamma_{j}:|h_{j}|(\text{d}x-\lambda_{j}(w)\text{d}t)=0$, and along
$\Gamma_{k}$:

$|h_{j}||\text{d}x-\lambda_{j}(w)\text{d}t|=|h_{j}||\lambda_{k}-\lambda_{j}|\text{d}t\geq
C_{0}|h_{j}||\text{d}t|,$

where $C_{0}>0$ is the given constant in Lemma 2.7.

Therefore, it follows from (2.33) that

$\begin{split}C_{0}\int_{\Gamma_{k}\cap\\{t_{0}\leq t\leq
t_{1}\\}}|h_{j}|\text{d}t\leq&\epsilon\Big{(}L(t_{0})+L(t_{1})+C\int^{t_{1}}_{t_{0}}R(s)\text{d}s\Big{)}<3\epsilon
L(t_{0})+\frac{3C_{1}}{2C_{0}}\epsilon L^{2}(t_{0})<4\epsilon
L(t_{0})<\frac{4C_{0}\epsilon}{3C_{1}}.\end{split}$

Thus, the proof of Lemma 2.8 is finished. ∎

Based on Lemma 2.8, (2.27) is shown. In the sequel, we prove the claim (2.26)
by the continuous induction argument. For any point $P_{0}(a(t),t)$ lying on
the $i$-th characteristics $\Gamma_{i,-}$, $P_{0}\notin R_{j,t_{0}}$ for
$j\not=i$ (see Figure $4$), by integrating (2.25) along the $j$-th
characteristics $\Gamma_{j}(j>i)$, we obtain

$\begin{split}h_{j}(x(t),t)|_{P_{0}}=&\epsilon\sum_{k,l\neq
i}\int^{t}_{0}\gamma_{jkl}h_{k}(x(s),s)h_{l}(x(s),s)\text{d}s,\\\
p_{j}(x(t),t)|_{P_{0}}=&\epsilon\sum_{k\neq i,1\leq l\leq
n}\int^{t}_{0}\zeta_{jkl}h_{k}(x(s),s)p_{l}(x(s),s)\text{d}s.\end{split}$

$x$$t=t_{0}$$(a_{0},t_{0})$$(b_{0},t_{0})$$R_{i,t_{0}}$$P_{0}$$\Gamma_{i,-}$$\Gamma_{j}$$\Gamma_{i,+}$Figure
$4$. The picture on the related characteristics

Assume that these estimates in (2.26) are valid for $t_{0}\leq t<t_{1}$. We
need to show they remain true for $t=t_{1}$.

  * 1.

When $k=l=j$, in terms of the expression of $\gamma_{jjj}$, then

$|\epsilon\int^{t_{1}}_{t_{0}}\gamma_{jjj}h^{2}_{j}\text{d}s|\leq
C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R^{c}_{j}}h^{2}_{j}\text{d}s+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{j}}h^{2}_{j}\text{d}s\leq CA^{2}_{0}\epsilon^{2}.$

At this time, $\zeta_{jjj}=0$.

  * 2.

When $k=l\neq j$, then $\gamma_{jkl}=0$, and

$\begin{split}&\epsilon|\int^{t_{1}}_{t_{0}}\zeta_{jkl}h_{k}p_{l}\text{d}s|\\\
\leq&C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R^{c}_{k,t_{0}}}|h_{k}p_{k}|\text{d}s+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}p_{k}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}p_{k}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{k}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})+C\epsilon^{2},\end{split}$

where $\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\subset R^{c}_{j,t_{0}}$,
$R_{k,t_{0}}\subset R^{c}_{j,t_{0}}$, $|p_{k}|\leq C$ for $k\neq i$ (see the
proof procedure of Lemma 4.3.2 in [13]), and $(x,t)\in R^{c}_{i,t_{0}}$. When
$\epsilon>0$ is small, one can let

$CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})<\frac{A_{1}}{2}.$

  * 3.

When $k\neq l$,

$\begin{split}&\epsilon|\int^{t_{1}}_{t_{0}}\gamma_{jkl}h_{k}h_{l}\text{d}s|\\\
\leq&C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap(R_{k,t_{0}}\cup
R_{l,t_{0}})^{c}}|h_{k}h_{l}|\text{d}s+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap(R_{k,t_{0}}\cup
R_{l,t_{0}})}|h_{k}h_{l}|\text{d}s\\\
\leq&CA^{2}_{0}\epsilon^{3}(t_{1}-t_{0})+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}h_{l}|\text{d}s+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{k}h_{l}|\text{d}s,\end{split}$

where we have used the assumption that

$|h_{j}(x,t)|\leq A_{0}\epsilon,\quad(x,t)\notin R_{j,t_{0}}.$

    * Case $1$.

$k=j$, $l\neq j$: $(x,t)\notin R_{j,t_{0}}$, $\zeta_{jkl}=0$, then

$\begin{split}&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}h_{l}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{k}h_{l}|\text{d}s\\\
=&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{j}h_{l}|\text{d}s\leq
A_{0}\epsilon^{2}\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{l}|\text{d}s\leq
CA_{0}\epsilon^{3},\end{split}$

where $|h_{j}(x,t)|\leq A_{0}\epsilon$ for $(x,t)\notin R_{j,t_{0}}$, and

$\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{l}|\text{d}s\leq C\epsilon,\quad
l\neq j.$

    * Case $2$.

$k\neq j$, $l=j$: $(x,t)\notin R_{j,t_{0}}$, $\Gamma_{j}\cap
R_{l,t_{0}}\subset R^{c}_{j,t_{0}}\cap R_{j,t_{0}}=\emptyset$, then

$\begin{split}&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{k}h_{l}|\text{d}s=0,\\\
&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}h_{l}|\text{d}s\leq
CA_{0}\epsilon^{2}\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}|\text{d}s\leq
CA_{0}\epsilon^{2}\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{k}|\text{d}s,\\\
&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}h_{l}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{k}h_{l}|\text{d}s\leq CA_{0}\epsilon^{3}.\end{split}$

Meanwhile,

$\begin{split}&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|\zeta_{jkl}h_{k}p_{l}|\text{d}s\\\
=&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap(R_{k,t_{0}}\cup
R_{j,t_{0}})^{c}}|\zeta_{jkl}h_{k}p_{j}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|\zeta_{jkl}h_{k}p_{j}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|\zeta_{jkl}h_{k}p_{j}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})+CA_{1}\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R^{c}_{j,t_{0}}}|h_{k}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})+CA_{1}\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{k}|\text{d}s\leq
C\epsilon,\end{split}$

where $R_{k,t_{0}}\cap R_{j,t_{0}}=\emptyset$, $\Gamma_{j}\cap
R_{j,t_{0}}=\emptyset$, $R_{k,t_{0}}\subset R^{c}_{j,t_{0}}$ for
$t_{0}<t<t_{1}$, and

$\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{j,t_{0}}}\zeta_{jkl}h_{k}p_{j}\text{d}s=0.$

    * Case $3$.

$k\neq j$, $l\neq j$, $k\neq l$: $(x,t)\notin R_{j,t_{0}}$, $\Gamma_{j}\cap
R_{k,t_{0}}\subset\Gamma_{j}\cap R_{l,t_{0}}^{c}$, $\Gamma_{j}\cap
R_{l,t_{0}}\subset\Gamma_{j}\cap R^{c}_{k,t_{0}}$, then

$\begin{split}&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{k}h_{l}|\text{d}s\leq
CA_{0}\epsilon^{2}\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{l}|\text{d}s,\quad
l\neq j,\\\ &\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}h_{l}|\text{d}s\leq
CA_{0}\epsilon^{2}\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{k}|\text{d}s,\quad
k\neq j,\\\ &\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}h_{l}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{k}h_{l}|\text{d}s\leq CA_{0}\epsilon^{3}.\end{split}$

Meanwhile,

$\begin{split}&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|\zeta_{jkl}h_{k}p_{l}|\text{d}s\\\
=&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap(R_{k,t_{0}}\cup
R_{l,t_{0}})^{c}}|\zeta_{jkl}h_{k}p_{l}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|\zeta_{jkl}h_{k}p_{l}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|\zeta_{jkl}h_{k}p_{l}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})+CA_{1}\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}|\text{d}s+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{l,t_{0}}}|h_{k}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{2}(t_{1}-t_{0})+C\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{k}|\text{d}s\leq
C\epsilon.\end{split}$

In particular, when $k\neq l$, $k\neq j$, and $l=i$, we have
$|\zeta_{jki}|\leq C\epsilon$, and

$\begin{split}&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|\zeta_{jkl}h_{k}p_{i}|\text{d}s\\\
=&\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap(R_{k,t_{0}}\cup
R_{i,t_{0}})^{c}}|\zeta_{jki}h_{k}p_{i}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|\zeta_{jki}h_{k}p_{i}|\text{d}s+\epsilon\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{i,t_{0}}}|\zeta_{jki}h_{k}p_{i}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{3}(t_{1}-t_{0})+CA_{1}\epsilon^{2}\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}\cap
R_{k,t_{0}}}|h_{k}|\text{d}s\\\
\leq&CA_{0}A_{1}\epsilon^{3}(t_{1}-t_{0})+C\epsilon^{2}\int_{\Gamma_{j}\cap\\{t_{0}<t<t_{1}\\}}|h_{k}|\text{d}s\leq
C\epsilon^{2}.\end{split}$

On the other hand, under the transformation $w(u)=\epsilon
D_{\epsilon}\omega$, the $i$-th equation of (2.1) can be written as

$l_{ii}(\epsilon
D_{\epsilon}\omega)(\partial_{t}\omega_{i}+\lambda_{i}\partial_{x}\omega_{i})+\epsilon\sum_{j\neq
i}l_{ij}(\epsilon
D_{\epsilon}\omega)(\partial_{t}\omega_{j}+\lambda_{i}\partial_{x}\omega_{j})=0,$
(2.34)

where $l_{ii}(\epsilon D_{\epsilon}\omega)\neq 0$. Then the equation (2.34)
can be reduced into

$\frac{\text{d}\omega_{i}}{\text{d}_{i}t}:=\partial_{t}\omega_{i}+\lambda_{i}\partial_{x}\omega_{i}=\epsilon\sum_{j\neq
i}p_{ij}(\epsilon
D_{\epsilon}\omega)\frac{\text{d}\omega_{j}}{\text{d}_{i}t}.$

Integrating this along $x=a(t)$ in the interval $[t_{0},t]$ ($t\leq t_{1}$)
yields

$\begin{split}&\omega_{i}(a(t),t)\\\
=&\omega_{i}(a_{0},t_{0})+\epsilon\sum_{j\neq i}p_{ij}(\epsilon
D_{\epsilon}\omega)\omega_{j}(a(t),t)\\\ &-\epsilon^{3}\sum_{j,k\neq
i}\int^{t}_{t_{0}}\partial_{\omega_{k}}p_{ij}(\epsilon
D_{\epsilon}\omega)(p_{ik}(\epsilon
D_{\epsilon}\omega)+1)\Big{(}\partial_{t}\omega_{k}+\lambda_{i}\partial_{x}\omega_{k}\Big{)}\omega_{j}(a(s),s)\text{d}s.\end{split}$

Due to $p_{ij}(0)\Big{|}_{j\neq i}=0$, then for $t_{0}\leq t\leq t_{1}$,

$\omega_{i}(a(t),t)=O(1).$

Furthermore, it follows from $\eqref{Y-3}_{3}$ and $\eqref{Y-3}_{2}$ that on
$x=a(t)$ with $t_{0}\leq t\leq t_{1}$ and for $j=i+1,\cdots,n$, it holds that

$\begin{split}&\partial_{t}h_{j}=O(1)\epsilon,\;\;\partial^{2}_{t}h_{j}=O(1)\epsilon,\;\;\partial^{3}_{t}h_{j}=O(1)\epsilon,\;\;\tilde{l}_{j}\partial^{2}_{t}\omega=O(1)\epsilon,\;\;\tilde{l}_{j}\partial^{3}_{t}\omega=O(1)\epsilon.\end{split}$

Similarly, we can also show that on $x=b(t)$ with $t_{0}\leq t\leq t_{1}$, the
following estimates hold for $j=1,\cdots,i-1$,

$\epsilon\omega_{j}(b(t),t)=O(1),\;\;h_{j}(b(t),t)=O(1)(\epsilon),\;\;\omega_{i}(b(t),t)=O(1),\;\;(\partial_{t}+\lambda_{i}\partial_{x})\omega_{i}(b(t),t)=O(1)(\epsilon),$

and

$\begin{split}&\partial_{t}h_{j}=O(1)\epsilon,\;\;\partial^{2}_{t}h_{j}=O(1)\epsilon,\;\;\partial^{3}_{t}h_{j}=O(1)\epsilon,\;\;\tilde{l}_{j}\partial^{2}_{t}\omega=O(1)\epsilon,\;\;\tilde{l}_{j}\partial^{3}_{t}\omega=O(1)\epsilon.\end{split}$

Return to the coordinate $(y,t)$, we now study the solvability of the
following initial-boundary value problem for the blowup system (2.24) in
$t\in[t_{0},t_{1}]$

$\begin{cases}\eqref{Y-3},\qquad\qquad\qquad\qquad\qquad\quad\;\text{ in
}D=\\{(y,t):a_{0}\leq y\leq b_{0},t_{0}\leq t\leq t_{1}\\},\\\\[3.0pt]
K=1,\;\omega_{j}=\omega^{0}_{j}(y),\;h_{j}=h^{0}_{j}(y),\qquad\text{for
}t=t_{0},\;j=1,\cdots,n,\\\\[3.0pt]
\epsilon\omega_{j}=\epsilon\omega_{j}^{*}(t)=O(1),\;h_{j}=h_{j}^{*}(t)=O(1)(\epsilon),\;j=i+1,\cdots,n,\;\text{
on }y=a_{0},\\\\[3.0pt]
\epsilon\omega_{j}=\epsilon\omega_{j}^{*}(t)=O(1),\;h_{j}=h_{j}^{*}(t)=O(1)(\epsilon),\;j=1,\cdots,i-1,\;\text{
on }y=b_{0},\end{cases}$ (2.35)

where $\partial_{y}^{k}\omega^{0}_{j}(y)=O(1)$ and
$\partial_{y}^{k}h^{0}_{j}(y)=O(1)(\epsilon)$ for $j\not=i$ and $k\geq 1$,
$\omega^{0}_{i}(y)=O(1)$ and $h^{0}_{i}(y)=O(1)$,
$\partial_{t}^{k}\omega_{j}^{*}|_{\text{$y=a_{0}$ or $y=b_{0}$}}=O(1)$
($j\not=i$) for $k\geq 1$. Moreover, the compatibility conditions of all
orders hold on the corners $(a_{0},t_{0})$ and $(b_{0},t_{0})$ for the
initial-boundary values of (2.35).

###  2.3 Solvability of the blowup system

In this subsection, we will show the existence of smooth solution
$(\omega,K,h)$ to problem (2.35) for $t\in[t_{0},t_{1}]$. Note that due to the
degeneracy of $K$ near the blowup time $T_{\epsilon}$, one can sometimes think
$y$ and $t$ as the new “time variable” and “space variable”, respectively.

First of all, we start to construct the first approximate solution
$(\varphi^{(0)},K^{(0)},\omega^{(0)})$ of (2.18) such that the nondegenerate
condition holds at some point.

Let $\tau:=\epsilon t$. Then it follows from $\eqref{eq:2.2}$ that the
corresponding solution for $\epsilon=0$ is

$\begin{split}&\text{$\bar{\omega}_{i}(x,\tau)=w^{i}_{0}(y)$ with
$x=\bar{\varphi}(y,\tau)=y+\partial_{u_{i}}\lambda_{i}(0)w^{i}_{0}(y)\tau$},\quad\text{$\bar{\omega}_{j}(y,\tau)=0$
for $j\not=i$},\\\
&\bar{K}(y,\tau)=1+\partial_{u_{i}}\lambda_{i}(0)\Big{(}w^{i}_{0}(y)\Big{)}^{\prime}\tau.\\\
\end{split}$ (2.36)

Choose a cut-off function $\chi(s)\in C^{\infty}(\mathbb{R})$ such that

$\chi(s)=1\;\text{ for }s\leq\frac{1}{2};\quad\chi(s)=0\;\text{ for
}s\geq\frac{3}{4}.$

Since there exists a local smooth solution
$(\varphi^{\epsilon},K^{\epsilon},\omega^{\epsilon})$ to the blowup system
(2.18) for $t_{0}\leq t<T_{\epsilon}$, we can glue the local smooth solution
$(\varphi^{\epsilon},K^{\epsilon},\omega^{\epsilon})$ and (2.36) to get the
first approximate solution $(\varphi^{(0)},K^{(0)},\omega^{(0)})$ as

$\begin{cases}\varphi^{(0)}(y,t)=\chi(\frac{t}{T_{\epsilon}})\varphi^{\epsilon}(y,t)+(1-\chi(\frac{t}{T_{\epsilon}}))\bar{\varphi}(y,t),\\\\[3.0pt]
K^{(0)}(y,t)=\chi(\frac{t}{T_{\epsilon}})K^{\epsilon}(y,t)+(1-\chi(\frac{t}{T_{\epsilon}}))\bar{K}(y,t),\\\\[5.0pt]
\omega^{(0)}(y,t)=\chi(\frac{t}{T_{\epsilon}})\omega^{\epsilon}(y,t)+(1-\chi(\frac{t}{T_{\epsilon}}))\bar{\omega}(y,t).\end{cases}$
(2.37)

Let $\omega^{(m+1)}$, $h^{(m+1)}$ and $K^{(m+1)}$ for $m\in\mathbb{N}_{0}$
satisfy the following linearized system of (2.24)

$\displaystyle\partial_{t}K^{(m+1)}=$ $\displaystyle\epsilon L^{(1)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}K^{(m)}+\epsilon\nabla\lambda_{i}\cdot
r_{i}(\epsilon D_{\epsilon}\omega^{(m)})h^{(m)}_{i},$ (2.38)
$\displaystyle\partial_{t}h^{(m+1)}_{i}=$
$\displaystyle\epsilon(\tilde{h}^{(m)})^{\top}Q^{(1)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}K^{(m)}+\epsilon L^{(2)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}h^{(m)}_{i},$ (2.39)
$\displaystyle\big{(}K^{(m)}\partial_{t}+$
$\displaystyle(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\big{)}h^{(m+1)}_{j}+\epsilon
K^{(m)}\Big{(}(\tilde{h}^{(m)})^{\top}Q^{(2)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m+1)}+L^{(3)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}h^{(m+1)}_{j}\Big{)}$
$\displaystyle+\epsilon h^{(m)}_{i}\Big{(}r^{\top}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})Q^{(3)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m+1)}+\nabla\lambda_{i}\cdot
r_{i}(\epsilon D_{\epsilon}\omega^{(m)})h^{(m+1)}_{j}\Big{)}=0,$ (2.40)
$\displaystyle\tilde{l}_{i}(\epsilon D_{\epsilon}\omega^{(m)})$
$\displaystyle\partial_{t}\omega^{(m+1)}=0,$ (2.41)
$\displaystyle\tilde{l}_{j}(\epsilon D_{\epsilon}\omega^{(m)})$
$\displaystyle\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\omega^{(m+1)}=0,\quad j\neq i$
(2.42)

with the initial-boundary values

$\begin{cases}\epsilon\omega^{(m+1)}_{j}=\epsilon\omega_{j}^{*}(t),\;h^{(m+1)}_{j}=h_{j}^{*}(t),\qquad\qquad\;j=i+1,\cdots,n,\;\text{
on }y=a_{0},\\\\[3.0pt]
\epsilon\omega^{(m+1)}_{j}=\epsilon\omega_{j}^{*}(t),\;h^{(m+1)}_{j}=h_{j}^{*}(t),\qquad\qquad\;j=1,\cdots,i-1,\;\text{
on }y=b_{0},\\\\[3.0pt]
K^{(m+1)}=1,\;\omega_{j}^{(m+1)}=\omega^{0}_{j}(y),\;h^{(m+1)}_{j}=h^{0}_{j}(y),\;\;\qquad\qquad\text{for
}t=t_{0},\;j=1,\cdots,n.\end{cases}$

Note that the iterative scheme (2.40) is delicately chosen. If $h^{(m+1)}$ is
replaced by $h^{(m)}$ in the second and third terms of (2.40), then it is
difficult for us to directly get the uniform boundedness of $h_{j}^{(m+1)}$
($j\not=i$) due to the appearance of $O(|h^{(m)}|^{2})$.

In the following, we will make the estimates of the solutions
$(K^{(m+1)},h^{(m+1)},\omega^{(m+1)})$ by five steps.

Step 1. Estimates of $K^{(m+1)}$, $h^{(m+1)}_{i},\tilde{h}^{(m+1)}$ and
$\omega^{(m+1)}$

We now draw the estimates of $K^{(m+1)}$, $h^{(m+1)}_{i},\tilde{h}^{(m+1)}$,
and $\omega^{(m+1)}$ by the following conclusion.

######  Lemma 2.9.

It holds that

$|K^{(m)}|\leq K,\;\;|h^{(m)}_{i}|\leq
H_{i},\;\;|\tilde{h}^{(m)}|\leq\tilde{H}=\tilde{C}\epsilon,\;\;,\text{$|\epsilon\omega^{(m)}_{j}|\leq
W$ for $j\not=i$},\;\;|\omega^{(m)}_{i}|\leq W,$ (2.43)

where $K$, $H_{i}$, $\tilde{C}$ and $W$ are some positive constants to be
determined later.

###### Proof.

The proof will be carried out by continuous induction. At first, (2.43) holds
true for $m=0$ by (2.37). Assume that (2.43) holds for $m$, then it suffices
to show it is still valid for $m+1$.

Integrating (2.38)-(2.39) for $t\in[t_{0},t_{1}]$ yields that

$|K^{(m+1)}|\leq 1+C_{1}(\epsilon W)\tilde{H}K+C_{2}(\epsilon W)H_{i},$ (2.44)
$|h^{(m+1)}_{i}|\leq H_{i0}+C_{3}(\epsilon W)\tilde{H}^{2}K+C_{4}(\epsilon
W)\tilde{H}H_{i},$ (2.45)

where $H_{i0}:=\displaystyle\max_{y\in[a_{0},b_{0}]}|h^{0}_{i}(y)|$, and
$C_{k}(\epsilon W)$ is a smooth function depending on $\epsilon W$ ($1\leq
k\leq 4$).

Before evaluating $h^{(m+1)}_{j}(j\neq i)$, we need to figure out the trend of
the first order operator
$K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}$ on the boundaries $y=a_{0}$ and
$y=b_{0}$, especially, check the signs of $K^{(m)}$ on the boundaries.

From $\eqref{YD-1}_{1}$, one has

$\begin{split}\partial_{t}K^{(m)}(y,t)=&\epsilon^{2}\sum_{j\neq
i}\partial_{u_{j}}\lambda_{i}(\epsilon
D_{\epsilon}\omega^{(m-1)})\partial_{y}\omega^{(m-1)}_{j}+\epsilon\partial_{u_{i}}\lambda_{i}(\epsilon
D_{\epsilon}\omega^{(m-1)})\partial_{y}\omega^{(m-1)}_{i}.\end{split}$ (2.46)

Note that

$\epsilon\omega^{(m-1)}_{j}(a_{0},t)=O(1),\quad
h^{(m-1)}_{j}(a_{0},t)=O(1)\epsilon\quad\text{ for }\;i+1\leq j\leq n,$

and

$\epsilon\omega^{(m-1)}_{j}(b_{0},t)=O(1),\quad
h^{(m-1)}_{j}(b_{0},t)=O(1)\epsilon\quad\text{ for }1\leq j\leq i-1.$

Along with (2.20) and (2.34), we have

$\begin{split}&\partial_{t}\omega^{(m-1)}_{j}(a_{0},t)=O(1),\;\;\;\partial_{y}\omega^{(m-1)}_{j}(a_{0},t)=O(1)\;\text{
for }j\neq i,\\\\[3.0pt]
&\partial_{t}\omega^{(m-1)}_{i}(a_{0},t)=O(1)\epsilon,\;\;\;\partial_{y}\omega_{i}^{(m-1)}(a_{0},t)=O(1)\epsilon.\end{split}$

Thus

$|\partial_{t}K^{(m)}(a_{0},t)|\leq C\epsilon^{2},$ (2.47)

which yields for $t\in[t_{0},t_{1}]$ and small $\epsilon>0$,

$K^{(m)}(a_{0},t)\geq 1-C\epsilon^{2}(t_{1}-t_{0})>\frac{1}{2}.$

Similarly,

$K^{(m)}(b_{0},t)\geq 1-C\epsilon^{2}(t_{1}-t_{0})>\frac{1}{2}.$

Meanwhile, for $t\in[t_{0},t_{1}]$ and small $\epsilon>0$,

$K^{(m)}(a_{0},t)<\frac{3}{2},\qquad K^{(m)}(b_{0},t)<\frac{3}{2}.$

Therefore, we can apply the characteristics method to (2.40), and derive that

$\begin{split}\sum_{j\neq
i}|h^{(m+1)}_{j}(y,t)|\leq&\tilde{H}_{0}+\Big{(}\epsilon C_{5}(\epsilon
W)\tilde{H}+\epsilon C_{6}(\epsilon W)H_{i}\Big{)}\sum_{k\neq
i}|h^{(m+1)}_{k}|,\qquad j\neq i,\end{split}$ (2.48)

where

$\tilde{H}_{0}=\max\Big{(}\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|h^{0}_{j}(y)|,\max_{t\in[t_{0},t_{1}]}\sum^{n}_{j=i+1}|h_{j}^{(m+1)}(a_{0},t)|,\max_{t\in[t_{0},t_{1}]}\sum^{i-1}_{j=1}|h_{j}^{(m+1)}(b_{0},t)|\Big{)}.$

Then it yields that for small $\epsilon>0$,

$\sum_{j\neq i}|h^{(m+1)}_{j}|\leq C\tilde{H}_{0}.$

Set

$p_{j}^{(m+1)}=\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\omega^{(m+1)},j=1,\cdots,n.$

Then

$\omega^{(m+1)}=\sum^{n}_{k=1}p^{(m+1)}_{k}\tilde{r}_{k}(\epsilon
D_{\epsilon}\omega^{(m)}).$

It follows from (2.41) and (2.42) that

$\displaystyle\partial_{t}p^{(m+1)}_{i}+\epsilon(\tilde{h}^{(m)})^{\top}M_{1}(\epsilon
D_{\epsilon}\omega^{(m)})p^{(m+1)}=0,$ (2.49)
$\displaystyle\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}p^{(m+1)}_{j}+\epsilon
K^{(m)}(\tilde{h}^{(m)})^{\top}M_{2}(\epsilon
D_{\epsilon}\omega^{(m)})p^{(m+1)}$ $\displaystyle\qquad\quad+\epsilon
h^{(m)}_{i}(r^{(m)}_{i})^{\top}M_{3}(\epsilon
D_{\epsilon}\omega^{(m)})p^{(m+1)}=0,\qquad\quad j\neq i,$ (2.50)

where and below $M_{k}(\epsilon D_{\epsilon}\omega^{(m)})$ stands for the
$n\times n$ matrix depending on $\epsilon D_{\epsilon}\omega^{(m)}$.

Integrating (2.49) with respect to $t$ on the interval $[t_{0},t]$ yields

$|p^{(m+1)}_{i}|\leq|p^{(m+1)}_{i}(y,t_{0})|+C_{7}(\epsilon
W)\tilde{H}|p^{(m+1)}|.$ (2.51)

Along with (2.51), we apply the characteristics method to (2.50), and obtain
that

$\max_{y\in[a_{0},b_{0}]}|p^{(m+1)}|\leq P_{0}+\epsilon\Big{(}C_{7}(\epsilon
W)+C_{8}(\epsilon W)\tilde{H}K+C_{9}(\epsilon
W)H_{i}\Big{)}\max_{y\in[a_{0},b_{0}]}|p^{(m+1)}|,$

where

$P_{0}:=\displaystyle\max\Big{(}\max_{y\in[a_{0},b_{0}]}\sum^{n}_{j=1}|p^{(m+1)}_{j}(y,t_{0})|,\max_{t\in[t_{0},t_{1}]}\sum^{n}_{j=i+1}|p^{(m+1)}_{j}(a_{0},t)|,\max_{t\in[t_{0},t_{1}]}\sum^{i-1}_{j=1}|p^{(m+1)}_{j}(b_{0},t)|\Big{)}.$

This derives

$|p^{(m+1)}|\leq CP_{0}.$ (2.52)

Let $W=CP_{0}$, then it satisfies that in $D$,

$\epsilon|\omega^{(m+1)}_{j}(y,t)|\leq W\quad\text{for
$j\not=i$},\quad|\omega^{(m+1)}_{i}(y,t)|\leq W$

and then $|v^{(m+1)}(y,t)|=|\epsilon D_{\epsilon}\omega^{(m+1)}|\leq\epsilon
W$. Based on (2.44), (2.45), and (2.52), when $\epsilon>0$ is small, one can
choose

$K=1+CH_{0},\;\;H=CH_{0},\;\;\tilde{H}=C\epsilon,\;\;H_{0}=H_{i0}+\tilde{H}_{0},\;\;W=CP_{0},$

such that (2.43) still holds for $m+1$, where $C>1$ is a positive constant
independent of $\epsilon$.

∎

Step 2. Estimates of $\nabla K^{(m+1)},\nabla
h^{(m+1)}_{i},\nabla\tilde{h}^{(m+1)}$ and  $\nabla\omega^{(m+1)}$

Let

$\bar{h}^{(m+1)}_{i}=\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\omega^{(m+1)},\qquad\bar{h}^{(m+1)}_{j}=\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{t}\omega^{(m+1)}\;\;\text{for $j\neq i$}.$
(2.53)

Meanwhile, one has

$\begin{split}\partial_{y}\omega^{(m+1)}(y,t)=&-\sum_{j\neq
i}\frac{K^{(m)}\bar{h}^{(m+1)}_{j}}{(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})}\tilde{r}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})+\bar{h}^{(m+1)}_{i}\tilde{r}_{i}(\epsilon
D_{\epsilon}\omega^{(m)}),\\\ \partial_{t}\omega^{(m+1)}(y,t)=&\sum_{j\neq
i}\bar{h}^{(m+1)}_{j}\tilde{r}_{j}(\epsilon
D_{\epsilon}\omega^{(m)}).\end{split}$

When $t=t_{0}$, we have

$\begin{split}&\partial_{t}K^{(m+1)}=O(1)\epsilon,\;\;\partial_{y}K^{(m+1)}=0,\;\partial_{y}h_{i}^{(m+1)}=O(1),\;\;\partial_{t}h^{(m+1)}_{i}=O(1)\epsilon,\;\;\bar{h}^{(m+1)}_{i}=O(1),\\\
&\bar{h}^{(m+1)}_{j}=O(1)\epsilon,\;\;\partial_{t}h^{(m+1)}_{j}=O(1)\epsilon,\;\;\partial_{y}h^{(m+1)}_{j}=O(1)\epsilon,\quad
j\neq i.\end{split}$

######  Lemma 2.10.

The following estimates hold

$|\nabla K^{(m)}|\leq K^{\prime},\;|\nabla h^{(m)}|\leq
H^{\prime},\;|\nabla\omega^{(m)}|\leq W^{\prime},$ (2.54)

where $|\partial_{t}K^{(m)}|\leq C\epsilon$, $|\partial_{t}h^{(m)}_{i}|\leq
C\epsilon$, $|\nabla h^{(m)}_{i}|\leq
H^{\prime}_{i},\;|\nabla\tilde{h}^{(m)}|\leq\tilde{H}^{\prime}\leq
C^{\prime}\epsilon$, $K^{\prime}$, $H^{\prime}$, $W^{\prime}$,
$H^{\prime}_{i}$, $C$ and $C^{\prime}$ are some positive constants to be
determined later.

###### Proof.

We will show that (2.54) still holds for $m+1$. Taking the first order
derivative of (2.38) yields

$\begin{split}\partial_{t}\nabla
K^{(m+1)}=&\epsilon\nabla\Big{(}L^{(1)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}\Big{)}K^{(m)}+\epsilon
L^{(1)}(\epsilon D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}\nabla K^{(m)}\\\
&+\epsilon^{2}\nabla_{u}(\nabla\lambda_{i}\cdot r_{i}(\epsilon
D_{\epsilon}\omega^{(m)}))D_{\epsilon}\nabla\omega^{(m)}h^{(m)}_{i}+\epsilon\nabla\lambda_{i}\cdot
r_{i}(\epsilon D_{\epsilon}\omega^{(m)})\nabla h^{(m)}_{i}.\end{split}$ (2.55)

Integrating (2.55) with respect to $t$ on the interval $[t_{0},t_{1}]$ yields

$\begin{split}|\nabla K^{(m+1)}|\leq&K^{\prime}_{0}+\epsilon C_{10}(\epsilon
W)\tilde{H}W^{\prime}+C_{1}(\epsilon
W)(\tilde{H}^{\prime}+\tilde{H}K^{\prime})+C_{11}(\epsilon W)\epsilon
W^{\prime}+C_{2}(\epsilon W)H^{\prime}_{i}\\\
\leq&K^{\prime}_{0}+CH^{\prime}+C\epsilon(K^{\prime}+W^{\prime}),\end{split}$
(2.56)

where $K^{\prime}_{0}=\max_{y\in[a_{0},b_{0}]}|\nabla K^{(m+1)}(y,t_{0})|$. In
particular, $|\partial_{t}K^{(m+1)}|\leq C\epsilon$.

Similarly, it follows from (2.39) that

$\begin{split}\partial_{t}\partial_{y}h^{(m+1)}_{i}=&\epsilon\partial_{y}(\tilde{h}^{(m)}Q^{(1)})\tilde{h}^{(m)}K^{(m)}+\epsilon(\tilde{h}^{(m)})^{\top}Q^{(1)}\partial_{y}\tilde{h}^{(m)}K^{(m)}+\epsilon(\tilde{h}^{(m)})^{\top}Q^{(1)}\tilde{h}^{(m)}\partial_{y}K^{(m)}\\\
+&\epsilon^{2}(\nabla
L^{(2)}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\tilde{h}^{(m)}h^{(m)}_{i}+\epsilon
L^{(2)}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\tilde{h}^{(m)}h^{(m)}_{i}+\epsilon
L^{(2)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}\partial_{y}h^{(m)}_{i}.\end{split}$

Let $H^{\prime}_{i0}=\max_{y\in[a_{0},b_{0}]}|\nabla h^{(m+1)}_{i}(y,t_{0})|$.
Then one has

$\begin{split}|\partial_{y}h^{(m+1)}_{i}|\leq&H^{\prime}_{i0}+C_{3}(\epsilon
W)\tilde{H}\tilde{H}^{\prime}+\epsilon C_{12}(\epsilon
W)W^{\prime}\tilde{H}^{2}+C_{3}(\epsilon W)\tilde{H}^{2}K^{\prime}\\\
&+\epsilon\tilde{H}C_{13}(\epsilon W)W^{\prime}+C_{4}(\epsilon
W)(H_{i}\tilde{H}^{\prime}+\tilde{H}H^{\prime}_{i})\\\
\leq&H^{\prime}_{i0}+C(\tilde{H}^{\prime}+\tilde{H}H^{\prime}_{i}).\end{split}$
(2.57)

In addition, it follows from (2.39) that

$|\partial_{t}h^{(m+1)}_{i}(y,t)|\leq|\partial_{t}h^{(m+1)}_{i}(y,t_{0})|+C\tilde{H}^{\prime}\leq
C\epsilon.$ (2.58)

From the expression (2.53) of $\bar{h}_{i}^{(m+1)}$, we can calculate directly
that

$\begin{split}&\partial_{t}\bar{h}^{(m+1)}_{i}=\epsilon(\nabla\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})D_{\epsilon}\sum_{j\neq
i}h^{(m)}_{j}\tilde{r}^{(m)}_{j})^{\top}\Big{(}-\sum_{l\neq
i}\frac{K^{(m)}\bar{h}^{(m+1)}_{l}}{(\lambda_{l}-\lambda_{i})^{(m)}}\tilde{r}^{(m)}_{l}+\bar{h}^{(m+1)}_{i}\tilde{r}^{(m)}_{i}\Big{)}\\\
&-\epsilon\Big{(}\nabla\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})D_{\epsilon}(-\sum_{j\neq
i}\frac{K^{(m)}h^{(m)}_{j}}{(\lambda_{j}-\lambda_{i})^{(m)}}\tilde{r}^{(m)}_{j}+h^{(m)}_{i}\tilde{r}^{(m)}_{i})\Big{)}^{\top}(\sum_{l\neq
i}\bar{h}^{(m+1)}_{l}\tilde{r}_{l}^{(m)})\\\ =&\epsilon
K^{(m)}(\tilde{h}^{(m)})^{\top}Q^{(4)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{\bar{h}}^{(m+1)}+\epsilon(\tilde{h}^{(m)})^{\top}M_{4}(\epsilon
D_{\epsilon}\omega^{(m)})r^{(m)}_{i}\bar{h}^{(m+1)}_{i}+\epsilon(r^{(m)}_{i})^{\top}M_{5}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{\bar{h}}^{(m+1)}h^{(m)}_{i},\end{split}$
(2.59)

where $\tilde{\bar{h}}^{(m+1)}$ is the $1\times(n-1)$ vector
$(\bar{h}_{1}^{(m+1)},\bar{h}_{2}^{(m+1)},\cdots,\hat{\bar{h}}_{i}^{(m+1)},\cdots,\bar{h}_{n}^{(m+1)})$
that does not contain component $\bar{h}_{i}^{(m+1)}$.

Integrating both sides of equation (2.59) with respect to $t$ yields

$\begin{split}\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}|\bar{h}^{(m+1)}_{i}|\leq&\bar{H}_{i0}+C_{14}(\epsilon
W)H\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}|\tilde{\bar{h}}^{(m+1)}|+\epsilon C_{15}(\epsilon
W)\tilde{H}\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}|\bar{h}^{(m+1)}_{i}|\\\
\lesssim&\bar{H}_{i0}+CH\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}|\tilde{\bar{h}}^{(m+1)}|+C\epsilon\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}|\bar{h}^{(m+1)}_{i}|,\end{split}$ (2.60)

with $\bar{H}_{i0}=\max_{y\in[a_{0},b_{0}]}|\bar{h}^{(m+1)}_{i}(y,t_{0})|$,
here and below $A\lesssim B$ stands for $A\leq CB$ with $C$ being a generic
positive constant independent of $\epsilon$.

We next establish the estimate of $\bar{h}^{(m+1)}_{j}=\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{t}\omega^{(m+1)}$ for $j\neq i$.
Differentiating (2.42) with respect to $t$, one can obtain that

$\begin{split}&(K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y})\bar{h}^{(m+1)}_{j}+\epsilon(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\Big{(}(\nabla\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial_{y}\omega^{(m+1)}\\\
&-(\nabla\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\partial_{t}\omega^{(m+1)}\Big{)}+\partial_{t}K^{(m)}\bar{h}^{(m+1)}_{j}+\epsilon\nabla(\lambda_{j}-\lambda_{i})D_{\epsilon}\partial_{t}\omega^{(m)}\tilde{l}^{(m)}_{j}\partial_{y}\omega^{(m+1)}=0.\end{split}$

This can be rewritten as

$\begin{split}&\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\bar{h}^{(m+1)}_{j}+\epsilon
K^{(m)}\Big{(}(\tilde{h}^{(m)})^{\top}Q_{5}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{\bar{h}}^{(m+1)}+L^{(3)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}\bar{h}^{(m+1)}_{j}\Big{)}\\\\[5.0pt]
&\;\;+\epsilon(\tilde{h}^{(m)})^{\top}M_{6}(\epsilon
D_{\epsilon}\omega^{(m)})\bar{h}^{(m+1)}_{i}r^{(m)}_{i}+\epsilon(r^{(m)}_{i})^{\top}M_{7}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{\bar{h}}^{(m+1)}h^{(m)}_{i}+\partial_{t}K^{(m)}\bar{h}^{(m+1)}_{j}=0.\end{split}$
(2.61)

By the characteristics method, it yields that

$\begin{split}\sum_{j\neq
i}|\bar{h}^{(m+1)}_{j}|&\lesssim\tilde{\bar{H}}^{\prime}_{0}+C\epsilon\max_{s\in[t_{0},t]}|\tilde{\bar{h}}^{(m+1)}|+C\epsilon^{2}\max_{s\in[t_{0},t]}|\bar{h}^{(m+1)}_{i}|,\end{split}$
(2.62)

where

$\tilde{\bar{H}}^{\prime}_{0}:=\max\Big{(}\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\bar{h}^{(m+1)}_{j}(y,t_{0})|,\max_{t\in[t_{0},t_{1}]}\sum^{n}_{j=i+1}|\bar{h}^{(m+1)}_{j}(a_{0},t)|,\max_{t\in[t_{0},t_{1}]}\sum^{i-1}_{j=1}|\bar{h}^{(m+1)}_{j}(b_{0},t)|\Big{)}.$

Together with (2.60), we arrive at

$\max_{y\in[a_{0},b_{0}]\atop s\in[t_{0},t]}|\tilde{\bar{h}}^{(m+1)}|\leq
C\tilde{\bar{H}}^{\prime}_{0}\leq
C\epsilon\tilde{W}^{\prime}_{0},\quad|\bar{h}^{(m+1)}_{i}(y,t)|\leq
C\max_{y\in[a_{0},b_{0}]}|\bar{h}^{(m+1)}_{i}(y,t_{0})|,$ (2.63)

where

$\tilde{W}^{\prime}_{0}:=\max\Big{(}\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\nabla\omega^{(m+1)}_{j}(y,t_{0})|,\max_{t\in[t_{0},t_{1}]}\sum^{n}_{j=i+1}|\nabla\omega^{(m+1)}_{j}(a_{0},t)|,\max_{t\in[t_{0},t_{1}]}\sum^{i-1}_{j=1}|\nabla\omega^{(m+1)}_{j}(b_{0},t)|\Big{)}.$

In the following, we will estimate $|\nabla h^{(m+1)}_{j}|$ for $j\neq i$.
Differentiating (2.40) with respect to $t$ and taking the direct computations,
one has

$\begin{split}&|\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\partial_{t}h^{(m+1)}_{j}+\partial_{t}K^{(m)}\partial_{t}h^{(m+1)}_{j}+\epsilon\nabla(\lambda_{j}-\lambda_{i})D_{\epsilon}\partial_{t}\omega^{(m)}\partial_{y}h^{(m+1)}_{j}|\\\
\leq&C\epsilon^{3}+C\epsilon^{2}\tilde{H}^{\prime}+C\epsilon\sum_{j\neq
i}|\partial_{t}h^{(m+1)}_{j}|.\end{split}$

In addition, from (2.40), it yields that

$|\partial_{y}h^{(m+1)}_{j}|\leq C|\partial_{t}h^{(m+1)}_{j}|+C\epsilon^{2}.$

Together with the same arguments as in Step $1$, we obtain

$\max_{y\in[a_{0},b_{0}]\atop t\in[t_{0},t_{1}]}\sum_{j\neq
i}|\partial_{t}h^{(m+1)}_{j}|\lesssim\tilde{H}^{\prime}_{0}+C\epsilon^{2}\tilde{H}^{\prime}+C\epsilon\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\partial_{t}h^{(m+1)}_{j}|,$

where
$\tilde{H}^{\prime}_{0}=\max\Big{(}\displaystyle\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\partial_{t}h^{(m+1)}_{j}(y,t_{0})|,C\epsilon\Big{)}$.

Then this means

$|\partial_{t}h^{(m+1)}_{j}|\leq
C\tilde{H}^{\prime}_{0},\quad|\partial_{y}h^{(m+1)}_{j}|\leq
C\tilde{H}^{\prime}_{0}.$ (2.64)

Thus, for small $\epsilon>0$, by choosing

$K^{\prime}=K^{\prime}_{0}+CH^{\prime}_{0},\;\;H^{\prime}=CH^{\prime}_{0},\;\;W^{\prime}=CW^{\prime}_{0},\;\;H^{\prime}_{0}=\tilde{H}^{\prime}_{0}+\tilde{H}^{\prime}_{i0},\;\;W^{\prime}_{0}=\tilde{W}^{\prime}_{0}+\bar{H}_{i0},$

the estimates (2.56)-(2.58), (2.60), (2.63) and (2.64) hold for $m+1$.

In particular, we can derive from (2.56), (2.58) and $\eqref{eq:2.41}_{1}$
that

$\partial_{t}K^{(m+1)}(y,t)=O(1)\epsilon,$

which is crucial for establishing the boundedness of the higher order
derivatives of the solutions.

∎

Step $3$. Estimates of $\nabla^{2}K^{(m+1)}$, $\nabla^{2}h^{(m+1)}$, and
$\nabla^{2}\omega^{(m+1)}$

For $k=1,\cdots,n$, set

$\begin{split}&\bar{q}^{(m+1)}_{k}:=\tilde{l}_{k}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{2}_{t}\omega^{(m+1)},\\\\[3.0pt]
&\bar{z}^{(m+1)}_{k}:=\tilde{l}_{k}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{2}_{ty}\omega^{(m+1)},\\\\[3.0pt]
&E^{(m+1)}_{k}:=\tilde{l}_{k}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{2}_{y}\omega^{(m+1)}.\end{split}$

Based on (2.41) and direct calculations, we have

$\bar{q}^{(m+1)}_{i}=-\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial_{t}\omega^{(m+1)}$

and

$\partial^{2}_{t}\omega^{(m+1)}=\sum_{j\neq
i}\bar{q}^{(m+1)}_{j}\tilde{r}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})-\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial_{t}\omega^{(m+1)}\tilde{r}_{i}(\epsilon
D_{\epsilon}\omega^{(m)}).$

Similarly,

$\displaystyle\partial^{2}_{ty}\omega^{(m+1)}=$ $\displaystyle\sum_{j\neq
i}\bar{z}^{(m+1)}_{j}\tilde{r}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})-\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\partial_{t}\omega^{(m+1)}\tilde{r}_{i}(\epsilon
D_{\epsilon}\omega^{(m)}),$ $\displaystyle\partial^{2}_{y}\omega^{(m+1)}=$
$\displaystyle\sum_{j=1}^{n}E^{(m+1)}_{j}\tilde{r}_{j}(\epsilon
D_{\epsilon}\omega^{(m)}).$

Then we can obtain the estimates of $\nabla^{2}K^{(m)}$, $\nabla^{2}h^{(m)}$
and $\nabla^{2}\omega^{(m)}$ as follows.

######  Lemma 2.11.

The following estimates hold

$|\nabla^{2}K^{(m)}|\leq K^{\prime\prime},\;|\nabla^{2}h^{(m)}|\leq
H^{\prime\prime},\;|\nabla^{2}\omega^{(m)}|\leq W^{\prime\prime},$ (2.65)

where $|\partial^{2}_{t}K^{(m)}|\leq C\epsilon$,
$|\partial^{2}_{t}h^{(m)}|\leq C\epsilon$,
$|\nabla^{2}\tilde{h}^{(m)}|\leq\tilde{H}^{\prime\prime}$, $K^{\prime\prime}$,
$H^{\prime\prime}$, $W^{\prime\prime}$, $C$ and $\tilde{H}^{\prime\prime}$ are
some positive constants.

###### Proof.

We will show that (2.65) still holds for $m+1$. At first, we estimate
$\nabla^{2}K^{(m+1)}$. In fact, it suffices only to treat
$\partial^{2}_{t}K^{(m+1)}$ since the other second order derivatives of
$K^{(m+1)}$ can be similarly proved.

Taking the first and second order derivatives of (2.38) with respect to $t$,
one has

$\begin{split}\partial^{2}_{t}K^{(m+1)}=&\epsilon^{2}(\nabla L^{(1)}(\epsilon
D_{\epsilon}\omega^{(m)})D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\tilde{h}^{(m)}K^{(m)}+\epsilon
L^{(1)}\partial_{t}\tilde{h}^{(m)}K^{(m)}+\epsilon
L^{(1)}\tilde{h}^{(m)}\partial_{t}K^{(m)}\\\
&+\epsilon^{2}\nabla(\nabla\lambda_{i}\cdot
r_{i}^{(m)})D_{\epsilon}\partial_{t}\omega^{(n)}h_{i}^{(m)}+\epsilon\nabla\lambda_{i}\cdot
r_{i}(\epsilon D_{\epsilon}\omega^{(m)})\partial_{t}h^{(m)}_{i}\end{split}$
(2.66)

and

$\begin{split}|\partial^{3}_{t}K^{(m+1)}|\leq&C\epsilon|\partial^{2}_{t}h^{(m)}|+C\epsilon^{2}\Big{(}|\partial^{2}_{t}K^{(m)}|+|\partial^{2}_{t}\omega^{(m)}|\Big{)}.\end{split}$
(2.67)

Then we can similarly derive that

$|\partial^{3}_{tty}K^{(m+1)}|\leq
C\epsilon|\partial^{2}_{ty}h^{(m)}|+C\epsilon^{2}\Big{(}|\partial^{2}_{ty}K^{(m)}|+|\partial^{2}_{ty}\omega^{(m)}|\Big{)}+C\epsilon^{2}$
(2.68)

and

$|\partial^{3}_{tyy}K^{(m+1)}|\leq
C\epsilon|\partial^{2}_{y}h^{(m)}|+C\epsilon^{2}\Big{(}|\partial^{2}_{y}K^{(m)}|+|\partial^{2}_{y}\omega^{(m)}|\Big{)}+C\epsilon^{2}.$
(2.69)

Integrating (2.67)-(2.69) with respect to $t$ yields

$\max_{y\in[a_{0},b_{0}]\atop s\in[t_{0},t_{1}]}|\nabla^{2}K^{(m+1)}|\leq
K^{\prime\prime}_{0}+CH^{\prime\prime}\quad\text{ with
}K^{\prime\prime}_{0}:=\max_{y\in[a_{0},b_{0}]}|\nabla^{2}K^{(m+1)}(y,t_{0})|.$

Next, we derive the boundedness of $\nabla^{2}h^{(m+1)}$. It suffices only to
estimate $\partial^{2}_{y}h^{(m+1)}$. Differentiating (2.39) with respect to
$y$ twice, then $\partial^{2}_{y}h^{(m+1)}_{i}$ satisfies

$\begin{split}|\partial_{t}\partial^{2}_{y}h^{(m+1)}_{i}|\leq&C\epsilon^{2}|\partial^{2}_{y}h^{(m)}|+C\epsilon^{3}|\partial^{2}_{y}K^{(m)}|+C\epsilon^{3}|\partial^{2}_{y}\omega^{(m)}|+C\epsilon|\partial^{2}_{y}\tilde{h}^{(m)}|+C\epsilon^{2}.\end{split}$
(2.70)

Then integrating (2.70) with respect to $t$ in the interval $[t_{0},t]$ yields

$|\partial^{2}_{y}h^{(m+1)}_{i}|\leq|\partial^{2}_{y}h^{(m+1)}_{i}(y,t_{0})|+C|\partial^{2}_{y}\tilde{h}^{(m)}|+C\epsilon|\partial^{2}_{y}h^{(m)}|.$

Similarly, one can obtain

$\begin{split}|\partial^{3}_{t}h^{(m+1)}_{i}|&\leq
C\epsilon|\partial^{2}_{t}\tilde{h}^{(m)}|+C\epsilon^{2}|\partial^{2}_{t}h^{(m)}|+C\epsilon^{3}+C\epsilon^{3}|\partial^{2}_{t}K^{(m)}|+C\epsilon^{3}|\partial^{2}_{t}\omega^{(m)}|,\\\
|\partial^{3}_{tty}h^{(m+1)}_{i}|&\leq
C\epsilon|\partial^{2}_{ty}\tilde{h}^{(m)}|+C\epsilon^{2}|\partial^{2}_{ty}h^{(m)}|+C\epsilon^{2}+C\epsilon^{3}|\partial^{2}_{ty}K^{(m)}|+C\epsilon^{3}|\partial^{2}_{ty}\omega^{(m)}|.\end{split}$

Therefore, it holds that

$\max_{y\in[a_{0},b_{0}]\atop t\in[t_{0},t_{1}]}|\nabla^{2}h^{(m+1)}_{i}|\leq
H^{\prime\prime}_{i0}+C\tilde{H}^{\prime\prime}\;\;\text{ with
}H^{\prime\prime}_{i0}=\max_{y\in[a_{0},b_{0}]}|\nabla^{2}h^{(m+1)}_{i}(y,t_{0})|.$

In particular,

$|\partial^{2}_{t}h^{(m+1)}_{i}|\leq|\partial^{2}_{t}h^{(m+1)}_{i}(y,t_{0})|+C|\partial^{2}_{t}\tilde{h}^{(m)}|\leq
C\epsilon.$

In addition, it follows from direct but tedious computation that

$\begin{split}&\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\partial^{2}_{t}h^{(m+1)}_{j}+2\partial_{t}K^{(m)}\partial^{2}_{t}h^{(m+1)}_{j}+2\epsilon\nabla(\lambda_{j}-\lambda_{i})D_{\epsilon}\partial_{t}\omega^{(m)}\partial^{2}_{yt}h^{(m+1)}_{j}\\\\[2.0pt]
&+\partial^{2}_{t}K^{(m)}\partial_{t}h^{(m+1)}_{j}+\partial^{2}_{t}(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}h^{(m+1)}_{j}+\epsilon\partial^{2}_{t}\Big{(}K^{(m)}\left((\tilde{h}^{(m)})^{\top}Q^{(2)}\tilde{h}^{(m+1)}+L^{(3)}\tilde{h}^{(m)}h^{(m+1)}_{j}\right)\Big{)}\\\
&+\epsilon\partial^{2}_{t}\Big{(}h^{(m)}_{i}\left((r_{i}(\epsilon
D_{\epsilon}\omega^{(m)}))^{\top}Q^{(3)}\tilde{h}^{(m+1)}+\nabla\lambda_{i}\cdot
r_{i}(\epsilon
D_{\epsilon}\omega^{(m)})h^{(m+1)}_{j}\right)\Big{)}=0\end{split}$ (2.71)

and

$\begin{split}&\big{|}\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\partial^{2}_{t}h^{(m+1)}_{j}+2\partial_{t}K^{(m)}\partial^{2}_{t}h^{(m+1)}_{j}\\\
+&\epsilon h^{(m)}_{i}\left((r_{i}(\epsilon
D_{\epsilon}\omega^{(m)}))^{\top}Q^{(3)}\partial^{2}_{t}\tilde{h}^{(m+1)}+\nabla\lambda_{i}\cdot
r_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{2}_{t}h^{(m+1)}_{j}\right)\big{|}\\\
\leq&C\epsilon^{2}+C\epsilon^{2}|\partial^{2}_{t}h^{(m)}|+C\epsilon^{2}|\partial^{2}_{t}\tilde{h}^{(m+1)}|+C\epsilon^{2}|\partial^{2}_{t}\omega^{(m)}|,\end{split}$
(2.72)

where we have used the fact that

$|\partial^{2}_{t}K^{(m)}|\leq
C\epsilon,\;\;|\partial^{2}_{yt}h^{(m+1)}_{j}|\leq
C|\partial^{2}_{t}h^{(m+1)}_{j}|+C\epsilon^{2}.$ (2.73)

This derives that

$\sum_{j\neq
i}|\partial^{2}_{t}h^{(m+1)}_{j}|\leq\tilde{H}^{\prime\prime}_{0}+C\epsilon^{2}+C\epsilon^{2}(W^{\prime\prime}+H^{\prime\prime})+C\epsilon|\partial^{2}_{t}\tilde{h}^{(m+1)}|$

and

$\max_{y\in[a_{0},b_{0}]\atop t\in[t_{0},t_{1}]}\sum_{j\neq
i}|\partial^{2}_{t}h^{(m+1)}_{j}|\leq C\tilde{H}^{\prime\prime}_{0}\;\;\text{
with
}\tilde{H}^{\prime\prime}_{0}=\max\Big{(}\displaystyle\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\nabla^{2}h^{(m+1)}_{j}(y,t_{0})|,C\epsilon\Big{)}.$

Differentiating (2.40) with respect to $y$ and taking direct estimates yield
that

$|\partial^{2}_{y}h^{(m+1)}_{j}|\leq
C|\partial^{2}_{ty}h^{(m+1)}_{j}|+C\epsilon,$

along with (2.73), it satisfies that

$\max_{y\in[a_{0},b_{0}]\atop t\in[t_{0},t_{1}]}\sum_{j\neq
i}|\partial^{2}_{yt}h^{(m+1)}_{j}|\leq
C\tilde{H}^{\prime\prime}_{0},\quad\max_{y\in[a_{0},b_{0}\atop
t\in[t_{0},t_{1}]}\sum_{j\neq i}|\partial^{2}_{y}h^{(m+1)}_{j}|\leq
C\tilde{H}^{\prime\prime}_{0},\qquad\forall j\neq i.$

In the following, we estimate $\partial^{2}_{t}\omega^{(m+1)}_{j}(j\neq i)$.
Differentiating (2.42) with respect to $t$ yields

$\begin{split}&\epsilon(\nabla\tilde{l}_{j}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\Big{(}K^{(m)}\partial_{t}\omega^{(m+1)}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\omega^{(m+1)}\Big{)}+\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\Big{(}K^{(m)}\partial^{2}_{t}\omega^{(m+1)}\\\
+&\partial_{t}K^{(m)}\partial_{t}\omega^{(m+1)}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{2}_{yt}\omega^{(m+1)}+\epsilon\nabla(\lambda_{j}-\lambda_{i})D_{\epsilon}\partial_{t}\omega^{(m)}\partial_{y}\omega^{(m+1)}\Big{)}=0.\end{split}$
(2.74)

By differentiating (2.74) with respect to $t$ and taking the direct but
tedious computations, one has

$\begin{split}&\big{|}\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\bar{q}^{(m+1)}_{j}+2\partial_{t}K^{(m)}\bar{q}^{(m+1)}_{j}\big{|}\\\
\leq&C\epsilon^{2}\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}\Big{(}|\bar{q}^{(m+1)}_{j}|+|\bar{z}^{(m+1)}_{j}|\Big{)}+C\epsilon^{2}\sum_{j\neq
i}|\bar{q}^{(m)}_{j}|+C\epsilon|\partial^{2}_{t}K^{(m)}|.\end{split}$

Meanwhile, we can derive from (2.74) that $|\bar{z}^{(m+1)}_{j}|\leq
C|\bar{q}^{(m+1)}_{j}|+C\epsilon^{2}$. By the characteristics method, we have

$\sum_{j\neq
i}|\bar{q}^{(m+1)}_{j}|\leq\tilde{Q}^{\prime\prime}_{0}+C\epsilon\sum_{j\neq
i}|\bar{q}^{(m+1)}_{j}|+C\epsilon^{2}\tilde{Q}^{\prime\prime}+C\epsilon^{2},$
(2.75)

where

$\tilde{Q}^{\prime\prime}_{0}:=\max\Big{(}\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\bar{q}^{(m+1)}_{j}(\cdot,t_{0})|,C\epsilon\Big{)},\quad\tilde{Q}^{\prime\prime}:=\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}\sum_{j\neq i}|\bar{q}^{(m)}_{j}(y,t)|.$

Differentiating (2.42) with respect to $y$ yields that

$\begin{split}(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})E^{(m+1)}_{j}=&-K^{(m)}\bar{z}^{(m+1)}_{j}-\partial_{y}K^{(m)}\bar{h}^{(m+1)}_{j}-\epsilon\nabla(\lambda_{j}-\lambda_{i})D_{\epsilon}\partial_{y}\omega^{(m)}\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\\\
&-\epsilon(\nabla\tilde{l}_{j}D_{\epsilon}\partial_{y}\omega^{(m)})(K^{(m)}\partial_{t}\omega^{(m+1)}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\omega^{(m+1)}),\end{split}$

which implies that $|E^{(m+1)}_{j}|\leq C|\bar{z}^{(m+1)}_{j}|+C\epsilon$.

Thus

$\sum_{j\neq i}|\bar{q}^{(m+1)}_{j}|\leq C\tilde{Q}^{\prime\prime}_{0}\leq
C\epsilon\tilde{W}^{\prime\prime}_{0},\;\;\tilde{W}^{\prime\prime}_{0}:=\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\partial^{2}_{t}\omega^{(m+1)}(y,t_{0})|,$

and

$\sum_{j\neq i}|\bar{z}^{(m+1)}_{j}|\leq
C\epsilon\tilde{W}^{\prime\prime}_{0},\quad\sum_{j\neq i}|E^{(m+1)}_{j}|\leq
C\epsilon\tilde{W}^{\prime\prime}_{0}.$ (2.76)

Next, we establish the estimates of $E^{(m+1)}_{i},\bar{q}^{(m+1)}_{i}$ and
$\bar{z}^{(m+1)}_{i}$. By (2.41), we arrive at

$\begin{split}|\partial_{t}E^{(m+1)}_{i}|=&\big{|}\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\sum_{j=1}^{n}E^{(m+1)}_{j}\tilde{r}^{(m)}_{j}-\partial^{2}_{y}\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\sum_{k\neq
i}\bar{h}^{(m+1)}_{k}\tilde{r}_{k}^{(m)}\\\
&-2\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\Big{(}\sum_{k\neq
i}\bar{z}^{(m+1)}_{k}\tilde{r}^{(m)}_{k}-\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\partial_{t}\omega^{(m+1)}\tilde{r}^{(m)}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\Big{)}\big{|}\\\
\lesssim&\epsilon^{2}\sum_{k=1}^{n}|E^{(m+1)}_{k}|+C\epsilon\sum_{k\neq
i}|\bar{z}^{(m+1)}_{k}|+C\epsilon^{2}|E^{(m)}_{i}|.\end{split}$ (2.77)

Then

$|E^{(m+1)}_{i}|\lesssim|E^{(m+1)}_{i}(y,t_{0})|+C\epsilon\sum^{n}_{k=1}|E^{(m+1)}_{k}|+C\sum_{k\neq
i}|\bar{z}^{(m+1)}_{k}|+C\epsilon|E^{(m)}_{i}|.$ (2.78)

Meanwhile, one has

$\displaystyle\big{|}\partial_{t}\bar{q}^{(m+1)}_{i}\big{|}=$
$\displaystyle\big{|}\partial^{2}_{t}\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{t}\omega^{(m+1)}+\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial^{2}_{t}\omega^{(m+1)}\big{|}\lesssim\epsilon^{2}\sum^{n}_{j=1}|\bar{q}^{(m+1)}_{j}|+\epsilon^{2}W^{\prime\prime},$
(2.79) $\displaystyle\big{|}\partial_{t}\bar{z}^{(m+1)}_{i}\big{|}=$
$\displaystyle\big{|}\partial^{2}_{yt}\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{t}\omega^{(m+1)}+\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\partial^{2}_{t}\omega^{(m+1)}\big{|}\lesssim\epsilon\sum^{n}_{j=1}|\bar{q}^{(m+1)}_{j}|+\epsilon^{2}W^{\prime\prime}.$
(2.80)

Collecting (2.78), (2.79) and (2.80) yields

$|\bar{q}^{(m+1)}_{i}|+|\bar{z}^{(m+1)}_{i}|+|E^{(m+1)}_{i}|\leq
C\epsilon\tilde{Q}^{\prime\prime}_{0}+|\bar{q}^{(m+1)}_{i}(y,t_{0})|+|\bar{z}^{(m+1)}_{i}(y,t_{0})|+|E^{(m+1)}_{i}(y,t_{0})|.$

This means

$\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}\sum^{n}_{j=1}\Big{(}|\bar{q}^{(m+1)}_{j}|+|\bar{z}^{(m+1)}_{j}|+|E^{(m+1)}_{j}|\Big{)}\leq
C\epsilon\tilde{W}^{\prime\prime}_{0}+CW^{\prime\prime}_{i0},$

where
$W^{\prime\prime}_{i0}=\displaystyle\max_{y\in[a_{0},b_{0}]}(|\bar{q}^{(m+1)}_{i}(y,t_{0})|+|\bar{z}^{(m+1)}_{i}(y,t_{0})|+|E^{(m+1)}_{i}(y,t_{0})|)$.

In conclusion, when $\epsilon>0$ is small, we can choose

$K^{\prime\prime}=K^{\prime\prime}_{0}+CH^{\prime\prime},\;H^{\prime\prime}=CH^{\prime\prime}_{0},\;W^{\prime\prime}=CW^{\prime\prime}_{0},\;H^{\prime\prime}_{0}=\tilde{H}^{\prime\prime}_{0}+H^{\prime\prime}_{i0},\;W^{\prime\prime}_{0}=\tilde{W}^{\prime\prime}_{0}+W^{\prime\prime}_{i0}$

such that (2.65) hold for $m+1$.

Therefore, we complete the proof of this lemma. ∎

Step $4$. The boundedness of $\nabla^{3}K^{(m+1)}$, $\nabla^{3}h^{(m+1)}$ and
$\nabla^{3}\omega^{(m+1)}$

In the following, we establish the estimates of $\nabla^{3}K^{(m+1)}$,
$\nabla^{3}h^{(m+1)}$ and $\nabla^{3}\omega^{(m+1)}$.

######  Lemma 2.12.

It holds that

$|\nabla^{3}K^{(m)}|\leq K^{\prime\prime\prime},\;|\nabla^{3}h^{(m)}|\leq
H^{\prime\prime\prime},\;|\nabla^{3}\omega^{(m)}|\leq W^{\prime\prime\prime}.$
(2.81)

where $|\partial^{3}_{t}K^{(m)}|\leq C\epsilon$,
$|\partial^{3}_{t}h^{(m)}|\leq C\epsilon$,
$|\nabla^{3}\tilde{h}^{(m)}|\leq\tilde{H}^{\prime\prime\prime}$,
$K^{\prime\prime\prime}$, $H^{\prime\prime\prime}$, $W^{\prime\prime\prime}$,
$C$ and $\tilde{H}^{\prime\prime\prime}$ are positive constants determined
later.

###### Proof.

We need to show that (2.81) still holds valid for $m+1$. First of all, we will
establish the boundedness of $\partial^{3}_{y}K^{(m+1)}$,
$\partial^{3}_{yyt}K^{(m+1)}$, $\partial^{3}_{ytt}K^{(m+1)}$ and
$\partial^{3}_{t}K^{(m+1)}$.

It follows from (2.38) and direct computations that

$|\partial_{t}\partial^{3}_{y}K^{(m+1)}|\leq
C\epsilon|\partial^{3}_{y}h^{(m)}|+C\epsilon^{2}(|\partial^{3}_{y}\omega^{(m)}|+|\partial^{3}_{y}K^{(m)}|)+C\epsilon^{2}.$
(2.82)

Similarly, we can arrive at

$\begin{split}&|\partial^{4}_{t}K^{(m+1)}|\leq
C\epsilon|\partial^{3}_{t}h^{(m)}|+C\epsilon^{2}(|\partial^{3}_{t}\omega^{(m)}|+|\partial^{3}_{t}K^{(m)}|)+C\epsilon^{3},\\\\[5.0pt]
&|\partial_{t}\partial^{3}_{yyt}K^{(m+1)}|\leq
C\epsilon|\partial^{3}_{yyt}h^{(m)}|+C\epsilon^{2}(|\partial^{3}_{yyt}\omega^{(m)}|+|\partial^{3}_{yyt}K^{(m)}|)+C\epsilon^{2},\\\\[5.0pt]
&|\partial_{t}\partial^{3}_{tty}K^{(m+1)}|\leq
C\epsilon|\partial^{3}_{tty}h^{(m)}|+C\epsilon^{2}(|\partial^{3}_{tty}\omega^{(m)}|+|\partial^{3}_{tty}K^{(m)}|)+C\epsilon^{2}.\end{split}$

In conclusion, one has

$\max_{y\in[a_{0},b_{0}]\atop t\in[t_{0},t_{1}]}|\nabla^{3}K^{(m+1)}|\leq
K^{\prime\prime\prime}_{0}+CH^{\prime\prime\prime}\;\;\text{ with
}K^{\prime\prime\prime}_{0}:=\max_{y\in[a_{0},b_{0}]}|\nabla^{3}K^{(m+1)}(y,t_{0})|.$

In particular, it holds

$|\partial^{3}_{t}K^{(m+1)}|\leq|\partial^{3}_{t}K^{(m+1)}(y,t_{0})|+C|\partial^{3}_{t}h^{(m)}|\leq
C\epsilon.$ (2.83)

Note that

$\begin{split}|\partial_{t}\partial^{3}_{y}h^{(m+1)}_{i}|\leq
C\epsilon|\partial^{3}_{y}\tilde{h}^{(m)}|+C\epsilon^{2}|\partial^{3}_{y}h^{(m)}|+C\epsilon^{2}.\end{split}$
(2.84)

Then this derives

$|\partial^{3}_{y}h^{(m+1)}_{i}|\leq|\partial^{3}_{y}h^{(m+1)}_{i}(y,t_{0})|+C|\partial^{3}_{y}\tilde{h}^{(m)}|.$

Analogously,

$\begin{split}&|\partial^{3}_{t}h^{(m+1)}_{i}|\leq|\partial^{3}_{t}h^{(m+1)}_{i}(y,t_{0})|+C|\partial^{3}_{t}\tilde{h}^{(m)}|,\\\\[5.0pt]
&|\partial^{3}_{tty}h^{(m+1)}_{i}|\leq|\partial^{3}_{tty}h^{(m+1)}_{i}(y,t_{0})|+C|\partial^{3}_{tty}\tilde{h}^{(m)}|,\\\\[5.0pt]
&|\partial^{3}_{tyy}h^{(m+1)}_{i}|\leq|\partial^{3}_{tyy}h^{(m+1)}_{i}(y,t_{0})|+C|\partial^{3}_{tyy}\tilde{h}^{(m)}|.\end{split}$

Therefore, it holds that

$\max_{y\in[a_{0},b_{0}],t\in[t_{0},t_{1}]}|\nabla^{3}h^{(m+1)}_{i}(y,t)|\leq
H^{\prime\prime\prime}_{i0}+C\tilde{H}^{\prime\prime\prime}\;\;\text{ with
}H^{\prime\prime\prime}_{i0}=\max_{y\in[a_{0},b_{0}]}|\nabla^{3}h^{(m+1)}_{i}(y,t_{0})|.$

In the following, we derive the estimate of $\nabla^{3}h^{(m+1)}_{j}$ for
$j\neq i$. It only suffices to establish the boundedness of
$\partial^{3}_{t}h^{(m+1)}_{j}$. Differentiating (2.71) with respect to $t$
together with direct computation yields

$\begin{split}&\Big{|}\big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\big{)}\partial^{3}_{t}h^{(m+1)}_{j}+3\partial_{t}K^{(m)}\partial^{3}_{t}h^{(m+1)}_{j}\Big{|}\\\
\leq&C\epsilon^{2}+C\epsilon|\partial^{3}_{t}K^{(m)}|+C\epsilon|\partial^{3}_{t}\tilde{h}^{(m+1)}|+C\epsilon^{2}|\partial^{3}_{t}h^{(m)}|+C\epsilon^{2}|\partial^{3}_{t}\omega^{(m)}|,\end{split}$

where we have used the fact derived from (2.71) that
$|\partial^{3}_{ytt}h^{(m+1)}_{j}|\leq
C|\partial^{3}_{t}h^{(m+1)}_{j}|+C\epsilon^{2}$. Denote

$\tilde{H}^{\prime\prime\prime}_{0}=\max\Big{(}\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\partial^{3}_{t}h^{(m+1)}_{j}(y,t_{0})|,C\epsilon\Big{)},$

then we have

$\begin{split}\sum_{j\neq
i}|\partial^{3}_{t}h^{(m+1)}_{j}|\leq\tilde{H}^{\prime\prime\prime}_{0}+C\epsilon\sum_{j\neq
i}|\partial^{3}_{t}h^{(m+1)}_{j}|+C\epsilon^{2}(H^{\prime\prime\prime}+W^{\prime\prime\prime})+C\epsilon^{2}.\end{split}$

This derives

$\max_{y\in[a_{0},b_{0}],\atop t\in[t_{0},t_{1}]}\sum_{j\neq
i}|\partial^{3}_{t}h^{(m+1)}_{j}|\leq
C\tilde{H}^{\prime\prime\prime}_{0},\quad\max_{y\in[a_{0},b_{0}],\atop
t\in[t_{0},t_{1}]}\sum_{j\neq i}|\partial^{3}_{ytt}h^{(m+1)}_{j}|\leq
C\tilde{H}^{\prime\prime\prime}_{0}.$

Analogously, based on (2.40), we can deduce that

$\sum_{j\neq i}|\partial^{3}_{yyt}h^{(m+1)}_{j}|\leq
C|\partial^{3}_{tty}h^{(m+1)}_{j}|+C\epsilon,\quad\sum_{j\neq
i}|\partial^{3}_{y}h^{(m+1)}_{j}|\leq
C|\partial^{3}_{yyt}h^{(m+1)}_{j}|+C\epsilon.$

Therefore

$\max_{y\in[a_{0},b_{0}],t\in[t_{0},t_{1}]}|\nabla^{3}h^{(m+1)}_{j}|\leq\tilde{H}^{\prime\prime\prime}:=C\tilde{H}^{\prime\prime\prime}_{0}.$

Next, we derive the estimate of $\nabla^{3}\omega^{(m+1)}_{j}$ for $j\neq i$.
For convenience, introduce some notations as

$\begin{split}&F^{(m+1)}_{j}:=\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{3}_{y}\omega^{(m+1)},\;\;G^{(m+1)}_{j}:=\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{3}_{tyy}\omega^{(m+1)},\\\
&J^{(m+1)}_{j}:=\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{3}_{ytt}\omega^{(m+1)},\;\;L^{(m+1)}_{j}:=\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{3}_{t}\omega^{(m+1)}.\end{split}$

Differentiating (2.42) with respect to $t$ three times and taking direct
computations, it holds that for $j\neq i$,

$\begin{split}&(\lambda_{j}-\lambda_{i})J^{(m+1)}_{j}+K^{(m)}L^{(m+1)}_{j}+\partial^{2}_{t}K^{(m)}\bar{h}^{(m+1)}_{j}+2\epsilon\nabla(\lambda_{j}-\lambda_{i})D_{\epsilon}\partial_{t}\omega^{(m)}\bar{z}^{(m+1)}_{j}\\\
&+2\partial_{t}K^{(m)}\bar{q}^{(m+1)}_{j}+\partial^{2}_{t}\tilde{l}_{j}(K^{(m)}\partial_{t}\omega^{(m+1)}+(\lambda_{j}-\lambda_{i})^{(m)}\partial_{y}\omega^{(m+1)})+\partial^{2}_{t}(\lambda_{j}-\lambda_{i})^{(m)}\tilde{l}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\omega^{(m+1)}\\\
&+2\epsilon(\nabla\tilde{l}_{j}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial_{t}\Big{(}K^{(m)}\partial_{t}\omega^{(m+1)}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\omega^{(m+1)}\Big{)}=0,\end{split}$
(2.85)

and

$\begin{split}&|\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}L^{(m+1)}_{j}+3\partial_{t}K^{(m)}L^{(m+1)}_{j}|\\\
\leq&C\epsilon^{2}W^{\prime\prime\prime}+C\epsilon^{2}+C\epsilon^{2}\sum_{j\neq
i}(|L^{(m+1)}_{j}|+|J^{(m+1)}_{j}|).\end{split}$ (2.86)

From (2.85), we can arrive at

$|J^{(m+1)}_{j}|\leq C|L^{(m+1)}_{j}|+C\epsilon^{2},\text{ for }j\neq i.$

Let

$\tilde{Q}^{\prime\prime\prime}_{0}:=\max\Big{(}\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|L^{(m+1)}_{j}(y,t_{0})|,C\epsilon\Big{)},\;\;\tilde{Q}^{\prime\prime\prime}:=\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}\sum_{j\neq i}|L^{(m+1)}_{j}(y,t)|.$

Then we can deduce from (2.86) that

$\sum_{j\neq
i}|L^{(m+1)}_{j}|\leq\tilde{Q}^{\prime\prime\prime}_{0}+C\epsilon\tilde{Q}^{\prime\prime\prime}+C\epsilon^{2}W^{\prime\prime\prime}+C\epsilon^{2}$

and

$\sum_{j\neq i}|L^{(m+1)}_{j}|\leq C\tilde{Q}^{\prime\prime\prime}_{0}\leq
C\epsilon\tilde{W}^{\prime\prime\prime}_{0},\quad\sum_{j\neq
i}|J^{(m+1)}_{j}|\leq C\epsilon\tilde{W}^{\prime\prime\prime}_{0},$

where
$\tilde{W}^{\prime\prime\prime}_{0}:=\displaystyle\max_{y\in[a_{0},b_{0}]}\sum_{j\neq
i}|\partial^{3}_{t}\omega^{(m+1)}_{j}(y,t_{0})|$.

In addition, based on (2.42), one has

$|G^{(m+1)}_{j}|\leq C|J^{(m+1)}_{j}|+C\epsilon.$

Analogously, we can also derive from (2.42) that

$|F^{(m+1)}_{j}|\leq C|G^{(m+1)}_{j}|+C\epsilon.$

Therefore,

$\max_{y\in[a_{0},b_{0}],\atop t\in[t_{0},t_{1}]}\sum_{j\neq
i}\Big{(}|L^{(m+1)}_{j}|+|J^{(m+1)}_{j}|+|F^{(m+1)}_{j}|+|G^{(m+1)}_{j}|\Big{)}\leq
C\epsilon\tilde{W}^{\prime\prime\prime}_{0}.$ (2.87)

Meanwhile, we can derive the equation of $F^{(m+1)}_{i}$ from (2.41) that

$\begin{split}|\partial_{t}F^{(m+1)}_{i}|=&|\epsilon(\nabla\tilde{l}_{i}^{(m)}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial^{3}_{y}\omega^{(m+1)}-\partial^{3}_{y}\tilde{l}_{i}^{(m)}\partial_{t}\omega^{(m+1)}-3\partial^{2}_{y}\tilde{l}^{(m)}_{i}\partial^{2}_{yt}\omega^{(m+1)}-3\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\partial^{3}_{yyt}\omega^{(m+1)}|\\\
\leq&C\epsilon^{2}\sum_{k=1}^{n}|F^{(m+1)}_{k}|+C\epsilon\sum_{k=1}^{n}|G^{(m+1)}_{k}|+C\epsilon^{2}W^{\prime\prime\prime}+C\epsilon^{2}.\end{split}$
(2.88)

Based on (2.41), we can directly calculate that

$\displaystyle|\partial_{t}G^{(m+1)}_{i}|=$
$\displaystyle|\epsilon(\nabla\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial^{3}_{yyt}\omega^{(m+1)}+\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{4}_{yytt}\omega^{(m+1)}|$ (2.89)
$\displaystyle=$
$\displaystyle|\partial^{3}_{yyt}\tilde{l}_{i}^{(m)}\partial_{t}\omega^{(m+1)}+\partial^{2}_{y}\tilde{l}_{i}^{(m)}\partial^{2}_{t}\omega^{(m+1)}+2\partial^{2}_{yt}\tilde{l}^{(m)}_{i}\partial^{2}_{yt}\omega^{(m+1)}+2\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\partial^{3}_{ytt}\omega^{(m+1)}|$
$\displaystyle\leq$ $\displaystyle
C\epsilon\sum^{n}_{k=1}|J^{(m+1)}_{k}|+C\epsilon^{2}W^{\prime\prime\prime}+C\epsilon^{2},$
$\displaystyle|\partial_{t}J^{(m+1)}_{i}|=$
$\displaystyle|\partial^{3}_{ytt}\tilde{l}^{(m)}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{t}\omega^{(m+1)}+\partial^{2}_{t}\tilde{l}_{i}^{(m)}\partial^{2}_{yt}\omega^{(m+1)}+2\partial^{2}_{yt}\tilde{l}^{(m)}_{i}\partial^{2}_{t}\omega^{(m+1)}$
(2.90)
$\displaystyle+\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial^{3}_{ytt}\omega^{(m+1)}+\epsilon(\nabla\tilde{l}_{i}D_{\epsilon}\partial_{y}\omega^{(m)})^{\top}\partial^{3}_{t}\omega^{(m+1)}|$
$\displaystyle\leq$ $\displaystyle
C\epsilon^{2}\sum^{n}_{k=1}|J^{(m+1)}_{k}|+C\epsilon\sum^{n}_{k=1}|L^{(m+1)}_{k}|+C\epsilon^{2}W^{\prime\prime\prime},$
$\displaystyle|\partial_{t}L^{(m+1)}_{i}|=$
$\displaystyle|\partial^{3}_{t}\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{t}\omega^{(m+1)}+3\partial^{2}_{t}\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})\partial^{2}_{t}\omega^{(m+1)}$ (2.91)
$\displaystyle+2\epsilon(\nabla\tilde{l}_{i}(\epsilon
D_{\epsilon}\omega^{(m)})D_{\epsilon}\partial_{t}\omega^{(m)})^{\top}\partial^{3}_{t}\omega^{(m+1)}|$
$\displaystyle\leq$ $\displaystyle
C\epsilon^{2}\sum_{k=1}^{n}|L^{(m+1)}_{k}|+C\epsilon^{2}W^{\prime\prime\prime}+C\epsilon^{2}.$

Collecting (2.88)-(2.91) derives that

$\begin{split}&|F^{(m+1)}_{i}|+|G^{(m+1)}_{i}|+|L^{(m+1)}_{i}|+|J^{(m+1)}_{i}|\\\
\leq&C\Big{(}|F^{(m+1)}_{i}(y,t_{0})|+|G^{(m+1)}_{i}(y,t_{0})|+|L^{(m+1)}_{i}(y,t_{0})|+|J^{(m+1)}_{i}(y,t_{0})|\Big{)}+C\epsilon\tilde{W}^{\prime\prime\prime}_{0}.\end{split}$
(2.92)

Summing (2.87) and (2.92) together, we have

$\max_{y\in[a_{0},b_{0}]\atop
t\in[t_{0},t_{1}]}\sum^{n}_{k=1}\Big{(}|F^{(m+1)}_{k}|+|G^{(m+1)}_{k}|+|L^{(m+1)}_{k}|+|J^{(m+1)}_{k}|\Big{)}\leq
C\epsilon\tilde{W}^{\prime\prime\prime}_{0}+W^{\prime\prime\prime}_{i0},$
(2.93)

where

$W^{\prime\prime\prime}_{i0}=\max_{y\in[a_{0},b_{0}]}\Big{(}|F^{(m+1)}_{i}(y,t_{0})|+|G^{(m+1)}_{i}(y,t_{0})|+|L^{(m+1)}_{i}(y,t_{0})|+|J^{(m+1)}_{i}(y,t_{0})|\Big{)}.$

In conclusion, for small $\epsilon>0$, we can choose

$K^{\prime\prime\prime}=K^{\prime\prime\prime}_{0}+C\tilde{H}^{\prime\prime\prime},\;H^{\prime\prime\prime}=CH^{\prime\prime\prime}_{0},\;W^{\prime\prime\prime}=CW^{\prime\prime\prime}_{0},$

with
$H^{\prime\prime\prime}_{0}=\tilde{H}^{\prime\prime\prime}+H^{\prime\prime\prime}_{i0},\;W^{\prime\prime\prime}_{0}=\tilde{W}^{\prime\prime\prime}_{0}+W^{\prime\prime\prime}_{i0}$,
such that the estimates (2.81) hold for $m+1$. ∎

Step $5$. The convergence of the approximate solutions

By the uniform boundedness of the approximate solutions
$(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$ established in Step
1-Step 4, we now start to show the uniform convergence of
$(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$ in $D$. In this case, if
we set
$(K,h_{i},\tilde{h},\omega)=\displaystyle\lim_{m\to\infty}(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$,
then $(K,h_{i},\tilde{h},\omega)$ is a classical solution to problem (2.35).

At first, by the analogous argument in Step 1, one can obtain there there
exists a uniform constant $C>0$ such that for $(y,t)\in D$ and all
$m\in\mathbb{N}$

$|K^{(m)}-K^{(0)}|\leq C\epsilon.$ (2.94)

Since $T_{\epsilon}+1-t_{0}\sim\frac{1}{\epsilon}$ holds and one can only
obtain

$|K^{(m+1)}-K^{(m)}|\leq\bar{M}_{0}|h^{(m)}-h^{(m-1)}|+\text{``contractible
terms"}$ (2.95)

and

$|h^{(m+1)}-h^{(m)}|\leq\bar{M}_{0}|h^{(m)}-h^{(m-1)}|+\text{``contractible
terms"}$ (2.96)

from (2.38) and (2.39) by the direct integrals on the time $t$, respectively.
This will arise the difficulty for us to show the Cauchy sequences of
$(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$ in $D$ (note that it is
unknown whether the constant $\bar{M}_{0}<1$ in (2.95) and (2.96) holds or
not). In order to overcome this difficulty, our strategy is to divide the time
interval $[t_{0},T_{\epsilon}+1]$ into $N$ subintervals as

$\begin{split}&I_{1}=[t_{0},t_{0}+\frac{T_{\epsilon}+1-t_{0}}{N}],\;I_{2}=[t_{0}+\frac{T_{\epsilon}+1-t_{0}}{N},t_{0}+\frac{2(T_{\epsilon}+1-t_{0})}{N}],\;\cdots,\\\
&I_{N}=[t_{0}+\frac{N-1}{N}(T_{\epsilon}+1-t_{0}),T_{\epsilon}+1],\end{split}$

where $N\in\mathbb{N}$ with $N<\frac{T_{\epsilon}-t_{0}+1}{2}$ is a suitably
large constant independent of $\epsilon$, and prove that
$(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$ is a Cauchy sequence in
any subinterval $I_{k}$ ($1\leq k\leq N$) by utilizing the length
$|I_{k}|\sim\frac{1}{N\epsilon}$ and replacing $\bar{M}_{0}$ in (2.95)-(2.96)
by the constant $\frac{\bar{M}_{0}}{N}<1$ for large $N$.

For $t\in I_{1}$, set

$\mathcal{K}=K^{(m+1)}-K^{(m)},\;\mathcal{I}=h^{(m+1)}_{i}-h^{(m)}_{i},\;\mathcal{J}=h^{(m+1)}_{j}-h^{(m)}_{j},\;\tilde{\mathcal{J}}=\sum_{j\neq
i}(h^{(m+1)}_{j}-h^{(m)}_{j}).$

Then we have

$\begin{split}\begin{cases}\partial_{t}\mathcal{K}=\epsilon L^{(1)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}K^{(m)}-\epsilon L^{(1)}(\epsilon
D_{\epsilon}\omega^{(m-1)})\tilde{h}^{(m-1)}K^{(m-1)}+\epsilon\nabla\lambda_{i}\cdot
r_{i}(\epsilon D_{\epsilon}\omega^{(m)})h^{(m)}_{i}\\\\[3.0pt]
\quad\qquad-\epsilon\nabla\lambda_{i}\cdot r_{i}(\epsilon
D_{\epsilon}\omega^{(m-1)})h^{(m-1)}_{i},\\\\[3.0pt]
\mathcal{K}(y,t_{0})=0.\end{cases}\end{split}$

Integrating with respect to $t$ in $I_{1}$ yields

$|\mathcal{K}|\leq C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon^{2}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|+\frac{C}{N}|h^{(m)}-h^{(m-1)}|.$
(2.97)

Similarly, $\mathcal{I}$ satisfies that

$\begin{split}\begin{cases}\partial_{t}\mathcal{I}=\epsilon(\tilde{h}^{(m)})^{\top}Q^{(1)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}K^{(m)}+\epsilon L^{(2)}(\epsilon
D_{\epsilon}\omega^{(m)})\tilde{h}^{(m)}h^{(m)}_{i}\\\\[3.0pt]
\qquad\qquad\qquad\;-\epsilon(\tilde{h}^{(m-1)})^{\top}Q^{(1)}(\epsilon
D_{\epsilon}\omega^{(m-1)})\tilde{h}^{(m-1)}K^{(m-1)}-\epsilon
L^{(2)}(\epsilon
D_{\epsilon}\omega^{(m-1)})\tilde{h}^{(m-1)}h^{(m-1)}_{i},\\\\[3.0pt]
\mathcal{I}(y,t_{0})=0.\end{cases}\end{split}$

Thus, we have

$\begin{split}|\mathcal{I}|\leq&\frac{C}{N}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|h^{(m)}_{i}-h^{(m-1)}_{i}|+C\epsilon^{2}|K^{(m)}-K^{(m-1)}|\\\\[3.0pt]
&+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|.\end{split}$
(2.98)

Based on (2.40), we can derive that

$\begin{split}&|\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\mathcal{J}|\\\
\leq&C\epsilon|\tilde{\mathcal{J}}|+C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|\\\
&+C\epsilon^{2}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon^{2}|h^{(m)}_{i}-h^{(m-1)}_{i}|.\end{split}$
(2.99)

Together with $K^{(m)}|_{t=t_{0}}>0$ and the initial-boundary conditions for
$t\in I_{1}$

$\begin{cases}\mathcal{J}(a_{0},t)=0,\quad j=i+1,\cdots,n,\\\\[3.0pt]
\mathcal{J}(y,t_{0})=0,\end{cases}$

or

$\begin{cases}\mathcal{J}(b_{0},t)=0,\quad j=1,\cdots,i-1,\\\\[3.0pt]
\mathcal{J}(y,t_{0})=0,\end{cases}$

it follows from (2.99) and the characteristics method that

$\begin{split}|\mathcal{\tilde{J}}|\leq&C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|+C\epsilon^{2}|h^{(m)}-h^{(m-1)}|.\end{split}$
(2.100)

Next, we show that $\omega^{(m)}$ is a Cauchy sequence in $I_{1}$, which is
equivalent to prove the Cauchy sequence of $p^{(m)}$. Denote

$\mathcal{P}_{i}=p^{(m+1)}_{i}-p^{(m)}_{i},\;\mathcal{P}_{j}=p^{(m+1)}_{j}-p^{(m)}_{j},\;\mathcal{P}=p^{(m+1)}-p^{(m)}.$

From (2.49), we arrive at

$\begin{cases}\partial_{t}\mathcal{P}_{i}+\epsilon(\tilde{h}^{(m)})^{\top}M_{1}(\epsilon
D_{\epsilon}\omega^{(m)})\mathcal{P}+\epsilon((\tilde{h}^{(m)})^{\top}-(\tilde{h}^{(m-1)})^{\top})M_{1}(\epsilon
D_{\epsilon}\omega^{(m)})p^{(m)}\\\
\quad\;\;\;+\epsilon(\tilde{h}^{(m-1)})^{\top}(M_{1}(\epsilon
D_{\epsilon}\omega^{(m)})-M_{1}(\epsilon
D_{\epsilon}\omega^{(m-1)}))p^{(m)}=0,\\\\[5.0pt]
\mathcal{P}_{i}(y,t_{0})=0.\end{cases}$

Then this yields in $I_{1}$,

$|\mathcal{P}_{i}|\leq\frac{C}{N}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|\mathcal{P}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|.$
(2.101)

In addition, by (2.50), one has

$\begin{split}&|\Big{(}K^{(m)}\partial_{t}+(\lambda_{j}-\lambda_{i})(\epsilon
D_{\epsilon}\omega^{(m)})\partial_{y}\Big{)}\mathcal{P}_{j}|\\\
\leq&C\epsilon|\mathcal{P}|+C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|\\\
&+C\epsilon|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|h^{(m)}_{i}-h^{(m-1)}_{i}|.\end{split}$
(2.102)

Analogous to the estimate of $\mathcal{J}$, we can obtain

$\begin{split}\sum_{j\neq
i}|\mathcal{P}_{j}|\leq&C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon|h^{(m)}-h^{(m-1)}|+C\epsilon|\mathcal{P}|\\\
&+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|.\end{split}$
(2.103)

Due to
$\omega^{(m+1)}=\displaystyle\sum^{n}_{j=1}p^{(m+1)}_{j}\tilde{r}_{j}(\epsilon
D_{\epsilon}\omega^{(m)})$, then for $k\neq i$,

$\begin{split}&\epsilon|\omega^{(m+1)}_{k}-\omega^{(m)}_{k}|\\\
\leq&C\sum_{j\neq i}|\mathcal{P}_{j}|+C\epsilon^{2}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|\\\
\leq&C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon|h^{(m)}-h^{(m-1)}|+C\epsilon|\mathcal{P}|\\\
&+C\epsilon^{2}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|\end{split}$
(2.104)

and

$\begin{split}|\omega^{(m+1)}_{i}-\omega^{(m)}_{i}|\leq&|\mathcal{P}|+C\epsilon^{2}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+\epsilon|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|\\\
\leq&C\epsilon|K^{(m)}-K^{(m-1)}|+\frac{C}{N}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|h^{(m)}_{i}-h^{(m-1)}_{i}|+C\epsilon|\mathcal{P}|\\\
&+C\epsilon^{2}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|.\end{split}$
(2.105)

Collecting (2.97), (2.98), (2.100), (2.101) and (2.103)–(2.105) yields

$\begin{split}&|\mathcal{K}|+|\mathcal{I}|+|\mathcal{\tilde{J}}|+|\mathcal{P}|+\epsilon\sum_{j\neq
i}|\omega^{(m+1)}_{j}-\omega^{(m)}_{j}|+|\omega^{(m+1)}_{i}-\omega^{(m)}_{i}|\\\
\leq&C\epsilon|K^{(m)}-K^{(m-1)}|+C(\epsilon+\frac{1}{N})|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C(\epsilon+\frac{1}{N})|h^{(m)}_{i}-h^{(m-1)}_{i}|\\\
&+C\epsilon|\mathcal{P}|+C\epsilon^{2}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|.\end{split}$
(2.106)

Thus, provided that $\epsilon$ is sufficiently small and $N$ is suitably large
such that $C(\epsilon+\frac{1}{N})<1$,
$(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$ is a Cauchy sequence in
$I_{1}$. By the analogous method, if it has been shown that
$(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$ is a Cauchy sequence in
$I_{k}$ for $2\leq k\leq N-1$, we next show that
$(K^{(m)},h^{(m)}_{i},\tilde{h}^{(m)},\omega^{(m)})$ is a Cauchy sequence in
$I_{N}$.

By (2.94) and the expression (2.36), it easy to know that
$K^{(m)}|_{t=t_{0}+\frac{N-1}{N}(T_{\epsilon}+1-t_{0})}>0$ holds. As before,
for $t\in I_{N}$, set

$\mathcal{K}=K^{(m+1)}-K^{(m)},\;\mathcal{I}=h^{(m+1)}_{i}-h^{(m)}_{i},\;\mathcal{J}=h^{(m+1)}_{j}-h^{(m)}_{j},\;\tilde{\mathcal{J}}=\sum_{j\neq
i}(h^{(m+1)}_{j}-h^{(m)}_{j})$

and

$\mathcal{P}_{i}=p^{(m+1)}_{i}-p^{(m)}_{i},\;\mathcal{P}_{j}=p^{(m+1)}_{j}-p^{(m)}_{j},\;\mathcal{P}=p^{(m+1)}-p^{(m)}.$

Then we have that for $t\in I_{N}$,

$\begin{split}|\mathcal{K}|\leq&|(K^{(m+1)}-K^{(m)})\Big{(}y,t_{0}+\frac{N-1}{N}(T_{\epsilon}+1-t_{0})\Big{)}|+\frac{C}{N}|h^{(m)}-h^{(m-1)}|+C\epsilon|K^{(m)}-K^{(m-1)}|\\\
&+C\epsilon^{2}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|,\
\end{split}$ (2.107)

where $K^{(m)}\Big{(}y,t_{0}+\frac{k-1}{N}(T_{\epsilon}+1-t_{0})\Big{)}$ has
been shown to be a Cauchy sequence.

Similarly, one has that for $t\in I_{N}$,

$\begin{split}|\mathcal{I}|\leq&|(h^{(m+1)}_{i}-h^{(m)}_{i})(y,t_{0}+\frac{k-1}{N}(T_{\epsilon}+1-t_{0}))|+\frac{C}{N}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|h^{(m)}_{i}-h^{(m-1)}_{i}|\\\\[3.0pt]
&+C\epsilon^{2}|K^{(m)}-K^{(m-1)}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|,\\\\[3.0pt]
\end{split}$ (2.108)

where $h^{(m)}_{i}(y,t_{0}+\frac{k-1}{N}(T_{\epsilon}+1-t_{0}))$ is a Cauchy
sequence.

Analogously, we arrive at

$\begin{split}|\tilde{\mathcal{J}}|\leq&C\epsilon|\tilde{\mathcal{J}}|+C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|\\\
&+C\epsilon^{2}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon^{2}|h^{(m)}_{i}-h^{(m-1)}_{i}|\end{split}$
(2.109)

and

$\displaystyle|\mathcal{P}_{i}|\leq$
$\displaystyle|\mathcal{P}_{i}(y,t_{0}+\frac{k-1}{N}(T_{\epsilon}+1-t_{0}))|+\frac{C}{N}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|\mathcal{P}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|$
$\displaystyle+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|$
$\displaystyle\leq$
$\displaystyle\frac{C}{N}|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|\mathcal{P}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|,$
(2.110) $\displaystyle\sum_{j\neq i}|\mathcal{P}_{j}|\leq$ $\displaystyle
C\epsilon|\mathcal{P}|+C\epsilon|K^{(m)}-K^{(m-1)}|+C\epsilon^{3}\sum_{j\neq
i}|\omega^{(m)}_{j}-\omega^{(m-1)}_{j}|+C\epsilon^{2}|\omega^{(m)}_{i}-\omega^{(m-1)}_{i}|$
(2.111)
$\displaystyle+C\epsilon|\tilde{h}^{(m)}-\tilde{h}^{(m-1)}|+C\epsilon|h^{(m)}_{i}-h^{(m-1)}_{i}|.$

Thus, along with (2.107)–(2.111), we can also get the same estimate (2.106)
for $t\in I_{N}$. Therefore, the proof of Theorem 2.2 is completed.

######  Theorem 2.13.

Under the assumptions (1.2), (1.4) and (2.13), there exists a unique blowup
point $(y_{\epsilon},T_{\epsilon})$ for problem (2.18) such that

$\partial_{y}\varphi(y_{\epsilon},T_{\epsilon})=0,\;\;\partial^{2}_{y}\varphi(y_{\epsilon},T_{\epsilon})=0,\;\;\partial^{3}_{y}\varphi(y_{\epsilon},T_{\epsilon})>0,\;\;\partial^{2}_{yt}\varphi(y_{\epsilon},T_{\epsilon})<0.$
(2.112)

which is of cusp type for the characteristic envelope
$\\{(y,t):\partial_{y}\varphi(y,t)=0,t\geq T_{\epsilon}\\}$.

###### Proof.

Let

$w=\epsilon D_{\epsilon}\omega,\;\tau=\epsilon t\;\;\text{and
$w_{0}(x)=D_{\epsilon}\omega_{0}(x)$}.$

In addition, without loss of generality, $\lambda_{i}(0)=0$ is assumed
(otherwise, one can apply the translation $(t,x)\mapsto(t,x+\lambda_{i}(0)t)$
to achieve this). Note that $\eqref{eq:2.88}_{1}$ can be reduced into

$\partial_{\tau}\omega+\epsilon^{-1}D^{-1}_{\epsilon}AD_{\epsilon}\partial_{x}\omega=0.$

This yields that for $\epsilon\to 0$,

$\partial_{\tau}\omega_{i}+\partial_{u_{i}}\lambda_{i}(0)\omega_{i}\partial_{x}\omega_{i}=0.$
(2.113)

By $\eqref{eq:2.2}_{2}$, one has that for $\epsilon=0$,

$\omega_{i}(x,\tau)=w^{i}_{0}(y),\quad
x=\varphi(y,\tau)=y+\partial_{u_{i}}\lambda_{i}(0)w^{i}_{0}(y)\tau.$

Then for $\epsilon=0$,

$K=\partial_{y}\varphi(y,\tau)=1+\partial_{u_{i}}\lambda_{i}(0)\Big{(}w^{i}_{0}(y)\Big{)}^{\prime}\tau.$

Note that for $\tau>0$

$\partial_{\tau}K(x_{0},\tau)|_{\epsilon=0}<0,\;\partial_{y}^{2}K(x_{0},\tau)|_{\epsilon=0}>0.$

On the other hand, for
$\tau_{0}=\Big{(}\max(-\partial_{u_{i}}\lambda_{i}(0)(w^{i}_{0}(y))^{\prime})\Big{)}^{-1}$,

$K(x_{0},\tau_{0})|_{\epsilon=0}=0,\quad\partial_{y}K(x_{0},\tau_{0})|_{\epsilon=0}=0.$

Therefore, from the implicit function theorem, there exist a unique point
$p(\epsilon)=(y_{\epsilon},\tau_{\epsilon})$ such that

$K(p(\epsilon))=0,\;\partial_{y}K(p(\epsilon))=0,\;\partial_{\tau}K(p(\epsilon))<0,\;\partial^{2}_{y}K(p(\epsilon))>0,\quad\displaystyle\lim_{\epsilon\to
0}(y_{\epsilon},\tau_{\epsilon})=(y_{0},\tau_{0}).$

This implies that for $T_{\epsilon}=\frac{\tau_{\epsilon}}{\epsilon}$,

$\partial_{y}\varphi|_{(y_{\epsilon},T_{\epsilon})}=0,\;\;\partial^{2}_{y}\varphi|_{(y_{\epsilon},T_{\epsilon})}=0,\;\;\partial^{2}_{yt}\varphi|_{(y_{\epsilon},T_{\epsilon})}<0,\;\;\partial^{3}_{y}\varphi|_{(y_{\epsilon},T_{\epsilon})}>0.$

Thus, we get the desired result (2.112). ∎

Next, we state our main result in the paper.

######  Theorem 2.14.

There admits a weak entropy solution to problem (1.1) including a shock curve
$x=\phi(t)\in C^{1}[T_{\epsilon},T_{\epsilon}+1]$ starting from the blowup
point $(x_{\epsilon},T_{\epsilon})$. Moreover, close to the point
$(x_{\epsilon},T_{\epsilon})$, it holds that for the solution $w$ of (2.1),

$\begin{split}\phi(t)=&x_{\epsilon}+\lambda_{i}(w(x_{\epsilon},T_{\epsilon}))(t-T_{\epsilon})+O(1)(t-T_{\epsilon})^{2},\\\
w_{i}(t)=&w_{i}(x_{\epsilon},T_{\epsilon})+O(1)\Big{(}(t-T_{\epsilon})^{3}+\big{(}x-x_{\epsilon}-\lambda_{i}(w(x_{\epsilon},T_{\epsilon}))(t-T_{\epsilon})\big{)}^{2}\Big{)}^{\frac{1}{6}},\\\
w_{j}(t)=&w_{j}(x_{\epsilon},T_{\epsilon})+O(1)\Big{(}(t-T_{\epsilon})^{3}+\big{(}x-x_{\epsilon}-\lambda_{i}(w(x_{\epsilon},T_{\epsilon}))(t-T_{\epsilon})\big{)}^{2}\Big{)}^{\frac{1}{3}},\quad
j\neq i.\end{split}$

######  Remark 2.15.

The result in Theorem 2.14 can be extended into the following case of $n\times
n$ conservation laws in (1.1) with the corresponding Riemann invariants
$w=(w_{1},...,w_{n})$ in Lemma 2.1: the initial data $w_{i}(x,0)\not=0\in
C_{0}^{\infty}(\mathbb{R})$ is small but $w^{\prime}_{i}(x,0)$ may be large,
while $w_{j}(x,0)\equiv C_{j}$ ($j\not=i$) hold.

##  3 Analysis on the cusp singularity near the blowup point

As shown in Section $2$, at the point $(y_{\epsilon},T_{\epsilon})$ (note that
$(x_{\epsilon},T_{\epsilon})=(\varphi(y_{\epsilon},T_{\epsilon}),T_{\epsilon})$
is just the unique blowup point at time $T_{\epsilon}$ for problem (1.1)), it
holds

$\partial_{y}\varphi(y_{\epsilon},T_{\epsilon})=0,\;\;\partial^{2}_{y}\varphi(y_{\epsilon},T_{\epsilon})=0,\;\;\partial^{3}_{y}\varphi(y_{\epsilon},T_{\epsilon})>0,\;\;\partial^{2}_{ty}\varphi(y_{\epsilon},T_{\epsilon})<0.$
(3.1)

Based on (3.1), it is derived from the unfolding theorem (see Theorem $2.1$ in
[21]) that there exist some smooth functions $h(y,t),A(t)$ and $B(t)$ such
that

$\varphi(y,t)=h^{3}(y,t)-A(t)h(y,t)+B(t),$ (3.2)

where $\partial_{y}h(y_{\epsilon},T_{\epsilon})>0$,
$A^{\prime}(T_{\epsilon})>0$, and

$3h^{2}(y_{\epsilon},T_{\epsilon})-A(T_{\epsilon})=0,\;\;h(y_{\epsilon},T_{\epsilon})=A(T_{\epsilon})=0,\;\;\varphi(y_{\epsilon},T_{\epsilon})=B(T_{\epsilon}).$

Differentiating (3.2) with respect to $t$ yields that

$\partial_{t}\varphi=(3h^{2}(y,t)-A(t))\partial_{t}h(y,t)-A^{\prime}(t)h+B^{\prime}(t)$
(3.3)

and $\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})=B^{\prime}(T_{\epsilon})$.
Taking the first order derivative of (3.3) with respect to $y$, one has

$\partial^{2}_{ty}\varphi=6h\partial_{y}h\partial_{t}h+(3h^{2}-A)\partial^{2}_{ty}h-A^{\prime}(t)\partial_{y}h$

and
$\partial^{2}_{ty}\varphi(y_{\epsilon},T_{\epsilon})=-A^{\prime}(T_{\epsilon})\partial_{y}h(y_{\epsilon},T_{\epsilon})<0$.

Let

$\Sigma=\Big{\\{}(y,t):\partial_{y}\varphi(y,t)=0,T_{\epsilon}\leq t\leq
T_{\epsilon}+1\Big{\\}}.$

Since $\partial_{y}h(y_{\epsilon},T_{\epsilon})>0$, then there exist two
smooth functions $y=\eta_{\pm}(t)$ such that

$h(\eta_{\pm}(t),t)=\pm\sqrt{\frac{A(t)}{3}},\quad
y_{\epsilon}=\eta_{\pm}(T_{\epsilon}).$

Set $x_{\pm}(t)=x(\eta_{\pm}(t),t)$, whose intersection point forms a cusp
point. In the set $\Sigma$, we have

$\partial^{2}_{yt}\varphi(y,t)\partial_{y}t+\partial^{2}_{y}\varphi(y,t)=0.$
(3.4)

Together with (3.1), this yields

$\partial_{y}t(y_{\epsilon},T_{\epsilon})=0.$

Since $\partial^{2}_{yt}\varphi(y_{\epsilon},T_{\epsilon})<0$, then from the
implicit function theorem, there exists a unique $C^{2}$ function $t=t(y)$ in
the neighbourhood of $(y_{\epsilon},T_{\epsilon})$ satisfying

$\partial_{y}\varphi(y,t(y))=0.$

Differentiating (3.4) with respect to $y$ yields that

$2\partial^{3}_{yyt}\varphi\partial_{y}t+\partial^{3}_{ytt}\varphi(\partial_{y}t)^{2}+\partial_{yt}^{2}\varphi\partial_{y}^{2}t+\partial_{y}^{3}\varphi=0.$

Together with (3.1), we have $\partial_{y}^{2}t(y_{\epsilon},T_{\epsilon})>0$,
which means that $t=t(y)$ achieves the minimum value at the point
$(y_{\epsilon},T_{\epsilon})$.

Without loss of generality and for simplicity to write, from now on we set

$\partial_{y}h(y_{\epsilon},T_{\epsilon})=A^{\prime}(T_{\epsilon})=1.$

Using Taylor expansion formula, we have

$\begin{split}h(\eta_{\pm}(t),t)=&h(y_{\epsilon},T_{\epsilon})+(\partial_{t}h\partial_{y}t+\partial_{y}h)(y_{\epsilon},T_{\epsilon})(\eta_{\pm}(t)-y_{\epsilon})+O(1)(\eta_{\pm}(t)-y_{\epsilon})^{2}\\\
=&(\eta_{\pm}(t)-y_{\epsilon})+O(1)(\eta_{\pm}(t)-y_{\epsilon})^{2}.\end{split}$

On the other hand, one has

$h(\eta_{\pm}(t),t)=\pm\sqrt{\frac{A(t)}{3}}=\pm\sqrt{\frac{t-T_{\epsilon}}{3}}+O(1)(t-T_{\epsilon}).$

Then it holds

$\begin{split}&\eta_{\pm}(t)-y_{\epsilon}=\pm\sqrt{\frac{t-T_{\epsilon}}{3}}+O(1)(t-T_{\epsilon}),\\\
&x_{\pm}(t)=h^{3}(\eta_{\pm}(t),t)-A(t)h(\eta_{\pm}(t),t)+B(t)=\mp\frac{2}{9}\sqrt{3}A(t)^{\frac{3}{2}}+B(t)\\\
&\qquad=\mp\frac{2}{9}\sqrt{3}(t-T_{\epsilon})^{\frac{3}{2}}+\varphi(y_{\epsilon},T_{\epsilon})+\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})(t-T_{\epsilon})+O(1)(t-T_{\epsilon})^{2},\end{split}$
(3.5)

where

$\begin{split}A(t)&=(t-T_{\epsilon})+O(1)(t-T_{\epsilon})^{2},\\\
B(t)&=\varphi(y_{\epsilon},T_{\epsilon})+\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})(t-T_{\epsilon})+O(1)(t-T_{\epsilon})^{2}.\end{split}$

Therefore,

$x_{\pm}(t)-\varphi(y_{\epsilon},T_{\epsilon})-\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})(t-T_{\epsilon})\sim\mp\frac{2}{9}\sqrt{3}(t-T_{\epsilon})^{\frac{3}{2}}.$

Next, we present the singularity structure of the solutions to the blowup
system (2.15) under the coordinate $(x,t)$ (see Figure $5$).

$x$$y^{\epsilon}_{0}$$x=x_{+}^{\epsilon}(t)$$x=x_{-}^{\epsilon}(t)$$t=T_{\epsilon}$$y^{\epsilon}_{-}(x,t)$$y^{\epsilon}_{+}(x,t)$$\eta^{\epsilon}_{-}(t)$$\eta^{\epsilon}_{+}(t)$$t_{0}$Figure
$5$. Real roots of the equation $x=\varphi(y,t)$

######  Lemma 3.1.

  * $(i)$

For $t\in(T_{\epsilon},T_{\epsilon}+1]$, there exist two distinct real roots
$\eta^{\epsilon}_{\pm}(t)$ to the equation $\partial_{y}\varphi(y,t)=0$ in a
small neighbourhood of $(y_{\epsilon},T_{\epsilon})$, satisfying
$\eta^{\epsilon}_{-}(t)<y_{\epsilon}(t)<\eta^{\epsilon}_{+}(t)$ and
$\eta^{\epsilon}_{\pm}(t)\in C^{\infty}(T_{\epsilon},T_{\epsilon}+1]$.

  * $(ii)$

Let $x^{\epsilon}_{-}(t)=\varphi(\eta^{\epsilon}_{-}(t),t)$ and
$x^{\epsilon}_{+}(t)=\varphi(\eta^{\epsilon}_{+}(t),t)$, then

    * $(1)$

for $x\in(x^{\epsilon}_{+}(t),x^{\epsilon}_{-}(t))$, there exist three real
roots $y_{-}^{\epsilon}(x,t)<y^{\epsilon}_{0}<y^{\epsilon}_{+}(x,t)$ to the
equation $x=\varphi(y,t)$.

    * $(2)$

for $x\geq x^{\epsilon}_{-}(t)$, there exists a unique real root
$y^{\epsilon}_{+}(x,t)$ to the equation $x=\varphi(y,t)$.

    * $(3)$

for $x\leq x^{\epsilon}_{+}(t)$, there exists a unique real root
$y^{\epsilon}_{-}(x,t)$ to the equation $x=\varphi(y,t)$.

###### Proof.

Let

$F(h)=h^{3}(y,t)-A(t)h(y,t)+B(t)-x.$

Then $F^{\prime}(h)=3h^{2}-A(t)$, and $F(h)$ achieves its local maximum value
at $h=-\sqrt{\frac{A(t)}{3}}$. Moreover,

$F(-\sqrt{\frac{A(t)}{3}})=-\frac{2}{3}A(t)h+B(t)-x=\frac{2}{9}\sqrt{3}A(t)^{\frac{3}{2}}+B(t)-x.$

Meanwhile, $F(h)$ also obtains its local minimum value at
$h=\sqrt{\frac{A(t)}{3}}$, and

$F(\sqrt{\frac{A(t)}{3}})=-\frac{2}{3}A(t)h+B(t)-x=-\frac{2}{9}\sqrt{3}A(t)^{\frac{3}{2}}+B(t)-x.$

When $x^{\epsilon}_{+}(t)<x<x^{\epsilon}_{-}(t)$, we derive from
$\eqref{eq:3.6}_{2}$ that

$-\frac{2}{9}\sqrt{3}A(t)^{\frac{3}{2}}=B(t)-x^{\epsilon}_{-}(t)<B(t)-x<B(t)-x^{\epsilon}_{+}(t)=\frac{2}{9}\sqrt{3}A(t)^{\frac{3}{2}},$

which means

$\begin{split}F(-\sqrt{\frac{A(t)}{3}})=&B(t)-x+\frac{2}{9}\sqrt{3}A(t)^{\frac{3}{2}}>0,\\\
F(\sqrt{\frac{A(t)}{3}})=&B(t)-x-\frac{2}{9}\sqrt{3}A(t)^{\frac{3}{2}}<0.\end{split}$

Therefore, there exist three real roots
$y_{-}^{\epsilon}(x,t)<y^{\epsilon}_{0}<y^{\epsilon}_{+}(x,t)$ to the equation
$x=\varphi(y,t)$.

When $x\geq x^{\epsilon}_{-}(t)$, $F(-\sqrt{\frac{A(t)}{3}})\leq 0$ holds.
Then there exists a unique solution $y^{\epsilon}_{+}(x,t)$ to the equation
$x=\varphi(y,t)$. Similarly, when $x\leq x^{\epsilon}_{-}(t)$, we can also
show that $F(\sqrt{\frac{A(t)}{3}})\geq 0$ holds and there is a unique
solution $y^{\epsilon}_{-}(x,t)$ to the equation $x=\varphi(y,t)$.

In conclusion, we complete the proof of this lemma. ∎

###  3.1 The singular behavior of solutions in cusp domain

In this subsection, we will describe the singular behavior of solutions to
(2.15) in the cusp domain, which is important to construct the first
approximation of shock solutions.

Denote

$\begin{split}\Omega_{+}=&\\{(x,t)\in\Omega:x>x^{\epsilon}_{+}(t),\;T_{\epsilon}<t\leq
T_{\epsilon}+1\\},\\\
\Omega_{-}=&\\{(x,t)\in\Omega:x<x^{\epsilon}_{-}(t),\;T_{\epsilon}<t\leq
T_{\epsilon}+1\\},\\\
\Omega_{0}=&\\{(x,t)\in\Omega:x^{\epsilon}_{+}(t)<x<x^{\epsilon}_{-}(t),\;T_{\epsilon}<t\leq
T_{\epsilon}+1\\},\\\
d_{\epsilon}=&(t-T_{\epsilon})^{3}+\big{(}x-x_{\epsilon}-\lambda_{i}(w(x_{\epsilon},T_{\epsilon}))(t-T_{\epsilon})\big{)}^{2},\\\\[3.0pt]
\end{split}$

where the picture of $\Omega_{0}$ sees Figure 6 below.

Let the $i$-th characteristics of system $\eqref{eq:1.1}_{1}$ be
$x=\phi^{0}(t)\in C^{\infty}[T_{\epsilon},T_{\epsilon}+1]$ satisfying

$\begin{cases}\begin{split}\frac{d\phi^{0}(t)}{dt}=&\lambda_{i}\Big{(}\int^{1}_{0}(\partial_{u_{k}}f_{l})\Big{(}\theta
u(v(y^{\epsilon}_{+}(\phi^{0}(t),t),t))+(1-\theta)u(v(y^{\epsilon}_{-}(\phi^{0}(t),t),t))\Big{)}d\theta\Big{)},\\\
\phi^{0}(T_{\epsilon})=&x_{\epsilon},\end{split}\end{cases}$ (3.6)

where $x^{\epsilon}_{+}(t)<\phi^{0}(t)<x^{\epsilon}_{-}(t)$, and

$\phi^{0}(t)=x_{\epsilon}+\lambda_{i}(w(x_{\epsilon},T_{\epsilon}))(t-T_{\epsilon})+O(1)((t-T_{\epsilon})^{2}),\quad
t\in[T_{\epsilon},T_{\epsilon}+1].$

$x=\phi^{0}(t)$ is called the pres-shock wave of (1.1), whose picture sees
Figure $7$. In the following, we establish some properties of
$y^{\epsilon}_{\pm}(x,t)\in C^{\infty}(\Omega_{\pm})$.

$x$$x=x_{+}^{\epsilon}(t)$$x=x_{-}^{\epsilon}(t)$$t=T_{\epsilon}$$y_{-}^{\epsilon}(x,t)$$y_{+}^{\epsilon}(x,t)$$t_{0}$$\Omega_{0}$Figure
$6$. Cusp domain $\Omega_{0}$

$x$$x=x_{+}^{\epsilon}(t)$$x=x_{-}^{\epsilon}(t)$$t=T_{\epsilon}$$y_{-}^{\epsilon}(x,t)$$y_{+}^{\epsilon}(x,t)$$t_{0}$Pre-
shock curve $x=\phi^{0}(t)$Figure $7$. Pre-shock curve $x=\phi^{0}(t)$ in cusp
domain

######  Lemma 3.2.

It holds that in $\Omega_{\pm}$,

$\begin{split}&|y^{\epsilon}_{\pm}(x,t)-y_{\epsilon}|\leq
Cd^{\frac{1}{6}}_{\epsilon},\quad|\partial_{x}y^{\epsilon}_{\pm}(x,t)|\leq
Cd^{-\frac{1}{3}}_{\epsilon},\\\ &|\partial_{l}y^{\epsilon}_{\pm}(x,t)|\leq
Cd^{-\frac{1}{6}}_{\epsilon},\quad|\partial^{2}_{x}y^{\epsilon}_{\pm}(x,t)|\leq
Cd^{-\frac{5}{6}}_{\epsilon},\\\
&|\partial^{2}_{xt}y^{\epsilon}_{\pm}(x,t)|\leq
Cd^{-\frac{5}{6}}_{\epsilon},\quad|\partial^{2}_{t}y^{\epsilon}_{\pm}(x,t)|\leq
Cd^{-\frac{5}{6}}_{\epsilon},\end{split}$

where $l$ stands for the tangent direction of the $i$-th characteristics
passing through the point $(x_{\epsilon},T_{\epsilon})$.

###### Proof.

It suffices to prove the desired results for $y^{\epsilon}_{+}(x,t)$. By
direct calculations, one has

$A(t)^{3}+(B(t)-\varphi(y,t))^{2}\sim(t-T_{\epsilon})^{3}+(\varphi(y,t)-\varphi(y_{\epsilon},T_{\epsilon})-\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})(t-T_{\epsilon}))^{2}=d_{\epsilon}.$

For the real roots of (3.2), one has

$\begin{split}h(y^{\epsilon}_{+}(x,t),t)=&\Big{(}-\frac{B(t)-\varphi(y^{\epsilon}_{+}(x,t),t)}{2}+\sqrt{\frac{1}{4}(B(t)-\varphi(y^{\epsilon}_{+}(x,t),t))^{2}-\frac{1}{27}A(t)^{3}}\Big{)}^{\frac{1}{3}}\\\
&+\Big{(}-\frac{B(t)-\varphi(y^{\epsilon}_{+}(x,t),t)}{2}-\sqrt{\frac{1}{4}(B(t)-\varphi(y^{\epsilon}_{+}(x,t),t))^{2}-\frac{1}{27}A(t)^{3}}\Big{)}^{\frac{1}{3}},\end{split}$
(3.7)

and

$\begin{split}&|B(t)-\varphi(y^{\epsilon}_{+}(x,t),t)|>\frac{2\sqrt{3}}{9}A(t)^{\frac{3}{2}},\\\
&|\Big{(}-\frac{B(t)-\varphi(y^{\epsilon}_{+}(x,t),t)}{2}+\sqrt{\frac{1}{4}(B(t)-\varphi(y^{\epsilon}_{+}(x,t),t))^{2}-\frac{1}{27}A(t)^{3}}\Big{)}^{\frac{2}{3}}+\frac{1}{3}A(t)\\\
&\Big{(}-\frac{B(t)-\varphi(y^{\epsilon}_{+}(x,t),t)}{2}-\sqrt{\frac{1}{4}(B(t)-\varphi(y^{\epsilon}_{+}(x,t),t))^{2}-\frac{1}{27}A(t)^{3}}\Big{)}^{\frac{2}{3}}|\leq
Cd^{\frac{1}{3}}_{\epsilon}.\end{split}$

Therefore

$C_{1}d^{\frac{1}{6}}_{\epsilon}\leq|h(y^{\epsilon}_{+}(x,t),t)|\leq
C_{2}d^{\frac{1}{6}}_{\epsilon},$ (3.8)

which means $h(y^{\epsilon}_{+}(x,t),t)\sim d^{\frac{1}{6}}_{\epsilon}$ with
$C_{1}$ and $C_{2}$ being positive constants independent of $\epsilon$.

From (3.2), it holds

$h^{3}\big{(}y^{\epsilon}_{+}(x,t),t\big{)}-A(t)h(y^{\epsilon}_{+}(x,t),t)+B(t)=x=\varphi(y,t).$
(3.9)

Differentiating (3.9) with respect to $x$ yields

$\partial_{x}y^{\epsilon}_{+}(x,t)=\frac{1}{\partial_{y}h(y^{\epsilon}_{+}(x,t),t)\Big{(}3h^{2}(y^{\epsilon}_{+}(x,t),t)-A(t)\Big{)}}.$
(3.10)

On the other hand, based on (3.9), one has

$\begin{split}|3h^{2}(y^{\epsilon}_{+}(x,t),t)-A(t)|=&3|h^{2}(y^{\epsilon}_{+}(x,t),t)-A(t)|+2A(t)\\\
=&|h^{-1}(y^{\epsilon}_{+}(x,t),t)||\varphi(y^{\epsilon}_{+}(x,t),t)-B(t)|+2A(t)\\\
\geq&Cd^{-\frac{1}{6}}_{\epsilon}((\varphi(y^{\epsilon}_{+}(x,t),t)-B(t))^{2}+A(t)^{3})^{\frac{1}{2}}=Cd^{\frac{1}{3}}_{\epsilon},\end{split}$
(3.11)

where we have used the fact derived from the formula (3.7) that
$h(y^{\epsilon}_{+}(x,t),t)$ has the same sign with
$\varphi(y^{\epsilon}_{+}(x,t),t)-B(t)$, and then it follows from (3.9) that

$h^{2}(y^{\epsilon}_{+}(x,t),t)-A(t)=h^{-1}(y^{\epsilon}_{+}(x,t),t)(\varphi(y^{\epsilon}_{+}(x,t),t)-B(t))>0.$

Together with (3.10), this yields

$|\partial_{x}y^{\epsilon}_{+}(x,t)|\leq Cd^{-\frac{1}{3}}_{\epsilon}.$

Similarly, one has from (3.9) that

$\partial_{t}y_{+}^{\epsilon}=-\frac{(3h^{2}-A(t))\partial_{t}h-A^{\prime}(t)h+B^{\prime}(t)}{\partial_{y}h(y^{\epsilon}_{+}(x,t),t)(3h^{2}-A(t))}=-\frac{\partial_{t}\varphi(y^{\epsilon}_{+}(x,t),t)}{\partial_{y}h(y^{\epsilon}_{+}(x,t),t)(3h^{2}-A(t))},$
(3.12)

which implies $|\partial_{t}y_{+}^{\epsilon}|\leq
Cd^{-\frac{1}{3}}_{\epsilon}$ and

$|\partial_{l}y^{\epsilon}_{+}(x,t)|=\big{|}\partial_{t}y^{\epsilon}_{+}(x,t)+\lambda_{i}(w(y^{\epsilon}_{+}(x,t),t))\partial_{x}y^{\epsilon}_{+}(x,t)\big{|}=\big{|}\frac{\partial_{t}\varphi(y^{\epsilon}_{+}(x,t),t)-\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})}{\partial_{y}h(y^{\epsilon}_{+}(x,t),t)(3h^{2}-A)}\big{|}\leq
Cd^{-\frac{1}{6}}_{\epsilon},$

where

$\begin{split}&|\partial_{t}\varphi(y^{\epsilon}_{+}(x,t),t)-\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})|\\\
=&\big{|}\partial^{2}_{ty}\varphi(y_{\epsilon},T_{\epsilon})(y^{\epsilon}_{+}-y_{\epsilon})+\partial^{2}_{t}\varphi(y_{\epsilon},T_{\epsilon})(t-T_{\epsilon})+O(1)\Big{(}(y^{\epsilon}_{+}-y_{\epsilon})^{2}+(t-T_{\epsilon})^{2}\Big{)}\big{|}\leq
Cd^{\frac{1}{6}}_{\epsilon}.\end{split}$

Therefore,

$|y^{\epsilon}_{+}(x,t)-y_{\epsilon}|\leq
Cd^{-\frac{1}{6}}_{\epsilon}(t-T_{\epsilon})\leq
C(t-T_{\epsilon})^{\frac{1}{2}}\leq Cd^{\frac{1}{6}}_{\epsilon}.$

Next, we estimate $\partial^{2}_{x}y^{\epsilon}_{+}(x,t)$,
$\partial^{2}_{xt}y^{\epsilon}_{+}(x,t)$ and
$\partial^{2}_{t}y^{\epsilon}_{+}(x,t)$. It follows from (3.10) and direct
computation that

$\begin{split}|\partial^{2}_{x}y^{\epsilon}_{+}(x,t)|=&\big{|}\Big{(}\frac{\partial^{2}_{y}h(y^{\epsilon}_{+}(x,t),t)}{(\partial_{y}h)^{2}(y^{\epsilon}_{+}(x,t),t)(3h^{2}-A(t))}+\frac{6h(y^{\epsilon}_{+}(x,t),t)}{(3h^{2}-A(t))^{2}}\Big{)}\partial_{x}y^{\epsilon}_{+}(x,t)\big{|}\\\
\leq&C(d^{-\frac{1}{6}}_{\epsilon}d^{-\frac{1}{3}}_{\epsilon}+d^{\frac{1}{6}}_{\epsilon}d^{-\frac{2}{3}}_{\epsilon})d^{-\frac{1}{3}}_{\epsilon}\leq
Cd^{-\frac{5}{6}}_{\epsilon},\end{split}$

where we have used the facts that

$\begin{split}|h(y^{\epsilon}_{+}(x,t),t)|\leq&Cd^{\frac{1}{6}}_{\epsilon},\;\partial_{y}h(y_{\epsilon},T_{\epsilon})=1,\;|3h^{2}-A(t)|\geq
Cd^{\frac{1}{3}}_{\epsilon},\;|\partial_{x}y^{\epsilon}_{+}|\leq
Cd^{-\frac{1}{3}}_{\epsilon},\\\
|\partial^{2}_{y}\varphi(y^{\epsilon}_{+}(x,t),t)|=&\big{|}\partial^{2}_{y}\varphi(y_{\epsilon},T_{\epsilon})+\partial^{3}_{y}\varphi(y_{\epsilon},T_{\epsilon})(y^{\epsilon}_{+}(x,t)-y_{\epsilon})+\partial^{3}_{tyy}\varphi(y_{\epsilon},T_{\epsilon})(t-T_{\epsilon})\\\
&+O(1)\Big{(}(t-T_{\epsilon})^{2}+(y^{\epsilon}_{+}-y_{\epsilon})^{2}\Big{)}\big{|}\leq
Cd^{\frac{1}{6}}_{\epsilon},\\\
|\partial^{2}_{y}h(y^{\epsilon}_{+}(x,t),t)|=&\big{|}\frac{1}{3h^{2}(y^{\epsilon}_{+}(x,t),t)-A(t)}\Big{(}\partial^{2}_{y}\varphi(y^{\epsilon}_{+}(x,t),t)-6h(\partial_{y}h)^{2}\Big{)}\big{|}\leq
Cd^{-\frac{1}{6}}_{\epsilon}.\end{split}$

Taking the first order derivatives of (3.2) with respect to $t$ and $y$
respectively, we have

$\displaystyle\partial_{t}h(y,t)=$
$\displaystyle\frac{\partial_{t}\varphi+A^{\prime}(t)h-B^{\prime}(t)}{3h^{2}-A(t)},$
(3.13) $\displaystyle\partial_{y}h(y,t)=$
$\displaystyle\frac{\partial_{y}\varphi}{3h^{2}-A(t)}.$ (3.14)

From (3.13)–(3.14), one has

$\partial_{t}h+\partial_{t}y^{\epsilon}_{+}\partial_{y}h=\frac{1}{3h^{2}-A(t)}(\partial_{t}\varphi-B^{\prime}(t)+\partial_{y}\varphi\partial_{t}y^{\epsilon}_{+}+A^{\prime}(t)h),$
(3.15)

where $A^{\prime}(t)=1+O(1)(t-T_{\epsilon})$ by $A^{\prime}(T_{\epsilon})=1$.
Using Taylor expansion formula, we arrive at

$\begin{split}\partial_{t}\varphi(y^{\epsilon}_{+},t)=&\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})+\partial^{2}_{ty}\varphi(y_{\epsilon},T_{\epsilon})(y^{\epsilon}_{+}-y_{\epsilon})+\partial^{2}_{t}\varphi(y_{\epsilon},T_{\epsilon})(t-T_{\epsilon})+O(1)\Big{(}(t-T_{\epsilon})^{2}+(y^{\epsilon}_{+}-y_{\epsilon})^{2}\Big{)},\\\
B^{\prime}(t)=&\partial_{t}\varphi(y_{\epsilon},T_{\epsilon})+B^{\prime\prime}(T_{\epsilon})(t-T_{\epsilon})+O(1)(t-T_{\epsilon})^{2}.\end{split}$

This yields