# Global Finite-Energy Solutions of
the Compressible Euler-Poisson Equations
for General Pressure Laws with Spherical Symmetry

Gui-Qiang G. Chen  Mathematical Institute, University of Oxford, Oxford OX2
6GG, UK; Academy of Mathematics and Systems Science, Chinese Academy of
Sciences, Beijing 100190, China; School of Mathematical Sciences, Fudan
University, Shanghai 200433, China<EMAIL_ADDRESS>,  Feimin Huang
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, China; School of Mathematical Sciences, University of Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS>,  Tianhong Li
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, China; Hua Loo-Keng Key Laboratory of Mathematics, AMSS, CAS,
Beijing 100190, People’s Republic of China<EMAIL_ADDRESS>,  Weiqiang Wang
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, China<EMAIL_ADDRESS>and  Yong Wang  Academy of
Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,
China; School of Mathematical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS>
###### Abstract.

We are concerned with global finite-energy solutions of the three-dimensional
compressible Euler-Poisson equations with gravitational potential and general
pressure law, especially including the constitutive equation of white dwarf
stars. In this paper, we construct global finite-energy solutions with
spherical symmetry of the Cauchy problem for the Euler-Poisson equations as
the inviscid limit of the corresponding compressible Navier-Stokes-Poisson
equations. The strong convergence of the vanishing viscosity solutions is
achieved through entropy analysis, uniform estimates in $L^{p}$, and a more
general compensated compactness framework via several new main ingredients. A
key estimate is first established for the integrability of the density over
unbounded domains independent of the vanishing viscosity coefficient. Then a
special entropy pair is carefully designed via solving a Goursat problem for
the entropy equation such that a higher integrability of the velocity is
established, which is a crucial step. Moreover, the weak entropy kernel for
the general pressure law and its fractional derivatives of the required order
near vacuum ($\rho=0$) and far-field ($\rho=\infty$) are carefully analyzed.
Owing to the generality of the pressure law, only the
$W^{-1,p}_{\mathrm{loc}}$-compactness of weak entropy dissipation measures
with $p\in[1,2)$ can be obtained; this is rescued by the equi-integrability of
weak entropy pairs which can be established by the estimates obtained above,
so that the div-curl lemma still applies. Finally, based on the above analysis
of weak entropy pairs, the $L^{p}$ compensated compactness framework for the
compressible Euler equations with general pressure law is established. This
new compensated compactness framework and the techniques developed in this
paper should be useful for solving further nonlinear problems with similar
features.

###### Key words and phrases:

Euler-Poisson equations, white dwarf stars, finite-energy solutions, general
pressure law, spherical symmetry, entropy analysis, $L^{p}$ estimates,
compensated compactness framework, Goursat problem.

###### 2010 Mathematics Subject Classification:

35Q85, 85A30, 35L65, 35D30, 35Q31, 76N10

###### Contents

  1. 1 Introduction
  2. 2 Mathematical Problem and Main Theorems
  3. 3 Properties of the General Pressure Law and Related Internal Energy
  4. 4 Entropy Analysis: Weak Entropy Pairs
    1. 4.1 A special entropy pair
    2. 4.2 Estimates of the weak entropy pairs
    3. 4.3 Singularities of the entropy kernel and the entropy flux kernel
  5. 5 Uniform Estimates of Approximate Solutions
    1. 5.1 Basic estimates
    2. 5.2 Higher integrability of the density and the velocity
  6. 6 Existence of Global Weak Solutions of CNSPEs
  7. 7 $W_{\mathrm{loc}}^{-1,p}$–Compactness of Weak Entropy Dissipation Measures
  8. 8 $L^{p}$ Compensated Compactness Framework
  9. 9 Existence of Global Finite-Energy Solutions of CEPEs
  10. A Some Inequalities
    1. A.1 A sharp Sobolev inequality
    2. A.2 Some variants of the Grönwall inequality

##  1\. Introduction

We are concerned with global finite-energy solutions of the three-dimensional
(3-D) compressible Euler-Poisson equations (CEPEs) that take the form:

$\left\\{\begin{aligned} &\partial_{t}\rho+\nabla\cdot\cal{M}=0,\\\
&\partial_{t}\cal{M}+\nabla\cdot\Big{(}\frac{\cal{M}\otimes\cal{M}}{\rho}\Big{)}+\nabla
P+\rho\nabla\Phi=0,\\\ &\Delta\Phi={k_{g}\rho},\end{aligned}\right.$ (1.1)

for
$(t,\mathbf{x}):=(t,x_{1},x_{2},x_{3})\in\mathbb{R}_{+}\times\mathbb{R}^{3}:=(0,\infty)\times\mathbb{R}^{3}$.
The system is used to model the motion of compressible gaseous stars under a
self-consistent gravitational field (cf. [6]), where $\rho$ is the density,
$P=P(\rho)$ is the pressure, $\cal{M}\in\mathbb{R}^{3}$ is the momentum,
$\Phi$ represents the gravitational potential of gaseous stars as ${k_{g}}>0$,
$\nabla=(\partial_{x_{1}},\partial_{x_{2}},\partial_{x_{3}})$, and
$\Delta=\partial_{x_{1}x_{1}}+\partial_{x_{2}x_{2}}+\partial_{x_{3}x_{3}}$.
Without loss of generality by scaling, we take ${k_{g}}=1$ throughout this
paper.

The constitutive pressure-density relation $P(\rho)$ depends on the types of
gaseous stars. For simplicity, the class of polytropic gases, i.e.,

$P(\rho)=\kappa\rho^{\gamma}\qquad\text{ for $\kappa>0$ and
$\gamma\in(1,3)$},$ (1.2)

has been widely investigated in mathematics. From the point view of astronomy,
the constitutive pressure $P(\rho)$ for certain gaseous stars is not of the
polytropic form. For example, the pressure law of a white dwarf star takes the
following form (cf. [6, 64]):

$P(\rho)=\mathcal{C}_{1}\int_{0}^{\mathcal{C}_{2}\rho^{\frac{1}{3}}}\frac{s^{4}}{\sqrt{\mathcal{C}_{3}+s^{2}}}\,\mathrm{d}s\qquad\,\text{
for $\rho>0$},$ (1.3)

where $\mathcal{C}_{1},\mathcal{C}_{2}$ and $\mathcal{C}_{3}$ are positive
constants. It can be checked that $P(\rho)\cong\kappa_{1}\rho^{\frac{5}{3}}$
as $\rho\to 0$ and $P(\rho)\cong\kappa_{2}\rho^{\frac{4}{3}}$ as
$\rho\to\infty$ for some positive constants $\kappa_{1}$ and $\kappa_{2}$.

In this paper, we consider a general pressure law in which any pressure
function $P(\rho)$ satisfies the following conditions:

  * (i)

The pressure function $P(\rho)\in C^{1}([0,\infty))\cap C^{4}(\mathbb{R}_{+})$
and satisfies the hyperbolic and genuinely nonlinear conditions:

$P^{\prime}(\rho)>0,\quad 2P^{\prime}(\rho)+\rho
P^{\prime\prime}(\rho)>0\qquad\,\,\text{for }\rho>0.$ (1.4)

  * (ii)

There exists a constant $\rho_{*}>0$ such that

$P(\rho)=\kappa_{1}\rho^{\gamma_{1}}\big{(}1+\mathcal{P}_{1}(\rho)\big{)}\qquad\text{
for }\rho\in[0,\rho_{*}),$ (1.5)

with some constants $\gamma_{1}\in(1,3)$ and $\kappa_{1}>0$, and a function
$\mathcal{P}_{1}(\rho)\in C^{4}(\mathbb{R}_{+})$ satisfying that
$|\mathcal{P}_{1}^{(j)}(\rho)|\leq C_{*}\rho^{\gamma_{1}-1-j}$ for
$\rho\in(0,\rho_{*})$ and $j=0,\cdots,4$, where $C_{*}>0$ is a constant
depending only on $\rho_{*}$.

  * (iii)

There exists a constant $\rho^{*}>\rho_{*}>0$ such that

$P(\rho)=\kappa_{2}\rho^{\gamma_{2}}\big{(}1+\mathcal{P}_{2}(\rho)\big{)}\qquad\text{
for }\rho\in[\rho^{*},\infty),$ (1.6)

with some constants $\gamma_{2}\in(\frac{6}{5},\gamma_{1}]$ and
$\kappa_{2}>0$, and a function $\mathcal{P}_{2}(\rho)\in
C^{4}(\mathbb{R}_{+})$ satisfying that $|\mathcal{P}_{2}^{(j)}(\rho)|\leq
C^{*}\rho^{-\epsilon-j}$ for $\rho\in[\rho^{*},\infty)$ and $j=0,\cdots,4$,
where $\epsilon>0$, and $C^{*}>0$ is a constant depending only on $\rho^{*}$.

It is direct to see that the polytropic gases in (1.2) satisfy assumptions
(1.4)–(1.6). Moreover, the white dwarf star (1.3) is also included with

$\gamma_{1}=\frac{5}{3},\quad\kappa_{1}=\frac{1}{5\sqrt{\mathcal{C}_{3}}}\mathcal{C}_{1}\mathcal{C}_{2}^{5},\quad\gamma_{2}=\frac{4}{3},\quad\kappa_{2}=\frac{1}{4}\mathcal{C}_{1}\mathcal{C}_{2}^{4},\quad\epsilon=\frac{2}{3}.$
(1.7)

The restriction: $\gamma_{2}>\frac{6}{5}$ is necessary to ensure the global
existence of finite-energy solutions with finite total mass. Such a condition
is also needed for the existence of the Lane-Emden solutions; see [6, 45].

We consider the Cauchy problem of (1.1) with the initial data:

$(\rho,\mathcal{M})(0,\mathbf{x})=(\rho_{0},\cal{M}_{0})(\mathbf{x})\,\rightarrow\,(0,\mathbf{0})\qquad\,\,\text{
as $|\mathbf{x}|\to\infty$}$ (1.8)

subject to the far field condition:

$\Phi(t,\mathbf{x})\,\rightarrow\,0\qquad\,\,\text{ as
$|\mathbf{x}|\to\infty$}.$ (1.9)

The global existence of solutions of the Cauchy problem (1.1) and (1.8)–(1.9)
is a longstanding open problem. Many efforts have been made for the polytropic
gas case (1.2). Considerable progress has been made on the smooth or special
solutions under some restrictions on the initial data. One of the most famous
solutions of CEPEs (1.1) are the Lane-Emden steady solutions (cf. [45]), which
describe spherically symmetric gaseous stars in equilibrium and minimize the
energy among all possible configurations (cf. [44]). There exist expanding
solutions for the non-steady CEPEs (1.1). Hadzić-Jang [32] proved the
nonlinear stability of the affine solution (which is linearly expanding) under
small spherically symmetric perturbations for $\gamma=\frac{4}{3}$, while the
stability problem for $\gamma\neq\frac{4}{3}$ is still widely open. A class of
linearly expanding solutions for $\gamma=1+\frac{1}{k}$,
$k\in\mathbb{N}\backslash\\{1\\}$ or $\gamma\in(1,\frac{14}{13})$ was further
constructed in [33]. For $1<\gamma<\frac{4}{3}$, concentration (collapse)
phenomena may happen. Indeed, as $\gamma=\frac{4}{3}$, there exists an
homologous concentration solution; see [26, 28, 53]. More recently, Guo-
Hadzić-Jang [29] first observed a continued concentration solution for
$1<\gamma<\frac{4}{3}$; see also [35]. A kind of smooth radially symmetric
self-similar solutions exhibiting gravitational collapse for
$1\leq\gamma<\frac{4}{3}$ can be found in [30, 31]. We refer to [49, 52] for
the local well-posedness of smooth solutions.

Owing to the strong nonlinearity and hyperbolicity, the smooth solutions of
(1.1) with (1.2) may break down in a finite time, especially when the initial
data are large (cf. [15, 53]). Therefore, weak solutions have to be considered
for large initial data. For gaseous stars surrounding a solid ball, Makino
[54] obtained the local existence of weak solutions for
$\gamma\in(1,\frac{5}{3}]$ with spherical symmetry; also see Xiao [67] for
global weak solutions with a class of initial data. For this case, the
possible singularity at the origin is prevented since the domain was
considered outside a ball. Luo-Smoller [48] proved the conditional stability
of rotating and non-rotating white dwarfs and rotating supermassive stars; see
also Rein [59] for the conditional nonlinear stability of the Lane-Emden
steady solutions.

Another fundamental question is whether global solutions can be constructed
via the vanishing viscosity limit of the solutions of the compressible Navier-
Stokes-Poisson equations (CNSPEs):

$\left\\{\begin{aligned} &\partial_{t}\rho+\nabla\cdot\cal{M}=0,\\\
&\partial_{t}\cal{M}+\nabla\cdot\Big{(}\frac{\cal{M}\otimes\cal{M}}{\rho}\Big{)}+\nabla
P+\rho\nabla\Phi=\varepsilon\nabla\cdot\Big{(}\mu(\rho)D\Big{(}\frac{\cal{M}}{\rho}\Big{)}\Big{)}+\varepsilon\nabla\Big{(}\lambda(\rho)\nabla\cdot\Big{(}\frac{\cal{M}}{\rho}\Big{)}\Big{)},\\\
&\Delta\Phi=\rho,\end{aligned}\right.$ (1.10)

where
$D(\frac{\cal{M}}{\rho})=\frac{1}{2}\big{(}\nabla(\frac{\cal{M}}{\rho})+(\nabla(\frac{\cal{M}}{\rho}))^{\bot}\big{)}$
is the stress tensor, the Lamé (shear and bulk) viscosity coefficients
$\mu(\rho)$ and $\lambda(\rho)$ depend on the density (that may vanish on the
vacuum) and satisfy

$\mu(\rho)\geq 0,\quad\mu(\rho)+3\lambda(\rho)\geq 0\qquad\,\,\,\text{for
$\rho\geq 0$},$

and parameter $\varepsilon>0$ is the inverse of the Reynolds number. Formally,
as $\varepsilon\to 0$, the sequence of the solutions of CNSPEs (1.10)
converges to a corresponding solution of CEPEs (1.1). However, the rigorous
proof has been one of the most challenging problems in the mathematical fluid
dynamics; see Chen-Feldman [8] and Dafermos [20].

The limit problem with vanishing physical viscosity dates back to the
pioneering paper by Stokes [63]. Most of the known results were around the
inviscid limit from the compressible Navier-Stokes to the Euler equations for
the polytropic gas case (1.2). The first rigorous proof of the vanishing
viscosity limit from the Navier-Stokes to the Euler equations was provided by
Gilbarg [27], in which he established the existence and inviscid limit of the
Navier-Stokes shock layers. For the case of large data, due to the lack of
$L^{\infty}$ uniform estimate, the $L^{\infty}$ compensated compactness
framework [21, 22, 23, 34, 46, 47] fails to work directly in the inviscid
limit of the compressible Navier-Stokes equations. An $L^{p}$ compensated
compactness framework was first studied by LeFloch-Westdickenberg [41] for the
isentropic Euler equations for the case $\gamma\in(1,\frac{5}{3})$ in (1.2),
and was further developed by Chen-Perepelitsa [12] to all $\gamma>1$ for (1.2)
with a simplified proof; see also [16] for spherically symmetric solutions of
the M-D isentropic Euler equations. We also refer to [61, 62] for the 1-D case
of asymptotically isothermal gas, i.e., $\gamma_{2}=1$ in (1.6). More
recently, Chen-He-Wang-Yuan [9] established both the strong inviscid limit of
CNSPEs (1.10) and the global existence of spherically symmetric solutions of
CEPEs (1.1) with large data for polytropic gases (1.2).

The main purpose of this paper is to establish the global existence of
spherically symmetric finite-energy solutions of (1.1) with general pressure
law (1.4)–(1.6):

$\rho(t,\mathbf{x})=\rho(t,r),\quad\cal{M}(t,x)=m(t,r)\frac{\mathbf{x}}{r},\quad\Phi(t,\mathbf{x})=\Phi(t,r)\qquad\,\,\text{for
}r=|\mathbf{x}|,$ (1.11)

subject to the initial condition:

$(\rho,\cal{M})(0,\mathbf{x})=(\rho_{0},\cal{M}_{0})(\mathbf{x})=(\rho_{0}(r),m_{0}(r)\frac{\mathbf{x}}{r})\,\to\,(0,\mathbf{0})\qquad\,\,\text{as
}r\to\infty,$ (1.12)

and the asymptotic boundary condition:

$\Phi(t,\mathbf{x})=\Phi(t,r)\,\rightarrow\,0\qquad\,\,\text{as }r\to\infty.$
(1.13)

Systems (1.1) and (1.10) for spherically symmetric solutions take the
following respective forms:

$\left\\{\begin{aligned} &\rho_{t}+m_{r}+\frac{2}{r}m=0,\\\
&m_{t}+\Big{(}\frac{m^{2}}{\rho}+P(\rho)\Big{)}_{r}+\frac{2}{r}\frac{m^{2}}{\rho}+\rho\Phi_{r}=0,\\\
&\Phi_{rr}+\frac{2}{r}\Phi_{r}=\rho,\end{aligned}\right.$ (1.14)

and

$\left\\{\begin{aligned} &\rho_{t}+m_{r}+\frac{2}{r}m=0,\\\
&m_{t}+\Big{(}\frac{m^{2}}{\rho}+P(\rho)\Big{)}_{r}+\frac{2}{r}\frac{m^{2}}{\rho}+\rho\Phi_{r}=\varepsilon\Big{(}(\mu(\rho)+\lambda(\rho))\Big{(}\big{(}\frac{m}{\rho}\big{)}_{r}+\frac{2}{r}\frac{m}{\rho}\Big{)}\Big{)}_{r}-\frac{2\varepsilon}{r}\mu(\rho)_{r}\frac{m}{\rho},\\\
&\Phi_{rr}+\frac{2}{r}\Phi_{r}=\rho.\end{aligned}\right.$ (1.15)

The study of spherically symmetric solutions is motivated by many important
physical problems such as stellar dynamics including gaseous stars and
supernovae formation [6, 58, 66]. An important question is how the waves
behave as they move radially inward near the origin, especially under the
self-gravitational force for gaseous stars. The spherically symmetric
solutions of the compressible Euler equations may blow up near the origin [19,
42, 55, 66] at certain time in some situations. Considering the effect of
gravitation, a fundamental problem for CEPEs (1.1) is whether a concentration
(delta-measure) is formed at the origin. This problem was answered in [9] for
polytropic gases in (1.2) when the initial total-energy is finite that no
delta measure is formed for the density at the origin for the two cases: (i)
$\gamma>\frac{6}{5}$; (ii) $\gamma\in(\frac{6}{5},\frac{4}{3}]$ and the
initial total-energy is finite and the total mass is less than a critical
mass.

In this paper, we establish the global existence of finite-energy solutions of
the Cauchy problem (1.1) and (1.12)–(1.13) with spherical symmetry as the
inviscid limits of global weak solutions of CNSPEs (1.10) with general
pressure law (1.4)–(1.6), especially including the white dwarf star (1.3). The
$L^{p}$ compensated compactness framework for general pressure is also
established. Moreover, it is proved that no delta measure is formed for the
density at the origin in the limit, and the critical mass for the white dwarf
star is the same as the Chandrasekhar limit for the polytropic gas (1.2) with
$\gamma=\frac{4}{3}$. The precise statements of the main results are given in
§2.

To achieve these, the main strategy is to develop entropy analysis, uniform
estimates in $L^{p}$, and a more general compensated compactness framework to
prove that there exists a strongly convergent subsequence of solutions of
CNSPEs (1.10) and show that the limit is the finite-energy weak solution of
CEPEs (1.1) with general pressure law. This consists of the following three
steps:

  * •

Establish the uniform $L^{p}$ estimates of the solutions of CNSPEs (1.10)
independent of $\varepsilon$ for some $p>1$;

  * •

Show the compactness for weak entropy dissipation measures;

  * •

Prove the associated Young measure $\nu_{(t,r)}$ is the delta measure almost
everywhere which leads to a subsequence of solutions of CNSPEs (1.10) strongly
converges to the global finite-energy solution of CEPEs (1.1).

The generality of pressure $P(\rho)$ causes essential difficulties in the
analysis for all of the above steps. We now describe these difficulties and
show how they can be overcome:

(i) The crucial step in the $L^{p}$ estimates is to show that $\rho|u|^{3}$
($u:=\frac{m}{\rho}$ is the velocity) is uniformly bounded in $L^{1}_{\rm
loc}$. This estimate might be obtained through constructing appropriate
entropy $\hat{\eta}$, which is a solution of $(\rho,u)$ to the entropy
equation:

$\eta_{\rho\rho}-\frac{P^{\prime}(\rho)}{\rho^{2}}\eta_{uu}=0,$ (1.16)

with corresponding entropy flux $\hat{q}$. If $(\rho,u)$ is the solution of
(1.15), any entropy-entropy flux pair (entropy pair, for short)
$(\hat{\eta},\hat{q})$ satisfies

$\displaystyle\quad(\hat{\eta}r^{2})_{t}+(\hat{q}r^{2})_{r}+2r\,(-\hat{q}+\rho
u\hat{\eta}_{\rho}+\rho u^{2}\hat{\eta}_{m})=\varepsilon\,r^{2}\big{(}(\rho
u_{r})_{r}+2\rho\big{(}\frac{u}{r}\big{)}_{r}\big{)}\hat{\eta}_{m}-\rho\int_{a}^{r}\rho\,z^{2}\mathrm{d}z\,\hat{\eta}_{m};$

see (5.68) below. For the polytropic gas case (1.2), there is an explicit
formula of the entropy kernel $\chi(\rho,u)$ so that $\chi*\psi$ is the
entropy, where $*$ denotes the convolution and $\psi(s)$ is any smooth
function. By choosing $\psi(s)=\frac{1}{2}s|s|$ as in [9], the corresponding
entropy flux $\hat{q}$ satisfies that $\hat{q}\geq c_{0}\rho|u|^{3}$ and
$-\hat{q}+\rho u\hat{\eta}_{\rho}+\rho u^{2}\hat{\eta}_{m}\leq 0$. Then the
uniform bound of $\rho|u|^{3}r^{2}$ in $L^{1}_{\rm loc}$ follows (cf. [9]).

However, there is no explicit formula of the entropy kernel $\chi$ for the
general pressure satisfying (1.4)–(1.6). Even for the special entropy pair
generated by $\psi(s)=\frac{1}{2}s|s|$, it is difficult to prove that
$\hat{q}\geq c_{0}\rho|u|^{3}$ and $-\hat{q}+\rho u\hat{\eta}_{\rho}+\rho
u^{2}\hat{\eta}_{m}\leq 0$, due to the lack of explicit formula of the entropy
kernel $\chi$. Hence, the above approach does not apply directly, so we have
to seek a new method to establish the uniform local integrability of
$\rho|u|^{3}$. One of the novelties of this paper is that a special entropy
$\hat{\eta}$ is constructed by solving a Goursat problem of the entropy
equation (1.16) in the domain: $|u|\leq
k(\rho):=\int_{0}^{\rho}\sqrt{P^{\prime}(y)}/{y}\,\mathrm{d}y$, so that
$\hat{\eta}$ is chosen as the mechanical energy $\eta^{*}$ (see (2.13)) when
$u\geq k(\rho)$, $-\eta^{*}$ when $u\leq-k(\rho)$, and The boundary condition
for the Goursat problem is given on the characteristics curves: $u\pm
k(\rho)=0$. One advantage of such a special entropy pair
$(\hat{\eta},\hat{q})$ is that $\hat{q}\geq c_{0}\rho|u|^{3}$ as $|u|\geq
k(\rho)$, and $|\hat{q}|\leq C\rho^{\gamma_{2}+1}$ for large $\rho$ as
$|u|\leq k(\rho)$ via careful analysis for the Goursat problem; see Lemma 5.8
for details. Moreover, $-\hat{q}+\rho u\hat{\eta}_{\rho}+\rho
u^{2}\hat{\eta}_{m}$ vanishes as $|u|\geq k(\rho)$. Similarly, $|-\hat{q}+\rho
u\hat{\eta}_{\rho}+\rho u^{2}\hat{\eta}_{m}|\leq C\rho^{\gamma_{2}+1}$ for
large $\rho$ as $|u|\leq k(\rho)$.

To show $\rho|u|^{3}$ is uniformly bounded in $L^{1}_{\rm loc}$, it remains to
prove that

$\int_{0}^{T}\int_{d}^{\infty}\rho^{\gamma_{2}+1}\,r\mathrm{d}r\mathrm{d}t$
(1.17)

is uniformly bounded for any $T>0$ and $d>0$. It should be noted that the
local integrability
$\int_{0}^{T}\int_{d}^{D}\rho^{\gamma_{2}+1}\,\mathrm{d}r\mathrm{d}t\leq C$
was obtained in [9], but it is not enough yet to obtain the uniform
$L^{1}_{\rm loc}$ estimate for $\rho|u|^{3}$. Fortunately, we can obtain even
stronger estimate than (1.17), i.e.,

$\int_{0}^{T}\int_{d}^{\infty}\rho^{\gamma_{2}+1}\,r^{2}\mathrm{d}r\mathrm{d}t\leq
C,$ (1.18)

by an elaborate analysis; see Lemma 5.6 and Corollary 5.7 for details.

(ii) For the polytropic gas case in (1.2), Chen-Perepelitsa [12, 13] and Chen-
He-Wang-Yuan [9] proved the $H_{\rm loc}^{-1}$–compactness for weak entropy
dissipation measures via the explicit formula of the weak entropy kernel
$\chi$ by convolution with any test function of compact support, which also
implies that the entropy pair $(\eta,q)$ is in $L^{r}_{\rm loc},r>2$. However,
it is not clear how the $H_{\rm loc}^{-1}$–compactness for the general
pressure satisfying (1.4)–(1.6) can be shown by using the expansions of the
weak entropy kernel established in [10, 11]. Motivated by [62], we instead
show the $W_{\mathrm{loc}}^{-1,p}$–compactness for $1\leq p<2$, so that an
improved div-curl lemma (cf. [18]) applies, which leads to the commutation
identity for the entropy pairs. In fact, we can show that the entropy flux
function $q$ is bounded by $\rho^{\frac{\gamma_{2}+1}{2}}$ (see (4.81)) as
$\rho$ is large by careful analysis on the expansion of the entropy pair so
that $q\in L^{2}_{\rm loc}$. Then the interpolation compactness yields the
$W^{-1,p}$ compactness for $1\leq p<2$; see Lemma 7.1 for details.

(iii) The argument for the reduction of the associated Young measure
$\nu_{(t,r)}(\rho,u)$ introduced in [9, 12, 13] for the polytropic gas case in
(1.2), can be roughly stated as follows: Show first every connected subset of
the support of the Young measure is a bounded interval; then use the
$L^{\infty}$ reduction technique introduced in [7, 21, 23, 46] for a bounded
supported Young measure to show the Young measure is either a delta measure or
supported on the vacuum line. This method essentially relies on the explicit
formula of the weak entropy kernel $\chi$. For the general pressure law
satisfying (1.4)–(1.6), the above method does not apply directly, since it is
difficult to show that every connected subset of the support of the Young
measure is a bounded interval. Motivated by [10, 11, 46, 61, 62], we carefully
analyze the singularities of $\partial^{\lambda_{1}+1}\chi$ with
$\lambda_{1}=\frac{3-\gamma_{1}}{2(\gamma_{1}-1)}$ for large $\rho$ and fully
exploit the property: $(\rho^{\gamma_{2}+1},\rho|u|^{3})\in L^{1}({\rm
d}\nu_{(t,r)})$ so that the $\partial^{\lambda_{1}+1}-$derivatives can be
operated in the commutation relation; see Lemmas 4.11–4.14 for details. Then
we prove that the Young measure is either a delta measure or supported on the
vacuum line by similar arguments as in [10, 11, 46, 62]. This new compensated
compactness framework and the techniques developed in this paper should be
useful for solving further nonlinear problems with similar features.

Finally, we remark that there are some related results on CNSPEs (1.10) and
the compressible Euler equations. For weak solutions of CNSPEs (1.10), we
refer to [25, 36, 38, 39] with constant viscosity, and [24, 68] with density-
dependent viscosity. Recently, Luo-Xin-Zeng [49, 50, 51] proved the large-time
stability of the Lane-Emden solution for $\gamma\in(\frac{4}{3},2)$. We also
refer to the BD entropy developed in [2, 3, 4, 5], which provides a new
estimate for the gradient of the density. For the compressible Euler
equations, we refer to [7, 14, 37, 42, 60] and the references cited therein.

The rest of this paper is organized as follows: In §2, the finite-energy
solutions of the Cauchy problem (1.1) and (1.8)–(1.9) for CEPEs are
introduced, and the main theorems of this paper are given. In §3, some
elementary quantities and basic properties about the pressure and related
internal energy are provided, and then some remarks on $M_{\rm c}$ are also
given. The entropy analysis for weak entropy pairs for the general pressure
satisfying (1.4)–(1.6) is presented in §4, especially a special entropy pair
is constructed by solving a Goursat problem for the entropy equation (2.14).
In §5, a free boundary problem (5.1)–(5.6) for (1.15) is analyzed, and some
uniform estimates of solutions are derived, including the basic energy
estimate, the BD-type entropy estimate, and the higher integrabilities of the
density and the velocity. In §6, the global existence of weak solutions of
CNSPEs (1.10) is established, and some uniform $L^{p}$ estimates in Theorem
2.1 are also obtained. In §7, we prove the
$W_{\mathrm{loc}}^{-1,p}$–compactness of the entropy dissipation measures for
the weak solutions of (1.15) and complete the proof of Theorem 2.1. In §8, the
$L^{p}$–compensated compactness framework for the general pressure law
(1.4)–(1.6) (Theorem 2.2) is established, which leads to the proof of Theorem
2.3 by taking the inviscid limit of weak solutions of CNSPEs (1.10) in §9.
Appendix A is devoted to the presentation of both the sharp Sobolev inequality
that is used in §5 and some variants of Grönwall’s inequality which are used
in the proof of some estimates in §4.

Notations: Throughout this paper, we denote
$C^{\alpha}(\Omega),L^{p}(\Omega),W^{k,p}(\Omega)$, and $H^{k}(\Omega)$ as the
standard Hölder space, and the corresponding Sobolev spaces, respectively, on
domain $\Omega$ for $\alpha\in(0,1)$ and $p\in[1,\infty]$. $C_{0}^{k}(\Omega)$
represents the space of continuously differentiable functions up to the $k$th
order with compact support over $\Omega$, and
$\mathcal{D}(\Omega):=C_{0}^{\infty}(\Omega)$. We also use
$L^{p}(I;r^{2}\mathrm{d}r)$ or $L^{p}([0,T)\times
I;r^{2}\mathrm{d}r\mathrm{d}t)$ for an open interval $I\subset\mathbb{R}_{+}$
with measure $r^{2}\mathrm{d}r$ or $r^{2}\mathrm{d}r\mathrm{d}t$
correspondingly, and $L_{\mathrm{loc}}^{p}([0,\infty);r^{2}\mathrm{d}r)$ to
represent $L^{p}([0,R];r^{2}\mathrm{d}r)$ for any fixed $R>0$.

##  2\. Mathematical Problem and Main Theorems

The spherically symmetric initial data function
$(\rho_{0},\cal{M}_{0})(\mathbf{x})$ given in (1.12) is assumed to be of both
finite initial total-energy:

$E_{0}:=\int_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}\Big{|}\frac{\mathcal{M}_{0}}{\sqrt{\rho_{0}}}\Big{|}^{2}+\rho_{0}e(\rho_{0})\Big{)}(\mathbf{x})\,\mathrm{d}\mathbf{x}=\omega_{3}\int_{0}^{\infty}\Big{(}\frac{1}{2}\frac{m^{2}_{0}}{\rho_{0}}+\rho_{0}e(\rho_{0})\Big{)}(r)\,r^{2}\mathrm{d}r<\infty,$
(2.1)

and initial total-mass:

$M:=\int_{\mathbb{R}^{3}}\rho_{0}(\mathbf{x})\,\mathrm{d}\mathbf{x}=\omega_{3}\int_{0}^{\infty}\rho_{0}(r)\,r^{2}\mathrm{d}r<\infty,$
(2.2)

where the internal energy $e(\rho)$ is related to the pressure by

$e^{\prime}(\rho)=\frac{P(\rho)}{\rho^{2}},\qquad e(0)=0,$ (2.3)

and $\omega_{n}:=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}$
denotes the surface area of the unit sphere in $\mathbb{R}^{n}$. The initial
potential $\Phi_{0}(\mathbf{x})$ is determined by

$\Delta\Phi_{0}(\mathbf{x})=\rho_{0}(\mathbf{x}),\qquad\lim_{|\mathbf{x}|\to\infty}\Phi_{0}(\mathbf{x})=0.$
(2.4)

For $\gamma_{2}\in(\frac{6}{5},\frac{4}{3}]$, we define the critical mass
$M_{\rm c}(\gamma_{2})$ by

(i) When $\gamma_{2}=\frac{4}{3}$,

$\displaystyle M_{\rm c}:=M_{\rm ch},$ (2.5)

where $M_{\rm ch}$ is the Chandrasekhar limit that is the total mass of the
Lane–Emden steady solution $(\rho_{s}(|{\bf}x|),0)$ for
$P(\rho)=\kappa_{2}\rho^{\frac{4}{3}}$: $\rho_{s}(|{\bf}x|)$ has compact
support and is determined by the equations:

$\displaystyle\nabla_{{\bf x}}P(\rho_{s}|{\bf x}|)+\rho_{s}(|{\bf
x}|)\nabla_{{\bf x}}\Phi({\bf x})=0,\qquad\Delta_{{\bf x}}\Phi({\bf
x})=\rho_{s}(|{\bf x}|),\qquad P(\rho_{s}|{\bf x}|)=\kappa_{2}(\rho_{s}(|{\bf
x}|))^{\frac{4}{3}},$

with the center density $\rho_{s}(0)=\varrho$. It is well-known that $M_{\rm
ch}$ is a uniform constant with respect to the center density $\varrho$ (cf.
[6]).

(ii) When $\gamma_{2}\in(\frac{6}{5},\frac{4}{3})$,

$M_{\rm c}:=\sup_{\beta>0}M_{\rm c}(\beta)$ (2.6)

with

$\displaystyle(4-3\gamma_{2})\Big{(}\frac{B_{\beta}}{3(\gamma_{2}-1)}\Big{)}^{-\frac{3(\gamma_{2}-1)}{4-3\gamma_{2}}}M_{\rm
c}(\beta)^{-\frac{5\gamma_{2}-6}{4-3\gamma_{2}}}-\omega_{3}^{-1}\beta M_{\rm
c}(\beta)=E_{0},$ (2.7) $\displaystyle
B_{\beta}:=\frac{2}{3}\omega_{4}^{-\frac{2}{3}}\omega_{3}^{\frac{4-3\gamma_{2}}{3(\gamma_{2}-1)}}(C_{\max}(\beta))^{\frac{5\gamma_{2}-6}{3(\gamma_{2}-1)}},\quad
C_{\max}(\beta):=\sup_{\rho\geq
0}\big{(}\rho^{\gamma_{2}-1}(\beta+e(\rho))^{-1}\big{)}^{\frac{1}{5\gamma_{2}-6}}>0.$
(2.8)

It is clear in (2.6)–(2.8) that $M_{\rm c}(\beta)$ is well determined for
$\beta>0$ and $\gamma_{2}\in(\frac{6}{5},\frac{4}{3})$. Some useful properties
of $M_{\rm c}:=\sup_{\beta>0}M_{\rm c}(\beta)$ will be presented in
Proposition 3.3 below. We also point out that $M_{\rm c}$ in (2.5) is strictly
larger than the one obtained in [9, (2.8)] for $\gamma_{2}=\frac{4}{3}$ (cf.
[17]).

For the spherically symmetric initial data $(\rho_{0},m_{0},\Phi_{0})(r)$
imposed in (1.11)–(1.13) satisfying (2.1)–(2.2), using similar arguments as in
[9, Appendix A], we can construct a sequence of approximate initial data
functions
$(\rho_{0}^{\varepsilon},m_{0}^{\varepsilon},\Phi_{0}^{\varepsilon})(r)$
satisfying

$\displaystyle\int_{0}^{\infty}\rho_{0}^{\varepsilon}(r)\,r^{2}\mathrm{d}r=\frac{M}{\omega_{3}},\qquad\Phi_{0r}^{\varepsilon}=\frac{1}{r^{2}}\int_{0}^{r}\rho_{0}^{\varepsilon}(z)\,z^{2}\mathrm{d}z,$
(2.9)
$\displaystyle\,E_{0}^{\varepsilon}:=\omega_{3}\int_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}\Big{|}\frac{m_{0}^{\varepsilon}}{\sqrt{\rho_{0}^{\varepsilon}}}\Big{|}^{2}+\rho_{0}^{\varepsilon}e(\rho_{0}^{\varepsilon})\Big{)}\,r^{2}\mathrm{d}r\leq
C(E_{0}+1)<\infty,$
$\displaystyle\,E_{1}^{\varepsilon}:=\varepsilon^{2}\int_{0}^{\infty}\big{|}\partial_{r}\sqrt{\rho_{0}^{\varepsilon}(r)}\big{|}^{2}\,r^{2}\mathrm{d}r\leq
C\varepsilon(M+1)<\infty.$

Moreover, as $\varepsilon\to 0$,
$(E_{0}^{\varepsilon},E_{1}^{\varepsilon})\rightarrow(E_{0},0)$ and

$\displaystyle(\rho_{0}^{\varepsilon},\rho_{0}^{\varepsilon}u_{0}^{\varepsilon})(r)\rightarrow(\rho_{0},\rho_{0}u_{0})(r)\qquad\text{in
}L^{\tilde{q}}([0,\infty);r^{2}\mathrm{d}r)\times
L^{1}([0,\infty);r^{2}\mathrm{d}r),$
$\displaystyle\Phi_{0r}^{\varepsilon}\to\Phi_{0r}\qquad\text{in
}L^{2}([0,\infty);r^{2}\mathrm{d}r),$

where $\tilde{q}\in\\{1,\gamma_{2}\\}$. Furthermore, there exists
$\varepsilon_{0}\in(0,1]$ such that, for any
$\varepsilon\in(0,\varepsilon_{0}]$,

$M<M_{\rm c}^{\varepsilon}\qquad\text{for
}\gamma_{2}\in(\frac{6}{5},\frac{4}{3}],$ (2.10)

where $M_{\rm c}^{\varepsilon}$ is defined in (2.5)–(2.8) by replacing $E_{0}$
by $E_{0}^{\varepsilon}$.

Now we introduce the weak entropy pairs of the 1-D isentropic Euler system
(cf. [10, 40]):

$\left\\{\begin{aligned} &\rho_{t}+m_{r}=0,\\\
&m_{t}+\big{(}\frac{m^{2}}{\rho}+P(\rho)\big{)}_{r}=0.\end{aligned}\right.$
(2.11)

A pair of functions $(\eta(\rho,m),q(\rho,m))$ is called an entropy pair of
the 1-D Euler system (2.11) if

$\nabla q(\rho,m)=\nabla\eta(\rho,m)\nabla\Big{(}\begin{matrix}m\\\
\frac{m^{2}}{\rho}+P(\rho)\end{matrix}\Big{)}.$ (2.12)

Moreover, $\eta(\rho,m)$ is called a weak entropy if
$\eta(\rho,m)|_{\rho=0}=0$, and a convex entropy if
$\nabla^{2}\eta(\rho,m)\geq 0$. The mechanical energy and energy flux pair is
defined as

$\eta^{*}(\rho,m)=\frac{1}{2}\frac{m^{2}}{\rho}+\rho e(\rho),\qquad
q^{*}(\rho,m)=\frac{1}{2}\frac{m^{3}}{\rho}+m(\rho e(\rho))^{\prime},$ (2.13)

which is a convex weak entropy pair. From (2.12), any entropy satisfies

$\eta_{\rho\rho}-\frac{P^{\prime}(\rho)}{\rho^{2}}\eta_{uu}=0$ (2.14)

with $u=\frac{m}{\rho}$. It is known in [10, 11, 46, 47] that any regular weak
entropy can be generated by the convolution of a smooth function $\psi(x)$
with a fundamental solution $\chi(\rho,u,s)$ of the entropy equation (2.14),
i.e.,

$\eta^{\psi}(\rho,u)=\int_{\mathbb{R}}\chi(\rho,u,s)\psi(s)\,\mathrm{d}s.$
(2.15)

The corresponding entropy flux is generated from the flux kernel
$\sigma(\rho,u,s)$ (see (4.56)), i.e.,

$\displaystyle
q^{\psi}(\rho,u)=\int_{\mathbb{R}}\sigma(\rho,u,s)\psi(s)\,\mathrm{d}s.$
(2.16)

We first consider the Cauchy problem of CNSPEs (1.10) with approximate initial
data:

$(\rho,\cal{M},\Phi)|_{t=0}=(\rho_{0}^{\varepsilon},\cal{M}_{0}^{\varepsilon},\Phi_{0}^{\varepsilon})(\mathbf{x}):=(\rho_{0}^{\varepsilon}(r),m_{0}^{\varepsilon}(r)\frac{\mathbf{x}}{r},\Phi_{0}^{\varepsilon}(r)),$
(2.17)

subject to the far field condition:

$\Phi^{\varepsilon}(t,\mathbf{x})\longrightarrow 0\qquad\text{as
}|\mathbf{x}|\to\infty.$ (2.18)

For concreteness, we take $\varepsilon\in(0,1]$ and the viscosity coefficients
$(\mu,\lambda)=(\rho,0)$ in (1.10).

######  Definition 2.1.

A triple
$(\rho^{\varepsilon},\mathcal{M}^{\varepsilon},\Phi^{\varepsilon})(t,\mathbf{x})$
is said to be a weak solution of the Cauchy problem (1.10) and (2.17) if

  * (i)

$\rho^{\varepsilon}(t,\mathbf{x})\geq 0$ and
$(\mathcal{M}^{\varepsilon},\frac{\mathcal{M}^{\varepsilon}}{\sqrt{\rho^{\varepsilon}}})(t,\mathbf{x})=\mathbf{0}\,$
a.e. on the vacuum states
$\left\\{(t,\mathbf{x})\,:\,\rho^{\varepsilon}(t,\mathbf{x})=0\right\\}$,

$\displaystyle\rho^{\varepsilon}\in
L^{\infty}(0,T;L^{\gamma_{2}}(\mathbb{R}^{3})),\quad\nabla\sqrt{\rho^{\varepsilon}}\in
L^{\infty}(0,T;L^{2}(\mathbb{R}^{3})),$
$\displaystyle\frac{\mathcal{M}^{\varepsilon}}{\sqrt{\rho^{\varepsilon}}}\in
L^{\infty}(0,T;L^{2}(\mathbb{R}^{3})),\quad\Phi^{\varepsilon}\in
L^{\infty}(0,T;L^{6}(\mathbb{R}^{3})),\quad\nabla\Phi^{\varepsilon}\in
L^{\infty}(0,T;L^{2}(\mathbb{R}^{3})).$

  * (ii)

For any $t_{2}\geq t_{1}\geq 0$ and any $\zeta(t,\mathbf{x})\in
C_{0}^{1}([0,\infty)\times\mathbb{R}^{3})$, the mass equation
$\eqref{1.8}_{1}$ holds in the sense:

$\int_{\mathbb{R}^{3}}(\rho^{\varepsilon}\zeta)(t_{2},\mathbf{x})\,\mathrm{d}\mathbf{x}-\int_{\mathbb{R}^{3}}(\rho^{\varepsilon}\zeta)(t_{1},\mathbf{x})\,\mathrm{d}\mathbf{x}=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}(\rho^{\varepsilon}\zeta_{t}+\mathcal{M}^{\varepsilon}\cdot\nabla\zeta)(t,\mathbf{x})\,\mathrm{d}\mathbf{x}\mathrm{d}t.$

  * (iii)

For any
$\Psi=(\Psi_{1},\Psi_{2},\Psi_{3})(t,\mathbf{x})\in(C_{0}^{2}([0,\infty)\times\mathbb{R}^{3}))^{3}$,
the momentum equations $\eqref{1.8}_{2}$ hold in the sense:

$\displaystyle\int_{\mathbb{R}_{+}^{4}}\Big{(}\mathcal{M}^{\varepsilon}\cdot\Psi_{t}+\frac{\mathcal{M}^{\varepsilon}}{\sqrt{\rho^{\varepsilon}}}\cdot\big{(}\frac{\mathcal{M}^{\varepsilon}}{\sqrt{\rho^{\varepsilon}}}\cdot\nabla\big{)}\Psi+P(\rho^{\varepsilon})\nabla\cdot\Psi\Big{)}\,\mathrm{d}\mathbf{x}\mathrm{d}t+\int_{\mathbb{R}^{3}}\mathcal{M}_{0}^{\varepsilon}(\mathbf{x})\cdot\Psi(0,\mathbf{x})\,\mathrm{d}\mathbf{x}$
$\displaystyle=-\varepsilon\int_{\mathbb{R}_{+}^{4}}\Big{(}\frac{1}{2}\mathcal{M}^{\varepsilon}\cdot\big{(}\Delta\Psi+\nabla(\nabla\cdot\Psi)\big{)}+\frac{\mathcal{M}^{\varepsilon}}{\sqrt{\rho^{\varepsilon}}}\cdot\big{(}\nabla\sqrt{\rho^{\varepsilon}}\cdot\nabla\big{)}\Psi\Big{)}\,\mathrm{d}\mathbf{x}\mathrm{d}t$
$\displaystyle\quad-\varepsilon\int_{\mathbb{R}_{+}^{4}}\nabla\sqrt{\rho^{\varepsilon}}\cdot\big{(}\frac{\mathcal{M}^{\varepsilon}}{\sqrt{\rho^{\varepsilon}}}\cdot\nabla\big{)}\Psi\,\mathrm{d}\mathbf{x}\mathrm{d}t+\int_{\mathbb{R}_{+}^{4}}\big{(}\rho\nabla\Phi\cdot\Psi\big{)}(t,\mathbf{x})\,\mathrm{d}\mathbf{x}.$

  * (iv)

For any $t\geq 0$ and $\xi(\mathbf{x})\in C_{0}^{1}(\mathbb{R}^{3})$,

$\int_{\mathbb{R}^{3}}\nabla\Phi^{\varepsilon}(t,\mathbf{x})\cdot\nabla\xi(\mathbf{x})\,\mathrm{d}\mathbf{x}=-\int_{\mathbb{R}^{3}}\rho^{\varepsilon}(t,\mathbf{x})\xi(\mathbf{x})\,\mathrm{d}\mathbf{x}.$

Then we have

######  Theorem 2.1 (Global existence of spherically symmetric solutions for
CNSPEs).

Assume that the initial data function
$(\rho_{0}^{\varepsilon},\mathcal{M}_{0}^{\varepsilon},\Phi_{0}^{\varepsilon})(\mathbf{x})$
is given in (2.17)–(2.18) with
$(\rho_{0}^{\varepsilon},m_{0}^{\varepsilon},\Phi_{0}^{\varepsilon})(r)$
satisfying (2.9)–(2.10). Then, for each fixed $\varepsilon\in(0,1]$, there
exists a global weak solution
$(\rho^{\varepsilon},\mathcal{M}^{\varepsilon},\Phi^{\varepsilon})(t,\mathbf{x})$
of the Cauchy problem (1.10) and (2.17)–(2.18) in the sense of Definition 2.1
with following spherical symmetry form:

$(\rho^{\varepsilon},\mathcal{M}^{\varepsilon},\Phi^{\varepsilon})(t,\mathbf{x})=(\rho^{\varepsilon}(t,r),m^{\varepsilon}(t,r)\frac{\mathbf{x}}{r},\Phi^{\varepsilon}(t,r))\qquad\text{for
}r=|\mathbf{x}|,$ (2.19)

such that

$\int_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}\Big{|}\frac{\mathcal{M}^{\varepsilon}}{\sqrt{\rho^{\varepsilon}}}\Big{|}^{2}+\rho^{\varepsilon}e(\rho^{\varepsilon})-\frac{1}{2}|\nabla\Phi^{\varepsilon}|^{2}\Big{)}\,\mathrm{d}\mathbf{x}\leq\int_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}\Big{|}\frac{\mathcal{M}_{0}^{\varepsilon}}{\sqrt{\rho_{0}^{\varepsilon}}}\Big{|}^{2}+\rho_{0}^{\varepsilon}e(\rho_{0}^{\varepsilon})-\frac{1}{2}|\nabla\Phi_{0}^{\varepsilon}|^{2}\Big{)}\,\mathrm{d}\mathbf{x}\quad\text{for
$t\geq 0$}.$ (2.20)

Furthermore, for
$(\rho^{\varepsilon},m^{\varepsilon},\Phi^{\varepsilon})(t,r)$, there exists a
measurable function $u^{\varepsilon}(t,r)$ with

$u^{\varepsilon}(t,r):=\frac{m^{\varepsilon}(t,r)}{\rho^{\varepsilon}(t,r)}\qquad{\it
a.e.}\text{ on }\big{\\{}(t,r)\,:\,\rho^{\varepsilon}(t,r)\neq 0\big{\\}},$

and $u^{\varepsilon}(t,r):=0~{}{\it a.e.}\text{ on
}\big{\\{}(t,r)\,:\,\rho^{\varepsilon}(t,r)=0\text{ or }r=0\big{\\}}$ such
that $m^{\varepsilon}(t,r)=(\rho^{\varepsilon}u^{\varepsilon})(t,r)$ a.e. on
$\mathbb{R}_{+}^{2}$. In addition, the following properties hold:

$\displaystyle({\rm
i})~{}~{}\int_{0}^{\infty}\rho^{\varepsilon}(t,r)\,r^{2}\mathrm{d}r=\int_{0}^{\infty}\rho_{0}^{\varepsilon}(r)\,r^{2}\mathrm{d}r=M\qquad\text{for
}t\geq 0,$ (2.21) $\displaystyle({\rm
ii})~{}~{}\int_{0}^{\infty}\eta^{*}(\rho^{\varepsilon},m^{\varepsilon})(t,r)\,r^{2}\mathrm{d}r+\varepsilon\int_{\mathbb{R}_{+}^{2}}(\rho^{\varepsilon}|u^{\varepsilon}|^{2})(t,r)\,r^{2}\mathrm{d}r\mathrm{d}t+\|\Phi^{\varepsilon}\|_{L^{6}(\mathbb{R}^{3})}+\|\nabla\Phi^{\varepsilon}\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\qquad+\int_{0}^{\infty}\Big{(}\int_{0}^{r}\rho^{\varepsilon}(t,z)\,z^{2}\mathrm{d}z\Big{)}\rho^{\varepsilon}(t,r)\,r\mathrm{d}r\leq
C\left(M,E_{0}\right)\qquad\text{for }t\geq 0,$ (2.22) $\displaystyle({\rm
iii})~{}~{}\sup_{t\in[0,T]}\varepsilon^{2}\int_{0}^{\infty}\left|\left(\sqrt{\rho^{\varepsilon}}\right)_{r}\right|^{2}\,r^{2}\mathrm{d}r+\varepsilon\int_{0}^{T}\int_{0}^{\infty}\frac{P^{\prime}(\rho^{\varepsilon})}{\rho^{\varepsilon}}\left|\rho_{r}^{\varepsilon}\right|^{2}\,r^{2}\mathrm{d}r\mathrm{d}t\leq
C(M,E_{0},T),$ (2.23) $\displaystyle({\rm
iv})~{}~{}\int_{0}^{T}\int_{d}^{D}\rho^{\varepsilon}\left|u^{\varepsilon}\right|^{3}\,r^{2}\mathrm{d}r\mathrm{d}t\leq
C(d,D,M,E_{0},T),$ (2.24) $\displaystyle({\rm
v})~{}~{}\int_{0}^{T}\int_{d}^{\infty}(\rho^{\varepsilon})^{\gamma_{2}+1}\,r^{2}\mathrm{d}r\mathrm{d}t\leq
C(d,M,E_{0},T),$ (2.25)

for any $T\in\mathbb{R}_{+}$ and interval $[d,D]\Subset(0,\infty)$, where
$C(M,E_{0}),C(M,E_{0},T)$, and $C(d,D,M,E_{0},T)$ are three universal positive
constants independent of $\varepsilon$. In addition, for
$\varepsilon\in(0,1]$,

$\partial_{t}\eta^{\psi}\left(\rho^{\varepsilon},m^{\varepsilon}\right)+\partial_{r}q^{\psi}\left(\rho^{\varepsilon},m^{\varepsilon}\right)\quad\text{
is compact in }W_{\operatorname{loc}}^{-1,p}\left(\mathbb{R}_{+}^{2}\right)$
(2.26)

for any $p\in[1,2)$, where $\psi(s)$ is any smooth function with compact
support on $\mathbb{R}$.

Now we introduce the notion of finite-energy weak solutions of CEPEs (1.1).

######  Definition 2.2.

A measurable vector function $(\rho,\cal{M},\Phi)$ is said to be a finite-
energy solution of the Cauchy problem (1.1) and (1.8)–(1.9) provided that

  * (i)

$\rho(t,\mathbf{x})\geq 0$ a.e., and
$(\cal{M},\frac{\cal{M}}{\sqrt{\rho}})(t,\mathbf{x})=\mathbf{0}$ a.e. on the
vacuum states $\\{(t,\mathbf{x})\,:\,\rho(t,\mathbf{x})=0\\}$.

  * (ii)

For a.e. $t>0$, the total energy is finite:

$\left\\{\begin{aligned}
&\int_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}\Big{|}\frac{\mathcal{M}}{\sqrt{\rho}}\Big{|}^{2}+\rho
e(\rho)+\frac{1}{2}|\nabla\Phi|^{2}\Big{)}(t,\mathbf{x})\,\mathrm{d}\mathbf{x}\leq
C(E_{0},M),\\\
&\int_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}\Big{|}\frac{\mathcal{M}}{\sqrt{\rho}}\Big{|}^{2}+\rho
e(\rho)-\frac{1}{2}|\nabla\Phi|^{2}\Big{)}(t,\mathbf{x})\,\mathrm{d}\mathbf{x}\leq\int_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}\Big{|}\frac{\mathcal{M}_{0}}{\sqrt{\rho_{0}}}\Big{|}^{2}+\rho_{0}e(\rho_{0})-\frac{1}{2}|\nabla\Phi_{0}|^{2}\Big{)}\,\mathrm{d}\mathbf{x}.\end{aligned}\right.$
(2.27)

  * (iii)

For any $\zeta(t,\mathbf{x})\in C_{0}^{1}([0,\infty)\times\mathbb{R}^{3})$,

$\int_{\mathbb{R}_{+}^{4}}(\rho\zeta_{t}+\cal{M}\cdot\nabla\zeta)\,\mathrm{d}\mathbf{x}\mathrm{d}t+\int_{\mathbb{R}^{3}}(\rho_{0}\zeta)(0,\mathbf{x})\,\mathrm{d}\mathbf{x}=0.$
(2.28)

  * (iv)

For all
$\Psi(t,\mathbf{x})=(\Psi_{1},\Psi_{2},\Psi_{3})(t,\mathbf{x})\in(C_{0}^{1}([0,\infty)\times\mathbb{R}^{3}))^{3}$,

$\displaystyle\int_{\mathbb{R}_{+}^{4}}\Big{(}\cal{M}\cdot\partial_{t}\Psi+\frac{\cal{M}}{\sqrt{\rho}}\cdot(\frac{\cal{M}}{\sqrt{\rho}}\cdot\nabla)\Psi+P(\rho)\,\nabla\cdot\Psi\Big{)}\,\mathrm{d}\mathbf{x}\mathrm{d}t+\int_{\mathbb{R}^{3}}\cal{M}_{0}(\mathbf{x})\cdot\Psi(0,\mathbf{x})\,\mathrm{d}\mathbf{x}$
(2.29)
$\displaystyle=\int_{\mathbb{R}_{+}^{4}}\left(\rho\nabla\Phi\cdot\Psi\right)(t,\mathbf{x})\,\mathrm{d}\mathbf{x}.$

  * (v)

For all $\xi(\mathbf{x})\in C_{0}^{1}(\mathbb{R}^{3})$,

$\int_{\mathbb{R}^{3}}\nabla\Phi(t,\mathbf{x})\cdot\nabla\xi(\mathbf{x})\,\mathrm{d}\mathbf{x}=-\int_{\mathbb{R}^{3}}\rho(t,\mathbf{x})\xi(\mathbf{x})\,\mathrm{d}\mathbf{x}\qquad\text{for
{\it a.e.} }t\geq 0.$ (2.30)

To establish the strong convergence of the inviscid limit $\varepsilon\to 0$
of solutions
$(\rho^{\varepsilon},\mathcal{M}^{\varepsilon},\Phi^{\varepsilon})(t,\mathbf{x})$
of CNSPEs (1.10) obtained in Theorem 2.1, we establish the following $L^{p}$
compensated compactness framework for the 1-D Euler equations (2.11) with
general pressure law (1.4)–(1.6), in which restriction
$\gamma_{2}\in(\frac{6}{5},\gamma_{1}]$ in (1.6) can be relaxed to
$\gamma_{2}\in(1,\gamma_{1}]$.

######  Theorem 2.2 ($L^{p}$ compensated compactness framework).

Let
$(\rho^{\varepsilon},m^{\varepsilon})(t,r)=(\rho^{\varepsilon},\rho^{\varepsilon}u^{\varepsilon})(t,r)$
be a sequence of measurable functions with $\rho^{\varepsilon}\geq 0$ a.e. on
$\mathbb{R}_{+}^{2}$ satisfying the following two conditions:

  * •

For any $T>0$ and $K\Subset\mathbb{R}_{+}$, there exists $C(K,T)>0$
independent of $\varepsilon$ such that

$\int_{0}^{T}\int_{K}\big{(}(\rho^{\varepsilon})^{\gamma_{2}+1}+\rho^{\varepsilon}|u^{\varepsilon}|^{3}\big{)}\,\mathrm{d}r\mathrm{d}t\leq
C(K,T).$

  * •

For any entropy pair $(\eta^{\psi},q^{\psi})$ defined in (2.15)–(2.16) with
any smooth function $\psi(s)$ of compact support on $\mathbb{R}$,

$\partial_{t}\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})+\partial_{r}q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})\qquad\mbox{is
compact in $W_{\mathrm{loc}}^{-1,1}(\mathbb{R}_{+}^{2})$}.$

Then there exists a subsequence $($still denoted$)$
$(\rho^{\varepsilon},m^{\varepsilon})(t,r)$ and a vector function
$(\rho,m)(t,r)$ such that, as $\varepsilon\to 0$,

$\displaystyle\rho^{\varepsilon}(t,r)\to\rho(t,r)~{}~{}\mbox{in}~{}L^{q_{1}}_{\rm
loc}(\mathbb{R}_{+}^{2})\qquad\,\,\,\,\text{for }q_{1}\in[1,\gamma_{2}+1),$
(2.31) $\displaystyle m^{\varepsilon}(t,r)\to
m(t,r)~{}~{}\mbox{in}~{}L^{q_{2}}_{\rm loc}(\mathbb{R}_{+}^{2})\qquad\text{for
}q_{2}\in[1,\frac{3(\gamma_{2}+1)}{\gamma_{2}+3}),$

where $L_{\mathrm{loc}}^{p}(\mathbb{R}_{+}^{2})$ represents $L^{p}([0,T]\times
K)$ for any $T>0$ and compact set $K\Subset\mathbb{R}_{+}$.

Now, we are ready to state our main theorem.

######  Theorem 2.3 (Global existence of finite-energy solutions).

Let the pressure function $P(\rho)$ satisfy (1.4)–(1.6), and let the
spherically symmetric initial data
$(\rho_{0},\mathcal{M}_{0},\Phi_{0})(\mathbf{x})$ be given in (1.12)–(1.13)
with $(\rho_{0},m_{0},\Phi_{0})(r)$ satisfying (2.1)–(2.2) and (2.4). Assume
that $\gamma_{2}>\frac{4}{3}$ or $M<M_{\rm c}$ as
$\gamma_{2}\in(\frac{6}{5},\frac{4}{3}]$. Then there exists a global finite-
energy solution $(\rho,\mathcal{M},\Phi)(t,\mathbf{x})$ of (1.1) and
(1.12)–(1.13) with spherical symmetry form (1.11) in the sense of Definition
2.2.

######  Remark 2.1.

For the steady gaseous star problem, there is no white dwarf star if the total
mass is larger than the so-called Chandrasekhar limit when
$\gamma\in(\frac{6}{5},\frac{4}{3}]$; see [6]. Theorem 2.3 requires similar
restriction on the total mass when $\gamma_{2}\in(\frac{6}{5},\frac{4}{3}]$
for non-steady gaseous stars. Moreover, in view of (2.5), for the non-steady
white dwarf star, the critical mass is exactly the Chandrasekhar limit in the
case that $P(\rho)=\kappa_{2}\rho^{\frac{4}{3}}$. It would be interesting to
analyze whether the critical mass defined in (2.6)–(2.8) for
$\gamma_{2}\in(\frac{6}{5},\frac{4}{3})$ is optimal.

######  Remark 2.2.

Theorem 2.3 can be extended to the $3$-D compressible Euler equations, i.e.,
(1.1) with $\Phi=0$. Moreover, the inviscid limit from the compressible
Navier-Stokes equations to the compressible Euler equations with far-field
vacuum can also be justified.

######  Remark 2.3.

Theorem 2.3 also holds for the plasmas case, i.e., $k_{g}=-1$ in (1.1), by a
similar proof. In this case, the restriction: $M<M_{\rm c}$ can be removed,
and the condition: $\gamma_{2}>\frac{6}{5}$ can be relaxed to $\gamma_{2}>1$
if the additional assumption: $\rho_{0}\in L^{\frac{6}{5}}(\mathbb{R}^{3})$ is
imposed. We omit the proof in this paper for brevity and, instead, refer the
reader to [9] for the details.

##  3\. Properties of the General Pressure Law and Related Internal Energy

In this section, we present some useful estimates involving the general
pressure $P(\rho)$ with (1.4)–(1.6) and the corresponding internal energy
$e(\rho)$, which are used in the subsequent development.

Denote $c(\rho):=\sqrt{P^{\prime}(\rho)}$ as the speed of sound, and

$k(\rho):=\int_{0}^{\rho}\frac{\sqrt{P^{\prime}(y)}}{y}\,\mathrm{d}y.$ (3.1)

By direct calculation, we can obtain the following asymptotic behaviors of
$P(\rho)$, $e(\rho)$, and $k(\rho)$.

######  Lemma 3.1.

Assume that $\rho_{*}$ given in (1.5) is small enough and $\rho^{*}$ given in
(1.6) is large enough such that the following estimates hold:

  1. (i)

When $\rho\in(0,\rho_{*}]$,

$\left\\{\begin{aligned} &\underline{\kappa}_{1}\rho^{\gamma_{1}}\leq
P(\rho)\leq\bar{\kappa}_{1}\rho^{\gamma_{1}},\\\
&\underline{\kappa}_{1}\gamma_{1}\rho^{\gamma_{1}-1}\leq
P^{\prime}(\rho)\leq\bar{\kappa}_{1}\gamma_{1}\rho^{\gamma_{1}-1},\\\
&\underline{\kappa}_{1}\gamma_{1}(\gamma_{1}-1)\rho^{\gamma_{1}-2}\leq
P^{\prime\prime}(\rho)\leq\bar{\kappa}_{1}\gamma_{1}(\gamma_{1}-1)\rho^{\gamma_{1}-2},\end{aligned}\right.$
(3.2)

and when $\rho\in[\rho^{*},\infty)$,

$\left\\{\begin{aligned} &\underline{\kappa}_{2}\rho^{\gamma_{2}}\leq
P(\rho)\leq\bar{\kappa}_{2}\rho^{\gamma_{2}},\\\
&\underline{\kappa}_{2}\gamma_{2}\rho^{\gamma_{2}-1}\leq
P^{\prime}(\rho)\leq\bar{\kappa}_{2}\gamma_{2}\rho^{\gamma_{2}-1},\\\
&\underline{\kappa}_{2}\gamma_{2}(\gamma_{2}-1)\rho^{\gamma_{2}-2}\leq
P^{\prime\prime}(\rho)\leq\bar{\kappa}_{2}\gamma_{2}(\gamma_{2}-1)\rho^{\gamma_{2}-2},\end{aligned}\right.$
(3.3)

where we have denoted $\underline{\kappa}_{i}:=(1-\mathfrak{a}_{0})\kappa_{i}$
and $\bar{\kappa}_{i}:=(1+\mathfrak{a}_{0})\kappa_{i}$ with
$\mathfrak{a}_{0}=\frac{3-\gamma_{1}}{2(\gamma_{1}+1)}$ and $i=1,2$.

  2. (ii)

For $e(\rho)$ and $k(\rho)$, there exists $C>0$ depending on
$(\gamma_{1},\gamma_{2},\kappa_{1},\kappa_{2},\rho_{*},\rho^{*})$ such that

$\displaystyle C^{-1}\rho^{\gamma_{1}-1}\leq e(\rho)\leq
C\rho^{\gamma_{1}-1},\quad C^{-1}\rho^{\gamma_{1}-2}\leq e^{\prime}(\rho)\leq
C\rho^{\gamma_{1}-2}\quad\,\,\text{ for }\rho\in(0,\rho_{*}],$ (3.4)
$\displaystyle C^{-1}\rho^{\gamma_{2}-1}\leq e(\rho)\leq
C\rho^{\gamma_{2}-1},\quad C^{-1}\rho^{\gamma_{2}-2}\leq e^{\prime}(\rho)\leq
C\rho^{\gamma_{2}-2}\quad\,\,\text{ for }\rho\in[\rho^{*},\infty),$ (3.5)

and, for $i=0,1$,

$\displaystyle\qquad C^{-1}\rho^{\theta_{1}-i}\leq k^{(i)}(\rho)\leq
C\rho^{\theta_{1}-i},\,\,C^{-1}\rho^{\theta_{1}-2}\leq|k^{\prime\prime}(\rho)|\leq
C\rho^{\theta_{1}-2}\,\,\,\,\text{for }\rho\in(0,\rho_{*}],$ (3.6)
$\displaystyle\qquad C^{-1}\rho^{\theta_{2}-i}\leq k^{(i)}(\rho)\leq
C\rho^{\theta_{2}-i},\,\,C^{-1}\rho^{\theta_{2}-2}\leq|k^{\prime\prime}(\rho)|\leq
C\rho^{\theta_{2}-2}\,\,\,\,\text{for }\rho\in[\rho^{*},\infty),$ (3.7)

where $\theta_{1}=\frac{\gamma_{1}-1}{2}$ and
$\theta_{2}=\frac{\gamma_{2}-1}{2}$.

It follows from (3.2)–(3.3) that

$\Big{(}\frac{(3\gamma_{1}-1)(\gamma_{1}-1)}{2(5+\gamma_{1})}\Big{)}2P^{\prime}(\rho)\leq\rho
P^{\prime\prime}(\rho)\leq\Big{(}\frac{(5+\gamma_{1})(\gamma_{1}-1)}{2(3\gamma_{1}-1)}\Big{)}2P^{\prime}(\rho)<2P^{\prime}(\rho),$
(3.8)

when $\rho\in[0,\rho_{*}]\cup[\rho^{*},\infty)$. For later use, we denote

$\displaystyle\nu=1-\frac{(3\gamma_{1}-1)(\gamma_{1}-1)}{2(5+\gamma_{1})}=\frac{3(3-\gamma_{1})(\gamma_{1}-1)}{2(5+\gamma_{1})}<1,$
(3.9) $\displaystyle d(\rho):=2+\frac{\rho
k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}.$ (3.10)

Then it follows from (3.8) that

$0<\Big{|}\frac{\rho
k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}\Big{|}=1-\frac{\rho
P^{\prime\prime}(\rho)}{2P^{\prime}(\rho)}\leq\nu<1\qquad\text{for
}\rho\in(0,\rho_{*}]\cup[\rho^{*},\infty).$ (3.11)

Motivated by [62], we have

######  Lemma 3.2.

$0<d(\rho)\leq C$ for all $\rho>0$, and

$\big{|}d(\rho)-(1+\theta_{2})\big{|}\leq
C\rho^{-\epsilon}\qquad\mbox{for}\,\,\rho\gg 1.$ (3.12)

Proof. It follows from (1.4) that $d(\rho)=1+\frac{\rho
P^{\prime\prime}(\rho)}{2P^{\prime}(\rho)}>0.$ Moreover, by (3.11), it is
direct to see that $d(\rho)$ is bounded. Using (1.6), we see that, for
$\rho\geq\rho^{*}$,

$\left\\{\begin{aligned}
&P^{\prime}(\rho)=\gamma_{2}\kappa_{2}\rho^{\gamma_{2}-1}\big{(}1+\mathcal{P}_{2}(\rho)+\rho\mathcal{P}_{2}^{\prime}(\rho)\big{)},\\\
&P^{\prime\prime}(\rho)=\gamma_{2}(\gamma_{2}-1)\kappa_{2}\rho^{\gamma_{2}-2}\big{(}1+\mathcal{P}_{2}(\rho)+3\rho\mathcal{P}_{2}^{\prime}(\rho)+\rho^{2}\mathcal{P}_{2}^{\prime\prime}(\rho)).\end{aligned}\right.$

Then, for $\rho\geq\max\\{\rho^{*},(8C^{*})^{1/\epsilon}\\}$,

$\big{|}d(\rho)-(1+\theta_{2})\big{|}=\Big{|}\frac{\rho
P^{\prime\prime}(\rho)}{2P^{\prime}(\rho)}-\theta_{2}\Big{|}=\Big{|}\frac{\theta_{2}\big{(}2\rho\mathcal{P}_{2}^{\prime}(\rho)+\rho^{2}\mathcal{P}_{2}^{\prime\prime}(\rho)\big{)}}{1+P_{2}(\rho)+3\rho\mathcal{P}_{2}(\rho)+\rho^{2}\mathcal{P}_{2}^{\prime\prime}(\rho)}\Big{|}\leq
C(\theta_{2},C^{*})\rho^{-\epsilon},$

where we have used that $|\mathcal{P}_{2}^{(j)}(\rho)|\leq
C^{*}\rho^{-\epsilon-j}$ for $j\in\\{0,1,2\\}$ in the last inequality.
$\hfill\square$

Hereafter, for simplicity of notation, we assume that (3.12) holds for
$\rho\geq\rho^{*}$. Furthermore, using $\eqref{1.5}$ and
$e^{\prime}(\rho)=\frac{P(\rho)}{\rho^{2}}$, we obtain

$e(\rho)=\frac{\kappa_{2}}{\gamma_{2}-1}\big{(}\rho^{\gamma_{2}-1}-(\rho^{*})^{\gamma_{2}-1}\big{)}+\kappa_{2}\int_{\rho^{*}}^{\rho}s^{\gamma_{2}-2}\mathcal{P}_{2}(s)\,\mathrm{d}s+\int_{0}^{\rho^{*}}\frac{P(s)}{s^{2}}\,\mathrm{d}s\qquad\,\,\text{for
}\rho\geq\rho^{*},$ (3.13)

which, with $e(0)=0$ and $|\mathcal{P}_{2}(\rho)|\leq C^{*}\rho^{-\epsilon}$,
yields that, for any parameter $\beta>0$,

$\lim\limits_{\rho\to
0}\rho^{\frac{\gamma_{2}-1}{5\gamma_{2}-6}}(\beta+e(\rho))^{-\frac{1}{5\gamma_{2}-6}}=0,\qquad\lim\limits_{\rho\to\infty}\rho^{\frac{\gamma_{2}-1}{5\gamma_{2}-6}}(\beta+e(\rho))^{-\frac{1}{5\gamma_{2}-6}}=(\frac{\kappa_{2}}{\gamma_{2}-1})^{-\frac{1}{5\gamma_{2}-6}}.$

Then we see that

$\displaystyle C_{\max}(\beta):=\sup_{\rho\geq
0}\rho^{\frac{\gamma_{2}-1}{5\gamma_{2}-6}}(\beta+e(\rho))^{-\frac{1}{5\gamma_{2}-6}}\in[(\frac{\kappa_{2}}{\gamma_{2}-1})^{-\frac{1}{5\gamma_{2}-6}},\,\infty),$
(3.14)
$\displaystyle\rho^{\frac{6(\gamma_{2}-1)}{5\gamma_{2}-6}}(\beta\rho+\rho
e(\rho))^{-\frac{1}{5\gamma_{2}-6}}\leq C_{\max}(\beta)\rho\qquad\,\text{for
}\rho>0.$ (3.15)

With a careful analysis of $C_{\max}(\beta)$, we obtain some estimates on
$M_{\rm c}$ defined in (2.6).

######  Proposition 3.3.

Let $h(\rho)=P(\rho)\rho^{-1}-(\gamma_{2}-1)e(\rho)$, and let
$\accentset{\sim}{M}_{\rm c}$ be the critical mass obtained in [9, (2.8)] for
polytropic gases in (1.2) with $\gamma\in(\frac{6}{5},\frac{4}{3})$. Then
$M_{\rm c}$ defined in (2.6)–(2.8) satisfies that $M_{\rm
c}\leq\accentset{\sim}{M}_{\rm c}$; in particular, $M_{\rm
c}<\accentset{\sim}{M}_{\rm c}$ when $h^{\prime}(\rho)>0$. For example,

$P_{\delta}(\rho):=\int_{0}^{\rho^{\frac{1}{3}}}\frac{s^{4}}{\sqrt{\delta+s^{2+\epsilon_{0}}}}\,\mathrm{d}s\qquad\,\,\text{for
$\delta>0$ and $\epsilon_{0}\in(0,\frac{4}{5})$}$ (3.16)

satisfies conditions (1.4)–(1.6). If $M_{\rm c}(\delta)$ is the critical mass
defined in (2.6)–(2.8) for pressure $P_{\delta}(\rho)$, then $M_{\rm
c}(\delta)<\accentset{\sim}{M}_{\rm c}$ for any $\delta>0$.

Proof. For $\gamma_{2}\in(\frac{6}{5},\frac{4}{3})$, it follows from
(2.6)–(2.8) and (3.14) that, for any fixed $\beta>0$,

$\displaystyle M_{\rm c}(\beta)$
$\displaystyle=\Big{(}\frac{2}{9(\gamma_{2}-1)}(C_{\max}^{\beta})^{\frac{5\gamma_{2}-6}{3(\gamma_{2}-1)}}\omega_{3}^{-\frac{4-3\gamma_{2}}{3(\gamma_{2}-1)}}\omega_{4}^{-\frac{2}{3}}\Big{)}^{-\frac{3(\gamma_{2}-1)}{5\gamma_{2}-6}}\Big{(}\frac{E_{0}+\omega_{3}^{-1}\beta
M_{\rm
c}(\beta)}{4-3\gamma_{2}}\Big{)}^{-\frac{4-3\gamma_{2}}{5\gamma_{2}-6}}$
$\displaystyle\leq\Big{(}\frac{2}{9(\gamma_{2}-1)}\Big{(}\frac{\kappa_{2}}{\gamma_{2}-1}\Big{)}^{-\frac{1}{3(\gamma_{2}-1)}}\omega_{3}^{-\frac{4-3\gamma_{2}}{3(\gamma_{2}-1)}}\omega_{4}^{-\frac{2}{3}}\Big{)}^{-\frac{3(\gamma_{2}-1)}{5\gamma_{2}-6}}\Big{(}\frac{E_{0}}{4-3\gamma_{2}}\Big{)}^{-\frac{4-3\gamma_{2}}{5\gamma_{2}-6}}=\accentset{\sim}{M}_{\rm
c},$ (3.17)

which yields $M_{\rm c}\leq\accentset{\sim}{M}_{\rm c}$.

Let
$g(\rho):=\rho^{\frac{\gamma_{2}-1}{5\gamma_{2}-6}}(\beta+e(\rho))^{-\frac{1}{5\gamma_{2}-6}}$.
Then $C_{\max}(\beta)=\max_{\rho\geq 0}g(\rho)$. Since
$e^{\prime}(\rho)=\frac{P(\rho)}{\rho^{2}}$, a direct calculation shows that

$g^{\prime}(\rho)=\frac{1}{5\gamma_{2}-6}\rho^{5-4\gamma_{2}}\big{(}\beta+e(\rho)\big{)}^{-\frac{5(\gamma_{2}-1)}{5\gamma_{2}-6}}\big{(}(\gamma_{2}-1)\beta-h(\rho)\big{)}.$
(3.18)

If $h^{\prime}(\rho)>0$ for all $\rho>0$, then $h(\rho)\geq h(0)=0$. Let
$K_{0}:=\max_{\rho>0}h(\rho)>0$. For $\beta$ small enough such that
$0<\beta<\frac{K_{0}}{\gamma_{2}-1}$, there exists a unique point
$\rho_{\beta}>0$ such that $g^{\prime}(\rho_{\beta})=0$, i.e.,

$h(\rho_{\beta})=(\gamma_{2}-1)\beta,$ (3.19)

and
$C_{\max}(\beta)=g(\rho_{\beta})=(\gamma_{2}-1)^{\frac{1}{5\gamma_{2}-6}}\big{(}P(\rho_{\beta})\rho_{\beta}^{-\gamma_{2}}\big{)}^{-\frac{1}{5\gamma_{2}-6}}.$
Moreover, it follows from (3.19) that $\lim_{\beta\to 0+}\rho_{\beta}=0$.
Thus, we see from (1.5) that

$\lim\limits_{\beta\to
0+}C_{\max}(\beta)=(\gamma_{2}-1)^{\frac{1}{5\gamma_{2}-6}}\lim\limits_{\rho_{\beta}\to
0}\big{(}P(\rho_{\beta})\rho_{\beta}^{-\gamma_{2}}\big{)}^{-\frac{1}{5\gamma_{2}-6}}=\infty,$

which, with $\eqref{A.15}_{1}$, implies that $\lim_{\beta\to 0+}M_{\rm
c}(\beta)=0$.

On the other hand, it follows directly from (3.14) and $\eqref{A.15}_{1}$ that
$\lim_{\beta\to\infty}M_{\rm c}(\beta)=0$. Therefore, the maximum value
$M_{\rm c}$ of $M_{\rm c}(\beta)$ must be attained at some point
$\beta_{0}\in(0,\infty)$ with $M_{\rm c}=M_{\rm
c}(\beta_{0})<\accentset{\sim}{M}_{\rm c}$ due to $\eqref{A.15}_{1}$.

For the pressure function $P_{\delta}(\rho)$ in (3.16), it is direct to check
that conditions (1.4)–(1.6) are satisfied and
$\gamma_{2}=\frac{4}{3}-\frac{\epsilon_{0}}{6}\in(\frac{6}{5},\frac{4}{3})$.
Let $e_{\delta}(\rho)$ be the corresponding internal energy with
$e_{\delta}^{\prime}(\rho)=\frac{P_{\delta}(\rho)}{\rho^{2}}$ and
$e_{\delta}(0)=0$, and
$h_{\delta}(\rho):=P_{\delta}(\rho)\rho^{-1}-(\gamma_{2}-1)e_{\delta}(\rho)$.
By direct calculation, one has

$h_{\delta}^{\prime}(\rho)=\rho^{-2}\big{(}-\gamma_{2}P_{\delta}(\rho)+\rho
P_{\delta}^{\prime}(\rho)\big{)}=:\rho^{-2}T_{\delta}(\rho),$

$T_{\delta}(0)=0$, and
$T_{\delta}^{\prime}(\rho)=-(\gamma_{2}-1)P_{\delta}^{\prime}(\rho)+\rho
P_{\delta}^{\prime\prime}(\rho)=\frac{2+\epsilon_{0}}{18}\delta\rho^{\frac{2}{3}}(\delta+\rho^{\frac{2+\epsilon_{0}}{3}})^{-\frac{3}{2}}>0$
for $\rho>0$. Thus, $T_{\delta}(\rho)>0$ for $\rho>0$, which implies that
$h_{\delta}^{\prime}(\rho)>0$ for $\rho>0$, so that $M_{\rm
c}(\delta)<\accentset{\sim}{M}_{\rm c}$ for any $\delta>0$.$\hfill\square$

##  4\. Entropy Analysis: Weak Entropy Pairs

Compared with the polytropic gas case in [9], there is no explicit formula of
the entropy kernel for the general pressure law (1.4)–(1.6) so that we have to
analyze the entropy equation (2.14) carefully to obtain several desired
estimates.

###  4.1. A special entropy pair

In order to obtain the higher integrability of the velocity, we are going to
construct a special entropy pair such that $\rho|u|^{3}$ can be controlled by
the entropy flux. Indeed, such a special entropy $\hat{\eta}(\rho,u)$ is
constructed as

$\hat{\eta}(\rho,u)=\begin{cases}\frac{1}{2}\rho u^{2}+\rho
e(\rho)\quad&\text{for }u\geq k(\rho),\\\ -\frac{1}{2}\rho u^{2}-\rho
e(\rho)\,\,\,&\text{for }u\leq-k(\rho),\end{cases}$ (4.1)

for $k(\rho)=\int_{0}^{\rho}\frac{\sqrt{P^{\prime}(y)}}{y}\,\mathrm{d}y$ and,
in the intermediate region $-k(\rho)\leq u\leq k(\rho)$, $\hat{\eta}(\rho,u)$
is the unique solution of the Goursat problem of the entropy equation (2.14):

$\left\\{\begin{aligned}
\displaystyle&\eta_{\rho\rho}-k^{\prime}(\rho)^{2}\eta_{uu}=0,\\\
\displaystyle&\eta(\rho,u)|_{u=\pm k(\rho)}=\pm\big{(}\frac{1}{2}\rho
u^{2}+\rho e(\rho)\big{)}{\rm;}\end{aligned}\right.$ (4.2)

see Fig. 4.1. Set

$V_{1}=\frac{1}{2k^{\prime}(\rho)}(\eta_{\rho}+k^{\prime}(\rho)\eta_{u}),\qquad
V_{2}=\frac{1}{2k^{\prime}(\rho)}(\eta_{\rho}-k^{\prime}(\rho)\eta_{u}).$
(4.3)

Then (4.2) can be rewritten as

$\begin{cases}\displaystyle\frac{\partial
V_{1}}{\partial\rho}-k^{\prime}(\rho)\frac{\partial V_{1}}{\partial
u}=-\frac{k^{\prime\prime}(\rho)}{2k^{\prime}(\rho)}(V_{1}+V_{2}),\\\\[11.38109pt]
\displaystyle\frac{\partial
V_{2}}{\partial\rho}+k^{\prime}(\rho)\frac{\partial V_{2}}{\partial
u}=-\frac{k^{\prime\prime}(\rho)}{2k^{\prime}(\rho)}(V_{1}+V_{2}).\end{cases}$
(4.4)

The corresponding characteristic boundary conditions become

$\left\\{\begin{aligned} &V_{1}|_{u=\pm
k(\rho)}=\pm\frac{1}{2k^{\prime}(\rho)}\big{(}\frac{1}{2}u^{2}+e(\rho)+\rho
e^{\prime}(\rho)\big{)}\pm\frac{1}{2}\rho u,\\\ &V_{2}|_{u=\pm
k(\rho)}=\pm\frac{1}{2k^{\prime}(\rho)}\big{(}\frac{1}{2}u^{2}+e(\rho)+\rho
e^{\prime}(\rho)\big{)}\mp\frac{1}{2}\rho u.\end{aligned}\right.$ (4.5)

Since $\frac{k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}$ has the singularity at
vacuum $\rho=0$, the Goursat problem (4.4)–(4.5) is singular, which requires a
careful analysis.

It follows from (4.4) that there exist two characteristic curves originating
from the origin point $O(0,0)$ in the $(\rho,u)$–plane:

$\ell_{+}:=\\{(\rho,u)\,:\,u=k(\rho)\\},\qquad\ell_{-}:=\\{(\rho,u)\,:\
u=-k(\rho)\\}.$ (4.6)

For any given point $O_{1}(\rho_{0},u_{0})$ with $u_{0}=0$, we can draw two
backward characteristic curves $\ell_{0}^{\pm}$ through
$O_{1}(\rho_{0},u_{0})$; see Fig. 4.1). Let $O_{2}(\rho_{0}^{+},u_{0}^{+})$ be
the intersection point of $\ell_{0}^{+}$ and $\ell_{+}$, and let
$O_{3}(\rho_{0}^{-},u_{0}^{-})$ be the intersection point of $\ell_{0}^{-}$
and $\ell_{-}$. Let $\Sigma$ be the region surrounded by arc
$\widehat{OO_{2}O_{1}O_{3}}$, and let $\overline{\Sigma}$ be the closure of
$\Sigma$.

$\mathbf{Fig.}$ 4.1. The schematic diagram of the characteristic curves of
(4.4)

######  Lemma 4.1.

The Goursat problem (4.2) admits a unique solution $\hat{\eta}\in
C^{2}(\mathbb{R}_{+}\times\mathbb{R})$ such that

  * (i)

$|\hat{\eta}(\rho,u)|\leq C(\rho|u|^{2}+\rho^{\gamma(\rho)})$ for
$(\rho,u)\in\mathbb{R}_{+}\times\mathbb{R}$, where $\gamma(\rho)=\gamma_{1}$
if $\rho\in[0,\rho_{*}]$ and $\gamma(\rho)=\gamma_{2}$ if
$\rho\in(\rho_{*},\infty)$.

  * (ii)

If $\hat{\eta}$ is regarded as a function of $(\rho,u)$,

$\displaystyle\qquad|\hat{\eta}_{\rho}(\rho,u)|\leq
C(|u|^{2}+\rho^{2\theta(\rho)}),\quad|\hat{\eta}_{u}(\rho,u)|\leq
C(\rho|u|+\rho^{\theta(\rho)+1})\qquad\text{for
}(\rho,u)\in\mathbb{R}_{+}\times\mathbb{R},$

and, if $\hat{\eta}$ is regarded as a function of $(\rho,m)$,

$\displaystyle\qquad|\hat{\eta}_{\rho}(\rho,m)|\leq
C(|u|^{2}+\rho^{2\theta(\rho)}),\quad|\hat{\eta}_{m}(\rho,m)|\leq
C(|u|+\rho^{\theta(\rho)})\qquad\,\text{for
}(\rho,m)\in\mathbb{R}_{+}\times\mathbb{R},$

where $\theta(\rho):=\frac{\gamma(\rho)-1}{2}$.

  * (iii)

If $\hat{\eta}_{m}$ is regarded as a function of $(\rho,u)$,

$\displaystyle\qquad|\hat{\eta}_{m\rho}(\rho,u)|\leq
C\rho^{\theta(\rho)-1},\quad|\hat{\eta}_{mu}(\rho,u)|\leq C,$

and, if $\hat{\eta}_{m}$ is regarded as a function of $(\rho,m)$,

$|\hat{\eta}_{m\rho}(\rho,m)|\leq
C\rho^{\theta(\rho)-1},\quad|\hat{\eta}_{mm}(\rho,m)|\leq C\rho^{-1}.$

  * (iv)

If $\hat{q}$ is the corresponding entropy flux determined by (2.12), then
$\hat{q}\in C^{2}(\mathbb{R}_{+}\times\mathbb{R})$ and

$\displaystyle\hat{q}(\rho,u)=\frac{1}{2}\rho|u|^{3}\pm\rho u(e(\rho)+\rho
e^{\prime}(\rho))\qquad\quad\text{for }\pm u\geq k(\rho),$
$\displaystyle|\hat{q}(\rho,u)|\leq
C\rho^{\gamma(\rho)+\theta(\rho)}\qquad\qquad\qquad\qquad\,\,\,\,\,\,\text{for
}|u|<k(\rho),$
$\displaystyle\hat{q}(\rho,u)\geq\frac{1}{2}\rho|u|^{3}\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\text{for
}|u|\geq k(\rho),$ $\displaystyle|\hat{q}-u\hat{\eta}|\leq
C(\rho^{\gamma(\rho)}|u|+\rho^{\gamma(\rho)+\theta(\rho)})\qquad\qquad\text{for
}(\rho,u)\in\mathbb{R}_{+}\times\mathbb{R}.$

Proof. To prove that (4.2) has a unique $C^{2}$–solution $\hat{\eta}$ in
$\mathbb{R}_{+}\times\mathbb{R}$, it suffices to prove that (4.4)–(4.5) admits
a unique $C^{1}$–solution $(V_{1},V_{2})$ in $\Sigma$ for any given point
$O(\rho_{0},u_{0})$. We use the Picard iteration and divide the proof into six
steps to achieve this.

1\. For any point $A(\rho,u)\in\Sigma$, we can draw two backward
characteristic curves through $A$:

$\displaystyle\ell_{1}:=\big{\\{}(s,\,u^{(1)}(s))\,:\,u^{(1)}(s)=-k(s)+u+k(\rho),0<s\leq\rho\big{\\}},$
(4.7)
$\displaystyle\ell_{2}:=\big{\\{}(s,\,u^{(2)}(s))\,:\,u^{(2)}(s)=-k(\rho)+u+k(s),0<s\leq\rho\big{\\}}.$

Let $B(\rho_{1},u_{1})$ be the intersection point of $\ell_{+}$ and
$\ell_{1}$, and let $C(\rho_{2},u_{2})$ be the intersection point of
$\ell_{-}$ and $\ell_{2}$. It follows from (4.6)–(4.7) that

$u_{1}=k(\rho_{1})=\frac{k(\rho)+u}{2},\qquad
u_{2}=-k(\rho_{2})=\frac{u-k(\rho)}{2}.$ (4.8)

Using (4.4) and integrating $V_{1}$ and $V_{2}$ along the characteristic
curves $\ell_{1}$ and $\ell_{2}$ respectively, we have

$V_{i}(\rho,u)=V_{i}(\rho_{i},u_{i})-\int_{\rho_{i}}^{\rho}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\,\sum\limits_{j=1}^{2}V_{j}(s,{u}^{(i)}(s))\,\mathrm{d}s\qquad\mbox{for
$i=1,2$}.$ (4.9)

Denote $V_{i}^{(0)}(\rho,u):=V_{i}(\rho_{i},u_{i})$. It follows from (4.5) and
(4.8) that

$V_{i}^{(0)}(\rho,u)=(-1)^{i+1}\Big{(}\frac{1}{2}\rho_{i}k(\rho_{i})+\frac{1}{4}\frac{k^{2}(\rho_{i})}{k^{\prime}(\rho_{i})}+\frac{e(\rho_{i})+\rho_{i}e^{\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{)}\qquad\mbox{for
$i=1,2$}.$ (4.10)

We define the iterated scheme:

$V_{i}^{(n+1)}(\rho,u):=V_{i}(\rho_{i},u_{i})-\int_{\rho_{i}}^{\rho}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\,\sum\limits_{j=1}^{2}V_{j}^{(n)}(s,u^{(i)}(s))\,\mathrm{d}s\qquad\mbox{for
$i=1,2$}.$ (4.11)

Then we obtain two sequences $\\{V_{i}^{(n)}\\}_{n=0}^{\infty}$ for $i=1,2$.
We now prove that $\\{V_{i}^{(n)}\\}_{n=0}^{\infty}$ are uniformly convergent
in $\overline{\Sigma}$, which is equivalent to proving that

$V_{i}^{(0)}(\rho,u)+\sum\limits_{n=1}^{\infty}\big{(}V_{i}^{(n)}-V_{i}^{(n-1)}\big{)}(\rho,u),\quad\mbox{$i=1,2$},$
(4.12)

are uniformly convergent in $\overline{\Sigma}$.

From Lemma 3.1 and (4.10), we know that $V_{i}^{(0)},i=1,2$, are continuous in
$\overline{\Sigma}$ and there exists a constant $C_{1}>0$ depending only on
$\rho_{*}$ and $\rho^{*}$ such that

$\displaystyle|V_{i}^{(0)}(\rho,u)|\leq\begin{cases}C_{1}\rho_{i}^{1+\theta_{1}}\,\,\,&\text{for
}\rho_{i}\leq\rho_{*},\\\ C_{1}\rho_{i}^{1+\theta_{2}}\,\,\,&\text{for
}\rho_{i}\geq\rho_{*},\end{cases}\qquad i=1,2,$

which, with the fact that $\rho_{i}\leq\rho$, yields that

$\displaystyle|V_{i}^{(0)}(\rho,u)|$
$\displaystyle\leq\begin{cases}C_{1}\rho^{1+\theta_{1}}\quad&\text{for
}\rho\leq\rho_{*},\\\ \tilde{C}_{1}\rho^{1+M_{1}}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\
\hat{C}_{1}\rho^{1+\theta_{2}}\quad&\text{for
}\rho\geq\rho^{*},\end{cases}\qquad i=1,2,$ (4.13)

where $\tilde{C}_{1}\geq C_{1}(\rho_{*})^{\theta_{2}-M_{1}}$, $\hat{C}_{1}\geq
C_{1}$, and $M_{1}$ are positive constants to be chosen later.

It follows from (3.9) and (3.11) that there exist a constant $\nu<1$ and a
constant $C_{0}\gg 1$ depending on $\rho_{*}$ and $\rho^{*}$ such that

$\Big{|}\frac{k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}\Big{|}\leq\begin{cases}\nu\rho^{-1}\quad&\text{for
$0<\rho\leq\rho_{*}$ and $\rho\geq\rho^{*}$},\\\\[1.42262pt]
C_{0}\rho^{-1}\,\,\,&\text{for $\rho_{*}<\rho<\rho^{*}$},\end{cases}$ (4.14)

For the estimate of $|V_{i}^{(1)}-V_{i}^{(0)}|$, we divide it into six cases:

Case 1. $\rho_{i}\leq\rho\leq\rho_{*}$: It follows from (4.11) and
(4.13)–(4.14) that

$\displaystyle\big{|}(V_{i}^{(1)}-V_{i}^{(0)})(\rho,u)\big{|}$
$\displaystyle\leq\int_{\rho_{i}}^{\rho}C_{1}\nu\,s^{\theta_{1}}\,\mathrm{d}s\leq
C_{1}\rho^{1+\theta_{1}}\,\varpi_{1},$ (4.15)

where $\varpi_{1}:=\frac{\nu}{1+\theta_{1}}\in(0,1)$.

Case 2. $\rho_{i}\leq\rho_{*}\leq\rho\leq\rho^{*}$: Then

$\displaystyle\big{|}(V_{i}^{(1)}-V_{i}^{(0)})(\rho,u)\big{|}\leq\Big{(}\int_{\rho_{i}}^{\rho_{*}}+\int_{\rho_{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\big{|}V_{j}^{(0)}(s,u^{(i)}(s))\big{|}\,\mathrm{d}s$
$\displaystyle\leq\int_{\rho_{i}}^{\rho_{*}}\frac{\nu}{2s}\,(2C_{1}s^{1+\theta_{1}})\,\mathrm{d}s+\int_{\rho_{*}}^{\rho}\frac{C_{0}}{2s}\,(2\tilde{C}_{1}s^{1+M_{1}})\,\mathrm{d}s$
$\displaystyle\leq\tilde{C}_{1}\big{(}\rho^{1+M_{1}}-(\rho_{*})^{1+M_{1}}\big{)}\varpi_{M_{1}}+C_{1}(\rho_{*})^{1+\theta_{1}}\varpi_{1}\leq\tilde{C}_{1}\rho^{1+M_{1}}\,\varpi_{M_{1}},$
(4.16)

where $\varpi_{M_{1}}:=\frac{C_{0}}{1+M_{1}}$ and, in the last inequality of
(4.1), we have chosen

$\tilde{C}_{1}\geq
C_{1}(\rho_{*})^{\theta_{1}-M_{1}}\varpi_{1}(\varpi_{M_{1}})^{-1}.$ (4.17)

Case 3. $\rho_{*}\leq\rho_{i}\leq\rho\leq\rho^{*}$: It is direct to see that

$\displaystyle\big{|}(V_{i}^{(1)}-V_{i}^{(0)})(\rho,u)\big{|}\leq\int_{\rho_{i}}^{\rho}\tilde{C}_{1}C_{0}\,s^{M_{1}}\,{\rm
d}s\leq\tilde{C}_{1}\rho^{1+M_{1}}\,\varpi_{M_{1}}.$ (4.18)

Case 4. $\rho_{i}\leq\rho_{*}<\rho^{*}\leq\rho$: Then

$\displaystyle\big{|}(V_{i}^{(1)}-V_{i}^{(0)})(\rho,u)\big{|}\leq\Big{(}\int_{\rho_{i}}^{\rho_{*}}+\int_{\rho_{*}}^{\rho^{*}}+\int_{\rho^{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\big{|}V_{j}^{(0)}(s,u^{(i)}(s))\big{|}\,\mathrm{d}s$
$\displaystyle\leq\int_{\rho_{i}}^{\rho_{*}}\frac{\nu}{2s}\,(2C_{1}s^{1+\theta_{1}})\,\mathrm{d}s+\int_{\rho_{*}}^{\rho^{*}}\frac{C_{0}}{2s}\,(2\tilde{C}_{1}s^{1+M_{1}})\,\mathrm{d}s+\int_{\rho^{*}}^{\rho}\frac{\nu}{2s}\,(2\hat{C}_{1}s^{1+\theta_{2}})\,\mathrm{d}s$
$\displaystyle\leq\hat{C}_{1}\big{(}\rho^{1+\theta_{2}}-(\rho^{*})^{1+\theta_{2}}\big{)}\varpi_{2}+\tilde{C}_{1}\big{(}(\rho^{*})^{1+M_{1}}-(\rho_{*})^{1+M_{1}}\big{)}\varpi_{M_{1}}+C_{1}(\rho_{*})^{1+\theta_{1}}\varpi_{1}\leq\hat{C}_{1}\rho^{1+\theta_{2}}\,\varpi_{2},$
(4.19)

where $\varpi_{2}:=\frac{\nu}{1+\theta_{2}}\in(0,1)$ and, in the last
inequality of (4.1), we have used (4.17) and chosen

$\hat{C}_{1}\geq\tilde{C}_{1}(\rho^{*})^{M_{1}-\theta_{2}}\varpi_{M_{1}}\varpi_{2}^{-1}.$
(4.20)

Case 5. $\rho_{*}\leq\rho_{i}\leq\rho^{*}\leq\rho$: It follows similarly that

$\displaystyle|(V_{i}^{(1)}-V_{i}^{(0)})(\rho,u)|\leq\Big{(}\int_{\rho_{i}}^{\rho^{*}}+\int_{\rho^{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}|V_{j}^{(0)}(s,u^{(i)}(s))|\,\mathrm{d}s$
$\displaystyle\leq\hat{C}_{1}\big{(}\rho^{1+\theta_{2}}-(\rho^{*})^{1+\theta_{2}}\big{)}\varpi_{2}+\tilde{C}_{1}(\rho^{*})^{1+M_{1}}\varpi_{M_{1}}\leq\hat{C}_{1}\rho^{1+\theta_{2}}\,\varpi_{2},$
(4.21)

where we have used (4.20) in the last inequality of (4.1).

Case 6. $\rho^{*}\leq\rho_{i}\leq\rho$: We see that

$\displaystyle\big{|}(V_{i}^{(1)}-V_{i}^{(0)})(\rho,u)\big{|}$
$\displaystyle\leq\int_{\rho_{i}}^{\rho}\hat{C}_{1}\nu\,s^{\theta_{2}}\,\mathrm{d}s\leq\hat{C}_{1}\rho^{1+\theta_{2}}\,\varpi_{2}.$
(4.22)

Combining (4.15)–(4.22), we obtain

$\big{|}(V_{i}^{(1)}-V_{i}^{(0)})(\rho,u)\big{|}\leq\begin{cases}C_{1}\rho^{1+\theta_{1}}\,\varpi_{1}\quad&\text{for
}\rho\leq\rho_{*},\\\
\tilde{C}_{1}\rho^{1+M_{1}}\,\varpi_{M_{1}}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\
\hat{C}_{1}\rho^{1+\theta_{2}}\,\varpi_{2}\quad&\text{for
}\rho\geq\rho^{*},\end{cases}\quad i=1,2,$ (4.23)

if (4.17) and (4.20) hold.

To utilize the induction arguments, we make the induction assumption for
$n=k$:

$\big{|}(V_{i}^{(k)}-V_{i}^{(k-1)})(\rho,u)\big{|}\leq\begin{cases}C_{1}\rho^{1+\theta_{1}}\,\varpi_{1}^{k}\quad&\text{for
}\rho\leq\rho_{*},\\\
\tilde{C}_{1}\rho^{1+M_{1}}\,\varpi_{M_{1}}^{k}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\
\hat{C}_{1}\rho^{1+\theta_{2}}\,\varpi_{2}^{k}\quad&\text{for
}\rho\geq\rho^{*},\end{cases}\quad i=1,2.$ (4.24)

We now make the estimate for $n=k+1$. For estimate
$\big{|}V_{i}^{(k+1)}(\rho,u)-V_{i}^{(k)}(\rho,u)\big{|}$, it suffices to
consider the case: $\rho_{i}\leq\rho_{*}<\rho^{*}\leq\rho$, since the other
cases can be done by similar arguments as in (4.15)–(4.22). Noting (4.24) and
$\rho_{i}\leq\rho_{*}<\rho^{*}\leq\rho$, and using similar arguments in (4.1),
we have

$\displaystyle\big{|}(V_{i}^{(k+1)}-V_{k}^{(k)})(\rho,u)\big{|}$
$\displaystyle\leq\Big{(}\int_{\rho_{i}}^{\rho_{*}}+\int_{\rho_{*}}^{\rho^{*}}+\int_{\rho^{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\big{|}V_{j}^{(k)}(s,u^{(i)}(s))-V_{j}^{(k-1)}(s,u^{(i)}(s))\big{|}\,\mathrm{d}s$
$\displaystyle\leq\int_{\rho_{i}}^{\rho_{*}}\nu
C_{1}\varpi_{1}^{k}s^{\theta_{1}}\,\mathrm{d}s+\int_{\rho_{*}}^{\rho^{*}}C_{0}\tilde{C}_{1}\varpi_{M_{1}}^{k}s^{M_{1}}\,\mathrm{d}s+\int_{\rho^{*}}^{\rho}\nu\hat{C}_{1}\varpi_{2}^{k}s^{\theta_{2}}\,\mathrm{d}s$
$\displaystyle\leq\hat{C}_{1}\big{(}\rho^{1+\theta_{2}}-(\rho^{*})^{1+\theta_{2}}\big{)}\varpi_{2}^{k+1}+\tilde{C}_{1}\big{(}(\rho^{*})^{1+M_{1}}-(\rho_{*})^{1+M_{1}}\big{)}\varpi_{M_{1}}^{k+1}+C_{1}(\rho^{*})^{1+\theta_{1}}\,\varpi_{1}^{k+1}\leq\hat{C}_{1}\rho^{1+\theta_{2}}\varpi_{2}^{k+1},$

where we have chosen $\tilde{C}_{1}$ and $\hat{C}_{1}$ such that

$\tilde{C}_{1}\geq
C_{1}(\rho_{*})^{\theta_{1}-M_{1}}\big{(}\varpi_{1}\varpi_{M_{1}}^{-1}\big{)}^{k+1},\qquad\hat{C}_{1}\geq\tilde{C}_{1}(\rho^{*})^{M_{1}-\theta_{2}}\big{(}\varpi_{M_{1}}\varpi_{2}^{-1}\big{)}^{k+1}.$
(4.25)

Therefore, under assumption (4.25), we obtain

$\displaystyle\big{|}(V_{i}^{(k+1)}-V_{i}^{(k)})(\rho,u)\big{|}\leq\begin{cases}C_{1}\rho^{1+\theta_{1}}\,\varpi_{1}^{k+1}\quad&\text{for
}\rho\leq\rho_{*},\\\
\tilde{C}_{1}\rho^{1+M_{1}}\,\varpi_{M_{1}}^{k+1}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\
\hat{C}_{1}\rho^{1+\theta_{2}}\,\varpi_{2}^{k+1}\,\,\,&\text{for
}\rho\geq\rho^{*}.\end{cases}\quad i=1,2.$

Recalling that $\theta_{1}\geq\theta_{2}$, we can take $C_{0}$ and $M_{1}$
large enough such that

$0<\varpi_{1}\leq\varpi_{M_{1}}\leq\varpi_{2}<1.$ (4.26)

Combining (4.13) and (4.23)–(4.26), and taking

$\varpi_{M_{1}}=\varpi_{2},\qquad\tilde{C}_{1}=C_{1}(\rho_{*})^{\theta_{2}-M_{1}},\qquad\hat{C}_{1}=\tilde{C}_{1}(\rho^{*})^{M_{1}-\theta_{2}},$
(4.27)

by induction, we conclude that, for any $n\geq 1$,

$\displaystyle\big{|}(V_{i}^{(n)}-V_{i}^{(n-1)})(\rho,u)\big{|}\leq\begin{cases}C_{1}\rho^{1+\theta_{1}}\,\varpi_{1}^{n}\quad&\text{for
}\rho\leq\rho_{*},\\\
\tilde{C}_{1}\rho^{1+M_{1}}\,\varpi_{M_{1}}^{n}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\
\hat{C}_{1}\rho^{1+\theta_{2}}\,\varpi_{2}^{n}\,\,\,&\text{for
}\rho\geq\rho^{*}.\end{cases}\quad\,i=1,2.$ (4.28)

Noting that (4.26) and $\rho\leq\rho_{0}$ for $(\rho,u)\in\overline{\Sigma}$,
we have proved that the two sequences in (4.12), $i=1,2$, are uniformly
convergent in $\overline{\Sigma}$ so that sequence
$\\{(V_{1}^{(n)},V_{2}^{(n)})\\}$ is uniformly convergent in
$\overline{\Sigma}$. Let $(V_{1},V_{2})$ be the limit function of sequence
$(V_{1}^{(n)},V_{2}^{(n)})$. Noting the continuity and the uniform convergence
of $(V_{1}^{(n)},V_{2}^{(n)})$, $(V_{1},V_{2})$ is continuous in
$\overline{\Sigma}$. Taking the limit: $n\to\infty$ in (4.11), we conclude
that $(V_{1},V_{2})$ is the continuous solution of (4.9).

2\. It follows from (4.13) and (4.28) that, for $(\rho,u)\in\\{\rho\geq
0,\,|u|\leq k(\rho)\\}$ and $i=1,2$,

$\big{|}V_{i}(\rho,u)\big{|}\leq|V_{i}^{(0)}(\rho,u)|+\sum\limits_{n=1}^{\infty}\big{|}(V_{i}^{(n)}-V_{i}^{(n-1)})(\rho,u)\big{|}\leq\begin{cases}C\rho^{1+\theta_{1}}\quad&\text{for
}\rho\leq\rho_{*},\\\ C\rho^{1+M_{1}}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\ C\rho^{1+\theta_{2}}\quad&\text{for
}\rho\geq\rho^{*}.\end{cases}$ (4.29)

On the other hand, we see from (4.3) that, for $|u|\leq k(\rho)$,

$\displaystyle|\hat{\eta}_{\rho}(\rho,u)|=\left|k^{\prime}(\rho)(V_{1}(\rho,u)+V_{2}(\rho,u))\right|\leq
C\rho^{\gamma(\rho)-1},$ (4.30)
$\displaystyle|\hat{\eta}_{u}(\rho,u)|=|V_{1}(\rho,u)-V_{2}(\rho,u)|\leq
C\rho^{1+\theta(\rho)}.$

Hence, for $|u|\leq k(\rho)$, it holds that

$|\hat{\eta}(\rho,u)|\leq\int_{\bar{\rho}}^{\rho}|\hat{\eta}_{\rho}(s,u)|\,\mathrm{d}s+|\hat{\eta}(\bar{\rho},u)|\leq
C\rho^{\gamma(\rho)},\quad|\hat{\eta}_{m}(\rho,m)|=|\rho^{-1}\hat{\eta}_{u}(\rho,u)|\leq
C\rho^{\theta(\rho)},$ (4.31)

where $(\bar{\rho},u)$ is the point satisfying $k(\bar{\rho})=|u|$, and we
have used the boundary data in (4.2).

3\. We now show that $V_{1}$ and $V_{2}$ have continuous first-order
derivatives with respect to $(\rho,u)$. Using (4.7)–(4.8) and Lemma 3.1, we
have

$\frac{\partial u^{(i)}(s)}{\partial
u}=1,\qquad\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}=\frac{1}{2|k^{\prime}(\rho_{i})|}\leq
C_{2}\rho_{i}^{1-\theta(\rho_{i})}.$ (4.32)

Applying $\partial_{u}$ to (4.11) and using (4.32) yield that for $i=1,2$,

$\displaystyle\frac{\partial V_{i}^{(n+1)}}{\partial u}(\rho,u)=\frac{\partial
V_{i}(\rho_{i},u_{i})}{\partial
u}+\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\,\sum\limits_{j=1}^{2}V_{j}^{(n)}(\rho_{i},u_{i})\,\frac{\partial\rho_{i}}{\partial
u}-\int_{\rho_{i}}^{\rho}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\,\sum\limits_{j=1}^{2}\frac{\partial
V_{j}^{(n)}}{\partial u}(s,u^{(i)}(s))\,\mathrm{d}s.$ (4.33)

It follows from (4.10), (4.32), Lemma 3.1, and a direct calculation that, for
$i=1,2$,

$\displaystyle\Big{|}\frac{\partial V_{i}^{(0)}}{\partial u}(\rho,u)\Big{|}$
$\displaystyle=\Big{|}\frac{{\rm{d}}V_{i}(\rho_{i},u_{i})}{{\rm{d}}\rho_{i}}\frac{\partial\rho_{i}}{\partial
u}\Big{|}\leq C_{2}\rho_{i}\leq\begin{cases}\bar{C}_{2}\rho\qquad\,&\text{for
}\rho\leq\rho_{*},\\\ \tilde{C}_{2}\rho^{1+M_{2}}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\ \hat{C}_{2}\rho\qquad\,&\text{for
}\rho\geq\rho^{*},\end{cases}$ (4.34)

where $C_{2}$ is chosen to be a common, fixed, and large enough constant in
(4.32) and (4.34) depending only on $\rho_{*}$ and $\rho^{*}$, and
$\bar{C}_{2}\geq C_{2},\tilde{C}_{2}\geq
C_{2}(\rho_{*})^{-M_{2}},\hat{C}_{2}\geq C_{2}$, and $M_{2}$ are some large
positive constants to be chosen later.

For the estimate of $\big{|}(\frac{\partial V_{i}^{(1)}}{\partial
u})(\rho,u)-(\frac{\partial V_{i}^{(0)}}{\partial u})(\rho,u)\big{|}$, we
divide it into six cases:

Case 1. $\rho_{i}\leq\rho\leq\rho_{*}$: It follows from (4.13)–(4.14) and
(4.33)–(4.34) that

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(1)}-V_{i}^{(0)})}{\partial
u}(\rho,u)\Big{|}$
$\displaystyle\leq\int_{\rho_{i}}^{\rho}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}\Big{(}\frac{\partial
V_{j}^{(0)}}{\partial
u}\Big{)}(s,u^{(i)}(s))\Big{|}\,\mathrm{d}s+\Big{|}\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{|}\,\sum\limits_{j=1}^{2}|V_{j}^{(0)}(\rho_{i},u_{i})|\,\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}$
$\displaystyle\leq\int_{\rho_{i}}^{\rho}\frac{\nu}{2s}\,(2\bar{C}_{2}s)\,\mathrm{d}s+\frac{\nu}{2\rho_{i}}\,(2C_{1}\rho_{i}^{1+\theta_{1}})\,(C_{2}\rho_{i}^{1-\theta_{1}})$
$\displaystyle=\bar{C}_{2}\nu\rho-\bar{C}_{2}\nu\rho_{1}+C_{1}C_{2}\nu\rho_{i}\leq\bar{C}_{2}\nu\rho,$
(4.35)

where, in the last inequality of (4.1), we have chosen

$\bar{C}_{2}\geq C_{1}C_{2}.$ (4.36)

Case 2. $\rho_{i}\leq\rho_{*}\leq\rho\leq\rho^{*}$: Then, similarly, we have

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(1)}-V_{i}^{(0)})}{\partial
u}(\rho,u)\Big{|}$
$\displaystyle\leq\Big{(}\int_{\rho_{i}}^{\rho_{*}}+\int_{\rho_{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}\Big{(}\frac{\partial
V_{j}^{(0)}}{\partial
u}\Big{)}(s,u^{(i)}(s))\Big{|}\,\mathrm{d}s+\Big{|}\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{|}\,\sum\limits_{j=1}^{2}|V_{j}^{(0)}(\rho_{i},u_{i})|\,\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}$
$\displaystyle\leq\tilde{C}_{2}\big{(}\rho^{1+M_{2}}-(\rho_{*})^{1+M_{2}}\big{)}\varpi_{M_{2}}+\bar{C}_{2}\nu(\rho_{*}-\rho_{i})+C_{1}C_{2}\nu\rho_{i}\leq\tilde{C}_{2}\rho\,\varpi_{M_{2}},$
(4.37)

where $\varpi_{M_{2}}:=\frac{C_{0}}{1+M_{2}}$ and, in the last inequality of
(4.1), we have used (4.36) and chosen

$\tilde{C}_{2}\geq\bar{C}_{2}(\rho_{*})^{-M_{2}}\,\nu\varpi_{M_{2}}^{-1}.$
(4.38)

Case 3. $\rho_{*}\leq\rho_{i}\leq\rho\leq\rho^{*}$: It follows that

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(1)}-V_{i}^{(0)})}{\partial
u}(\rho,u)\Big{|}$
$\displaystyle\leq\int_{\rho_{i}}^{\rho}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}\Big{(}\frac{\partial
V_{j}^{(0)}}{\partial
u}\Big{)}(s,u^{(i)}(s))\Big{|}\,\mathrm{d}s+\Big{|}\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}V_{j}^{(0)}(\rho_{i},u_{i})\Big{|}\,\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}$
$\displaystyle\leq\tilde{C}_{2}\big{(}\rho^{1+M_{2}}-\rho_{i}^{1+M_{2}}\big{)}\varpi_{M_{2}}+\tilde{C_{1}}C_{2}C_{0}\rho_{i}^{1+M_{1}-\theta_{2}}\leq\tilde{C}_{2}\rho\,\varpi_{M_{2}},$
(4.39)

where, in the last inequality of (4.1), we have chosen

$M_{1}\geq
M_{2}+\theta_{2},\qquad\tilde{C}_{2}\geq\tilde{C}_{1}C_{2}(1+M_{2})(\rho^{*})^{M_{1}-M_{2}-\theta_{2}}.$
(4.40)

Case 4: $\rho_{i}\leq\rho_{*}<\rho^{*}\leq\rho$. For this case, similarly, we
have

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(1)}-V_{i}^{(0)})}{\partial
u}(\rho,u)\Big{|}$
$\displaystyle\leq\Big{(}\int_{\rho_{i}}^{\rho_{*}}+\int_{\rho_{*}}^{\rho^{*}}+\int_{\rho^{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}\Big{(}\frac{\partial
V_{j}^{(0)}}{\partial
u}\Big{)}(s,u^{(i)}(s))\Big{|}\,\mathrm{d}s+\Big{|}\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{|}\,\sum\limits_{j=1}^{2}|V_{j}^{(0)}(\rho_{i},u_{i})|\,\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}$
$\displaystyle\leq\hat{C}_{2}(\rho-\rho^{*})\nu+\tilde{C}_{2}\big{(}(\rho^{*})^{1+M_{2}}-(\rho_{*})^{1+M_{2}}\big{)}\varpi_{M_{2}}+\bar{C}_{2}(\rho_{*}-\rho_{i})\nu+C_{1}C_{2}\rho_{i}\nu\leq\hat{C}_{2}\rho\,\nu,$
(4.41)

where, in the last inequality of (4.1), we have used (4.36) and (4.38) and
chosen

$\hat{C}_{2}\geq\tilde{C}_{2}(\rho^{*})^{M_{2}}\,\nu^{-1}\varpi_{M_{2}}.$
(4.42)

Case 5. $\rho_{*}\leq\rho_{i}\leq\rho^{*}\leq\rho$: Then

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(1)}-V_{i}^{(0)})}{\partial
u}(\rho,u)\Big{|}$
$\displaystyle\leq\Big{(}\int_{\rho_{i}}^{\rho^{*}}+\int_{\rho^{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}\Big{(}\frac{\partial
V_{j}^{(0)}}{\partial
u}\Big{)}(s,u^{(i)}(s))\Big{|}\,\mathrm{d}s+\Big{|}\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{|}\,\sum\limits_{j=1}^{2}|V_{j}^{(0)}(\rho_{i},u_{i})|\,\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}$
$\displaystyle\leq\hat{C}_{2}(\rho-\rho^{*})\nu+\tilde{C}_{2}\big{(}(\rho^{*})^{1+M_{2}}-\rho_{i}^{1+M_{2}}\big{)}\varpi_{M_{2}}+\tilde{C}_{1}C_{2}C_{0}\rho_{i}^{1+M_{1}-\theta_{2}}\leq\hat{C}_{2}\rho\,\nu,$
(4.43)

where we have used (4.40) and (4.42) in the last inequality of (4.1).

Case 6. $\rho^{*}\leq\rho_{i}\leq\rho$: It follows similarly that

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(1)}-V_{i}^{(0)})}{\partial
u}(\rho,u)\Big{|}$
$\displaystyle\leq\int_{\rho_{i}}^{\rho}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}\Big{(}\frac{\partial
V_{j}^{(0)}}{\partial
u}\Big{)}(s,u^{(i)}(s))\Big{|}\,\mathrm{d}s+\Big{|}\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{|}\,\sum\limits_{j=1}^{2}|V_{j}^{(0)}(\rho_{i},u_{i})|\,\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}$
$\displaystyle\leq\hat{C}_{2}\rho\nu-\hat{C}_{2}\rho_{i}\nu+\hat{C}_{1}C_{2}\rho_{i}\nu\leq\hat{C}_{2}\rho\,\nu,$
(4.44)

where, in the last inequality of (4.1), we have chosen

$\hat{C}_{2}\geq\hat{C}_{1}C_{2}.$ (4.45)

Combining (4.1)–(4.45), we conclude that, for $i=1,2$,

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(1)}-V_{i}^{(0)})}{\partial
u}(\rho,u)\Big{|}\leq\begin{cases}\bar{C}_{2}\rho\,\nu\quad&\text{for
}\rho\leq\rho_{*},\\\
\tilde{C}_{2}\rho^{1+M_{2}}\,\varpi_{M_{2}}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho_{*},\\\ \hat{C}_{2}\rho\,\nu\quad&\text{for
}\rho\geq\rho_{*},\end{cases}$ (4.46)

provided (4.36), (4.38), (4.40), (4.42), and (4.45) hold.

To use the induction arguments, we make the induction assumption for $n=k$:
For $i=1,2$,

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(k)}-V_{i}^{(k-1)})}{\partial
u}(\rho,u)\Big{|}\leq\begin{cases}\bar{C}_{2}\rho\,\nu^{k}\quad&\text{for
}\rho\leq\rho_{*},\\\
\tilde{C}_{2}\rho^{1+M_{2}}\,\varpi_{M_{2}}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho_{*},\\\ \hat{C}_{2}\rho\,\nu^{k}\quad&\text{for
}\rho\geq\rho_{*}.\end{cases}$ (4.47)

To estimate $|\frac{\partial(V_{i}^{(k+1)}-V_{i}^{(k)})}{\partial
u}(\rho,u)|$, it suffices to consider the case:
$\rho_{i}\leq\rho_{*}<\rho^{*}\leq\rho$ for simplicity of presentation, since
the other cases can be estimated by similar arguments in (4.1)–(4.45). In
fact, for the case $\rho_{i}\leq\rho_{*}<\rho^{*}\leq\rho$, it follows from
(4.28) and (4.47) that

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(k+1)}-V_{i}^{(k)})}{\partial
u}(\rho,u)\Big{|}$
$\displaystyle\leq\Big{(}\int_{\rho_{i}}^{\rho_{*}}+\int_{\rho_{*}}^{\rho^{*}}+\int_{\rho^{*}}^{\rho}\Big{)}\Big{|}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\Big{|}\,\sum\limits_{j=1}^{2}\Big{|}\frac{\partial(V_{j}^{(k)}-V_{j}^{(k-1)})}{\partial
u}(s,u^{(i)}(s))\Big{|}\mathrm{d}s$
$\displaystyle\quad\,\,+\Big{|}\frac{k^{\prime\prime}(\rho_{i})}{2k^{\prime}(\rho_{i})}\Big{|}\,\Big{|}\frac{\partial\rho_{i}}{\partial
u}\Big{|}\,\sum\limits_{j=1}^{2}|(V_{j}^{(k)}-V_{j}^{(k-1)})(\rho_{i},u_{i})|$
$\displaystyle\leq\int_{\rho_{i}}^{\rho_{*}}\bar{C}_{2}\nu^{k+1}\,\mathrm{d}s+\int_{\rho_{*}}^{\rho^{*}}C_{0}\tilde{C}_{2}\varpi_{M_{2}}^{k}s^{+M_{2}}\,\mathrm{d}s+\int_{\rho^{*}}^{\rho}\hat{C}_{2}\nu^{k+1}\,\mathrm{d}s+C_{2}\nu\rho_{i}^{-\theta_{1}}\,C_{1}\varpi_{M_{2}}^{k}\rho_{i}^{1+\theta_{1}}$
$\displaystyle\leq\hat{C}_{2}(\rho-\rho^{*})\,\nu^{k+1}+\tilde{C}_{2}\big{(}(\rho^{*})^{1+M_{2}}-(\rho_{*})^{1+M_{2}}\big{)}\varpi_{M_{2}}^{k+1}+\bar{C}_{2}(\rho_{*}-\rho_{i})\nu^{k+1}+C_{1}C_{2}\rho_{i}\nu^{k+1}$
$\displaystyle\leq\hat{C}_{2}\rho\,\nu^{k+1},$

where we have chosen

$\displaystyle\begin{split}&M_{1}\geq M_{2}+\theta_{2},\qquad\bar{C}_{2}\geq
C_{1}C_{2},\\\
&\tilde{C}_{2}\geq\max\Big{\\{}\tilde{C}_{1}C_{2}(1+M_{2})(\rho^{*})^{M_{1}-M_{2}-\theta_{2}}\Big{(}\frac{1+M_{2}}{1+M_{1}}\Big{)}^{k},\,\bar{C}_{2}(\rho_{*})^{-M_{2}}\big{(}\nu\varpi_{M_{2}}^{-1}\big{)}^{k+1}\Big{\\}},\\\
&\hat{C}_{2}\geq\max\Big{\\{}\hat{C}_{1}C_{2},\,\tilde{C}_{2}(\rho^{*})^{M_{2}}\big{(}\nu^{-1}\varpi_{M_{2}}\big{)}^{k+1}\Big{\\}}.\end{split}$
(4.48)

Thus, under assumption (4.48), we conclude that, for $i=1,2$,

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(k+1)}-V_{i}^{(k)})}{\partial
u}(\rho,u)\Big{|}\leq\begin{cases}\bar{C}_{2}\rho\,\nu^{k+1}\quad&\text{for
}\rho\leq\rho_{*},\\\
\tilde{C}_{2}\rho^{1+M_{2}}\,\varpi_{M_{2}}^{k+1}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho_{*},\\\ \hat{C}_{2}\rho\,\nu^{k+1}\quad&\text{for
}\rho\geq\rho_{*}.\end{cases}$

Combining (4.27) with (4.46)–(4.48) and taking

$\displaystyle\varpi_{M_{2}}=\nu,\qquad\bar{C}_{2}=\max\\{C_{2},\,C_{1}C_{2}\\},$
$\displaystyle\tilde{C}_{2}=\max\big{\\{}\bar{C}_{2}\rho_{*}^{-M_{2}},\tilde{C}_{1}C_{2}(1+M_{2})(\rho^{*})^{M_{1}-M_{2}-\theta_{2}}\big{\\}},\qquad\hat{C}_{2}=\max\big{\\{}\tilde{C}_{2}(\rho^{*})^{M_{2}},\hat{C}_{1}C_{2}\big{\\}},$

we have proved that, for any $n\geq 1$ and $i=1,2$,

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(n)}-V_{i}^{(n-1)})}{\partial
u}(\rho,u)\Big{|}\leq\begin{cases}\bar{C}_{2}\rho\,\nu^{n}\quad&\text{for
}\rho\leq\rho_{*},\\\ \tilde{C}_{2}\rho^{1+M_{2}}\,\nu^{n}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho_{*},\\\ \hat{C}_{2}\rho\,\nu^{n}\quad\,&\text{for
}\rho\geq\rho_{*}.\end{cases}$ (4.49)

Noting that $\nu<1$ and $\rho\leq\rho_{0}$ for $(\rho,u)\in\overline{\Sigma}$,
we know that $\big{\\{}\frac{\partial V_{i}^{(n)}}{\partial u}\big{\\}}$ is
uniformly convergent in $\overline{\Sigma}$. It is direct to check that the
limit function is $\frac{\partial V_{i}}{\partial u}$. Due to the continuity
and uniform convergence of $\\{\frac{\partial V_{i}^{(n)}}{\partial u}\\}$, it
is clear that $\frac{\partial V_{i}}{\partial u}$ is continuous in
$\overline{\Sigma}$.

On the other hand, it follows from (4.9) that

$\left\\{\begin{aligned} &\frac{\partial
V_{1}^{(n)}}{\partial\rho}=k^{\prime}(\rho)\frac{\partial
V_{1}^{(n)}}{\partial
u}-\frac{k^{\prime\prime}(\rho)}{2k^{\prime}(\rho)}\big{(}V_{1}^{(n-1)}+V_{2}^{(n-1)}\big{)},\\\
&\frac{\partial V_{2}^{(n)}}{\partial\rho}=-k^{\prime}(\rho)\frac{\partial
V_{2}^{(n)}}{\partial
u}-\frac{k^{\prime\prime}(\rho)}{2k^{\prime}(\rho)}\big{(}V_{1}^{(n-1)}+V_{2}^{(n-1)}\big{)},\end{aligned}\right.$

which, with (4.14), (4.28), and (4.49), yields that, for $k\geq 0$ and
$i=1,2$,

$\displaystyle\Big{|}\frac{\partial(V_{i}^{(n)}-V_{i}^{(n-1)})}{\partial\rho}(\rho,u)\Big{|}$
$\displaystyle\leq
k^{\prime}(\rho)\Big{|}\frac{\partial(V_{i}^{(n)}-V_{i}^{(n-1)})}{\partial
u}(\rho,u)\Big{|}+\Big{|}\frac{k^{\prime\prime}(\rho)}{2k^{\prime}(\rho)}\Big{|}\,\sum\limits_{j=1}^{2}\big{|}(V_{j}^{(n)}-V_{j}^{(n-1)})(\rho,u)\big{|}$
$\displaystyle\leq\begin{cases}C\rho^{\theta_{1}}\,\nu^{n}\qquad&\text{for
}\rho\leq\rho_{*},\\\ C\rho^{M_{1}}\,\nu^{n}\,\,\,&\text{for
}\rho_{*}\leq\rho\leq\rho_{*},\\\ C\rho^{\theta_{2}}\,\nu^{n}\qquad&\text{for
}\rho\geq\rho_{*},\end{cases}$ (4.50)

for some large constant $C>0$. Thus, $\frac{\partial
V_{i}^{(n)}}{\partial\rho}$ converges uniformly to $\frac{\partial
V_{i}}{\partial\rho}$ in $\overline{\Sigma}$. It is clear that $\frac{\partial
V_{i}}{\partial\rho}$ is continuous. Therefore,
$(V_{1}(\rho,u),V_{2}(\rho,u))$ is a $C^{1}$–solution of the Goursat problem
(4.4)–(4.5), which implies that $\hat{\eta}$ is a $C^{2}$–solution of (4.2).

4\. From (4.34) and (4.49), we obtain that, for $i=1,2$,

$\Big{|}\frac{\partial V_{i}}{\partial
u}(\rho,u)\Big{|}\leq\Big{|}\frac{\partial V_{i}^{(0)}}{\partial
u}(\rho,u)\Big{|}+\sum\limits_{n=1}^{\infty}\Big{|}\frac{\partial(V_{i}^{(n)}-V_{i}^{(n-1)})}{\partial
u}(\rho,u)\Big{|}\leq C\rho$ (4.51)

for $\rho\geq 0$ and $|u|\leq k(\rho)$. Similarly, using (4.1), we see that,
for $i=1,2$,

$\Big{|}\frac{\partial V_{i}}{\partial\rho}\Big{|}\leq
C\rho^{\theta(\rho)}\qquad\text{for $\rho\geq 0$ and $|u|\leq k(\rho)$}.$

Therefore, for $|u|\leq k(\rho)$, it follows from (4.3) and (4.51) that

$\displaystyle|\hat{\eta}_{uu}(\rho,u)|=|\partial_{u}V_{1}(\rho,u)-\partial_{u}V_{2}(\rho,u)|\leq
C\rho,\,\,\,|\hat{\eta}_{\rho
u}(\rho,u)|=|k^{\prime}(\rho)(\partial_{u}V_{1}(\rho,u)+\partial_{u}V_{2}(\rho,u))|\leq
C\rho^{\theta(\rho)}.$

If $\hat{\eta}_{m}$ is regarded as a function of $(\rho,u)$, we have

$|\hat{\eta}_{m\rho}(\rho,u)|\leq
C\rho^{\theta(\rho)-1},\quad|\hat{\eta}_{mu}(\rho,u)|\leq C\qquad\,\text{ for
}|u|\leq k(\rho).$

If $\hat{\eta}_{m}$ is regarded as a function of $(\rho,m)$, we see that, for
$|u|\leq k(\rho)$,

$|\hat{\eta}_{m\rho}(\rho,m)|=|\hat{\eta}_{m\rho}(\rho,u)+u\hat{\eta}_{mu}(\rho,u)|\leq
C\rho^{\theta(\rho)-1},\quad|\hat{\eta}_{mm}(\rho,m)|=|\rho^{-1}\hat{\eta}_{mu}(\rho,u)|\leq
C\rho^{-1}.$

5\. We now prove the uniqueness of $\hat{\eta}$, which is equivalent to the
uniqueness of solutions of (4.4)–(4.5) in the class of $C^{1}$–solutions
satisfying (4.29). Suppose that there exist two $C^{1}$ solutions
$(V_{1},V_{2})$ and $(\tilde{V}_{1},\tilde{V}_{2})$ of (4.4)–(4.5) satisfying
the uniform estimate (4.29). Then it follows from (4.9) that

$V_{i}(\rho,u)-\tilde{V}_{i}(\rho,u)=-\int_{\rho_{i}}^{\rho}\frac{k^{\prime\prime}(s)}{2k^{\prime}(s)}\,\sum\limits_{j=1}^{2}\big{(}V_{j}(s,u^{(i)}(s))-\tilde{V}_{j}(s,u^{(i)}(s))\big{)}\,\mathrm{d}s\qquad\mbox{for
$i=1,2$}.$ (4.52)

Applying the uniform estimates (4.29) and similar arguments as in
(4.28)–(4.52) yields

$\displaystyle\max_{(\rho,u)\in\mathbb{R}_{+}\times\mathbb{R}\atop|u|\leq
k(\rho)}\big{|}V_{i}(\rho,u)-\tilde{V}_{i}(\rho,u)\big{|}\leq\left\\{\begin{aligned}
&C\rho^{1+\theta_{1}}\,\varpi_{1}^{n}\quad\,\,\,\,\,\,\,\text{for
}\rho\leq\rho_{*},\\\ &C\rho^{1+M_{1}}\,\varpi_{M_{1}}^{n}\quad\,\text{for
}\rho_{*}\leq\rho\leq\rho^{*},\\\
&C\rho^{1+\theta_{2}}\,\varpi_{2}^{n}\qquad\,\text{for
}\rho\geq\rho^{*},\end{aligned}\right.$

for any $n\geq 0$, where $C\gg 1$ is independent of $n$. Taking $n\to\infty$,
we obtain that $V_{i}(\rho,u)\equiv\tilde{V}_{i}(\rho,u)$ for $|u|\leq
k(\rho)$ which, with (4.3) and $\hat{\eta}(0,u)\equiv 0$, yields the
uniqueness of $\hat{\eta}$.

6\. We now estimate the entropy flux $\hat{q}$. It follows from (2.12) that,
for all entropy pairs,

$q_{\rho}=u\eta_{\rho}+\rho k^{\prime}(\rho)^{2}\eta_{u},\qquad
q_{u}=\rho\eta_{\rho}+u\eta_{u}.$ (4.53)

Then there exists an entropy flux $\hat{q}(\rho,u)\in
C^{2}(\mathbb{R}_{+}\times\mathbb{R})$ corresponding to the special entropy
$\hat{\eta}$:

$\hat{q}(\rho,u)=\frac{1}{2}\rho|u|^{3}\pm\rho u(e(\rho)+\rho
e^{\prime}(\rho))\qquad\text{for }\pm u\geq k(\rho).$

It follows from (4.30) and (4.53) that
$|\hat{q}_{\rho}(\rho,u)|=|u\hat{\eta}_{\rho}+\rho
k^{\prime}(\rho)^{2}\hat{\eta}_{u}|\leq C\rho^{\gamma(\rho)+\theta(\rho)-1}$
for $|u|\leq k(\rho)$, which implies

$|\hat{q}(\rho,u)|=\Big{|}\int_{\bar{\rho}}^{\rho}\hat{q}_{\rho}(s,u)\,\mathrm{d}s+\hat{q}(\bar{\rho},u)\Big{|}\leq
C\rho^{\gamma(\rho)+\theta(\rho)}\qquad\text{for }|u|\leq k(\rho),$ (4.54)

where $(\bar{\rho},u)$ is the point satisfying $k(\bar{\rho})=|u|$.

For $|u|\leq k(\rho)$, using (4.31) and (4.54), we have
$|\hat{q}-u\hat{\eta}|\leq|\hat{q}|+|u||\hat{\eta}|\leq
C\rho^{\gamma(\rho)+\theta(\rho)}$. In region $\\{(\rho,u)\,:\,|u|\geq
k(\rho)\\}$, it is direct to check that all the estimates in Lemma 4.1 hold by
using (4.1). Therefore, the proof of Lemma 4.1 is now complete.
$\hfill\square$

###  4.2. Estimates of the weak entropy pairs

In order to show the compactness of the weak entropy dissipation measures
below, we now derive some estimates of the weak entropy pairs. To achieve
this, from (2.15)–(2.16), it requires to analyze the entropy kernel and
entropy flux kernel, respectively.

The entropy kernel $\chi=\chi(\rho,u,s)$ is a fundamental solution of the
entropy equation (2.14):

$\left\\{\begin{aligned}
\displaystyle&\chi_{\rho\rho}-\frac{P^{\prime}(\rho)}{\rho^{2}}\chi_{uu}=0,\\\
\displaystyle&\chi|_{\rho=0}=0,\quad\chi_{\rho}|_{\rho=0}=\delta_{u=s}.\end{aligned}\right.$
(4.55)

As pointed out in [10] that equation (4.55) is invariant under the Galilean
transformation, which implies that
$\chi(\rho,u,s)=\chi(\rho,u-s,0)=\chi(\rho,0,s-u)$. For simplicity, we write
it as $\chi(\rho,u,s)=\chi(\rho,u-s)$ below when no confusion arises.

The corresponding entropy flux kernel $\sigma(\rho,u,s)$ satisfies the Cauchy
problem for $\sigma-u\chi$:

$\left\\{\begin{aligned}
\displaystyle&(\sigma-u\chi)_{\rho\rho}-\frac{P^{\prime}(\rho)}{\rho^{2}}(\sigma-u\chi)_{uu}=\frac{P^{\prime\prime}(\rho)}{\rho}\chi_{u},\\\
\displaystyle&(\sigma-u\chi)|_{\rho=0}=0,\quad(\sigma-u\chi)_{\rho}|_{\rho=0}=0.\end{aligned}\right.$
(4.56)

We recall from [10] that $\sigma-u\chi$ is also Galilean invariant. From
(1.4)–(1.6), $P(\rho)$ satisfies all the conditions in [10, 11].

For later use, we introduce the definition of fractional derivatives (cf. [7,
10, 46]). For any real $\alpha>0$, the fractional derivative
$\partial_{s}^{\alpha}f$ of a function $f=f(s)$ is

$\partial_{s}^{\alpha}f(s)=\Gamma(-\alpha)f*[s]_{+}^{-\alpha-1},$

where $\Gamma(x)$ is the Gamma function and the convolution should be
understood in the sense of distributions. The following formula:

$\partial_{s}^{\alpha}(sg(s))=s\partial_{s}^{\alpha+1}g+(\alpha+1)\partial_{s}^{\alpha}g$

holds for fractional derivatives. We now present two useful lemmas for the
entropy kernel $\chi(\rho,u)$ and the entropy flux kernel $\sigma(\rho,u)$
when $\rho$ is bounded.

######  Lemma 4.2 ([10, Theorems 2.1–2.2]).

The entropy kernel $\chi(\rho,u)$ admits the expansion:

$\chi(\rho,u)=a_{1}(\rho)G_{\lambda_{1}}(\rho,u)+a_{2}(\rho)G_{\lambda_{1}+1}(\rho,u)+g_{1}(\rho,u)\qquad\text{for
$\rho\in[0,\infty)$},$ (4.57)

where $k(\rho)=\int_{0}^{\rho}\frac{\sqrt{P^{\prime}(y)}}{y}\,\mathrm{d}y$ and

$\displaystyle
G_{\lambda_{1}}(\rho,u)=[k(\rho)^{2}-u^{2}]_{+}^{\lambda_{1}},\qquad\lambda_{1}=\frac{3-\gamma_{1}}{2(\gamma_{1}-1)}>0,$
(4.58) $\displaystyle
a_{1}(\rho)=M_{\lambda_{1}}k(\rho)^{-\lambda_{1}}k^{\prime}(\rho)^{-\frac{1}{2}}>0,\quad\,M_{\lambda_{1}}=\left(\frac{2\lambda_{1}}{\sqrt{2\lambda_{1}+1}}\int_{-1}^{1}(1-z^{2})^{\lambda_{1}}\,\mathrm{d}z\right)^{-1},$
$\displaystyle
a_{2}(\rho)=-\frac{1}{4(\lambda_{1}+1)}k(\rho)^{-\lambda_{1}-1}k^{\prime}(\rho)^{-\frac{1}{2}}\int_{0}^{\rho}k(s)^{\lambda_{1}}k^{\prime}(s)^{-\frac{1}{2}}a_{1}^{\prime\prime}(s)\,\mathrm{d}s.$

Moreover, $\operatorname{supp}\chi(\rho,u)\subset\\{(\rho,u)\,:\,|u|\leq
k(\rho)\\}$, and $\chi(\rho,u)>0$ in $\\{(\rho,u)\,:\,|u|<k(\rho)\\}$. The
remainder term $g_{1}(\rho,\cdot)$ and its fractional derivative
$\partial_{u}^{\lambda_{1}+1}g_{1}(\rho,\cdot)$ are Hölder continuous.
Furthermore, for any fixed $\rho_{\max}>0$, there exists $C(\rho_{\max})>0$
depending only on $\rho_{\rm max}$ such that

$|g_{1}(\rho,u-s)|\leq
C(\rho_{\max})[k(\rho)^{2}-(u-s)^{2}]_{+}^{\lambda_{1}+\alpha_{0}+1},$ (4.59)

for any $0\leq\rho\leq\rho_{\max}$ and some $\alpha_{0}\in(0,1)$. In addition,
for any $0\leq\rho\leq\rho_{\max}$,

$|a_{1}(\rho)|+\rho^{1-2\theta_{1}}|a_{1}^{\prime}(\rho)|+\rho^{2-2\theta_{1}}|a_{1}^{\prime\prime}(\rho)|+|a_{2}(\rho)|+\rho|a_{2}^{\prime}(\rho)|+\rho^{2}|a_{2}^{\prime\prime}(\rho)|\leq
C(\rho_{\max}).$ (4.60)

Proof. Since (4.57)–(4.59) have been derived in [10, Theorem 2.2], it suffices
to prove (4.60). From (3.6), we find that $|a_{1}(\rho)|\leq C(\rho_{\max})$
for $0\leq\rho\leq\rho_{\max}$. For $|a_{1}^{\prime}(\rho)|$, a direct
calculation shows that

$a_{1}^{\prime}(\rho)=-\lambda_{1}M_{\lambda_{1}}k(\rho)^{-\lambda_{1}-1}k^{\prime}(\rho)^{\frac{1}{2}}-\frac{1}{2}M_{\lambda_{1}}k(\rho)^{-\lambda_{1}}k^{\prime}(\rho)^{-\frac{3}{2}}k^{\prime\prime}(\rho).$

It follows from (1.5) that
$k(\rho)=C_{1}\rho^{\theta_{1}}\big{(}1+O(\rho^{2\theta_{1}})\big{)}$ as
$\rho\in[0,\rho_{\max}]$ for some constant $C_{1}>0$ that may depend on
$\kappa_{1}$ and $\gamma_{1}$. Then, by direct calculation, we observe that
the term involving $\rho^{-1}$ in $a_{1}^{\prime}(\rho)$ vanishes so that
$|a_{1}^{\prime}(\rho)|\leq C(\rho_{\max})\rho^{2\theta_{1}-1}$. Similarly, we
obtain that $|a_{1}^{\prime\prime}(\rho)|\leq
C(\rho_{\max})\rho^{2\theta_{1}-2}$. Finally, using $\eqref{6.5-1}_{3}$, we
can obtain $\eqref{6.5}_{2}$ by a direct calculation. This completes the
proof. $\hfill\square$

######  Lemma 4.3 ([10, Theorem 2.3]).

The entropy flux kernel $\sigma(\rho,u)$ admits the expansion

$(\sigma-u\chi)(\rho,u)=-u\big{(}b_{1}(\rho)G_{\lambda_{1}}(\rho,u)+b_{2}(\rho)G_{\lambda_{1}+1}(\rho,u)\big{)}+g_{2}(\rho,u)\qquad\text{for
$\rho\in[0,\infty)$},$

where

$\displaystyle b_{1}(\rho)=M_{\lambda_{1}}\rho
k(\rho)^{-\lambda_{1}-1}k^{\prime}(\rho)^{\frac{1}{2}}>0,$ (4.61)
$\displaystyle\begin{aligned} b_{2}(\rho)&=-\frac{1}{4(\lambda_{1}+1)}\rho
k^{\prime}(\rho)^{\frac{1}{2}}k(\rho)^{-(\lambda_{1}+2)}\int_{0}^{\rho}k(s)^{\lambda_{1}}k^{\prime}(s)^{-\frac{1}{2}}a_{1}^{\prime\prime}(s)\,\mathrm{d}s\\\
&\quad-\frac{1}{4(\lambda_{1}+1)}k(\rho)^{-(\lambda_{1}+2)}k^{\prime}(\rho)^{-\frac{1}{2}}\int_{0}^{\rho}k(s)^{\lambda_{1}+1}k^{\prime}(s)^{-\frac{1}{2}}b_{1}^{\prime\prime}(s)\,\mathrm{d}s\\\
&\quad+\frac{1}{4(\lambda_{1}+1)}k(\rho)^{-(\lambda_{1}+2)}k^{\prime}(\rho)^{-\frac{1}{2}}\int_{0}^{\rho}sk(s)^{\lambda_{1}}k^{\prime}(s)^{\frac{1}{2}}a_{1}^{\prime\prime}(s)\,\mathrm{d}s.\end{aligned}$

The remainder term $g_{2}(\rho,\cdot)$ and its fractional derivative
$\partial_{u}^{\lambda_{1}+1}g_{2}(\rho,\cdot)$ are Hölder continuous.
Moreover, for any fixed $\rho_{\max}>0$, there exists $C(\rho_{\max})>0$
depending only on $\rho_{max}$ such that

$|g_{2}(\rho,u-s)|\leq
C(\rho_{\max})[k(\rho)^{2}-(u-s)^{2}]_{+}^{\lambda_{1}+\alpha_{0}+1},$

for any $0\leq\rho\leq\rho_{\max}$ and some $\alpha_{0}\in(0,1)$. Furthermore,
similar to the proof of (4.60), for any $0\leq\rho\leq\rho_{\max}$,

$|b_{1}(\rho)|+\rho^{1-2\theta_{1}}|b_{1}^{\prime}(\rho)|+\rho^{2-2\theta_{1}}|b_{2}^{\prime\prime}(\rho)|+|b_{2}(\rho)|+\rho|b_{2}^{\prime}(\rho)|+\rho^{2}|b_{2}^{\prime\prime}(\rho)|\leq
C(\rho_{\max}).$ (4.62)

######  Remark 4.1.

In [10, Theorem 2.2], it is proved that $a_{2}(\rho)$ and $b_{2}(\rho)$
satisfy $|a_{2}(\rho)|+|b_{2}(\rho)|\leq C\rho k(\rho)^{-2}$ for the pressure
law given in [10, (2.1)]. In this paper, we have improved them to be
$\eqref{6.5}$ and $\eqref{6.8}$ under conditions (1.4)–(1.6).

For later use, we recall a useful representation formula for $\chi(\rho,u)$.

######  Lemma 4.4 (First representation formula, [62, Lemma 3.4]).

Given any $(\rho,u)$ with $|u|\leq k(\rho)$ and $0\leq\rho_{0}<\rho$,

$\displaystyle\chi(\rho,u)=$
$\displaystyle\,\frac{1}{2(\rho-\rho_{0})k^{\prime}(\rho)}\int_{\rho_{0}}^{\rho}k^{\prime}(s)\,\tilde{d}(s)\big{(}\chi(s,u+k(\rho)-k(s))+\chi(s,u-k(\rho)+k(s))\big{)}\,\mathrm{d}s$
$\displaystyle\,-\frac{1}{2(\rho-\rho_{0})k^{\prime}(\rho)}\int_{-(k(\rho)-k(\rho_{0}))}^{k(\rho)-k(\rho_{0})}\chi(\rho_{0},u-s)\,\mathrm{d}s,$

where
$\,\tilde{d}(\rho):=2+(\rho-\rho_{0})\frac{k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}.$

######  Remark 4.2.

In the statement of [62, Lemma 3.4], $\rho_{0}$ is positive. However, the
proof of [62, Lemma 3.4] is also valid for $\rho_{0}=0$ without modification;
see also [10, (3.38)].

Given any $\psi\in C_{0}^{2}(\mathbb{R})$, a regular weak entropy pair
$(\eta^{\psi},\,q^{\psi})$ can be given by

$\displaystyle\eta^{\psi}(\rho,u)=\int_{\mathbb{R}}\psi(s)\,\chi(\rho,u,s)\,\mathrm{d}s,\qquad
q^{\psi}(\rho,u)=\int_{\mathbb{R}}\psi(s)\,\sigma(\rho,u,s)\,\mathrm{d}s.$
(4.63)

It follows from (4.55) that

$\left\\{\begin{aligned}
\displaystyle&\eta_{\rho\rho}^{\psi}-k^{\prime}(\rho)^{2}\eta_{uu}^{\psi}=0,\\\
\displaystyle&\eta^{\psi}|_{\rho=0}=0,\quad\eta_{\rho}^{\psi}|_{\rho=0}=\psi(u).\end{aligned}\right.$
(4.64)

Using Lemmas 4.2–4.4, we can obtain the following lemma for the weak entropy
pair $(\eta^{\psi},q^{\psi})$.

######  Lemma 4.5.

For any weak entropy $(\eta^{\psi},q^{\psi})$ defined in (4.63), there exists
a constant $C_{\psi}>0$ depending only on $\rho^{*}$ and $\psi$ such that, for
all $\rho\in[0,2\rho^{*}]$,

$|\eta^{\psi}(\rho,u)|+|q^{\psi}(\rho,u)|\leq C_{\psi}\rho.$

If $\eta^{\psi}$ is regarded as a function of $(\rho,m)$, then

$\displaystyle|\eta_{m}^{\psi}(\rho,m)|+|\rho\eta_{mm}^{\psi}(\rho,m)|\leq
C_{\psi},\qquad|\eta_{\rho}^{\psi}(\rho,m)|\leq
C_{\psi}(1+\rho^{\theta_{1}}).$

Moreover, if $\eta_{m}^{\psi}$ is regarded as a function of $(\rho,u)$, then

$|\eta_{mu}^{\psi}(\rho,u)|+|\rho^{1-\theta_{1}}\eta_{m\rho}^{\psi}(\rho,u)|\leq
C_{\psi}.$

Proof. All the estimates can be found in [61, Lemma 3.8] or [62, Lemma 4.13]
except the estimate of $\eta_{\rho}^{\psi}(\rho,m)$. In fact, applying Lemma
4.4 to (4.64) and using $\displaystyle d(\rho):=2+\frac{\rho
k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}$, we have

$\displaystyle\eta^{\psi}(\rho,u)=$ $\displaystyle\,\frac{1}{2\rho
k^{\prime}(\rho)}\int_{0}^{\rho}k^{\prime}(s)\,d(s)\,\eta^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s$
$\displaystyle\,+\frac{1}{2\rho
k^{\prime}(\rho)}\int_{0}^{\rho}k^{\prime}(s)\,d(s)\,\eta^{\psi}(s,u-k(\rho)+k(s))\,\mathrm{d}s:=I_{1}+I_{2}.$
(4.65)

We regard $\eta^{\psi}$ as a function of $(\rho,m)$. Then we have

$\partial_{\rho}\eta^{\psi}(\rho,m)=\partial_{\rho}\eta^{\psi}(\rho,u)-\frac{u}{\rho}\partial_{u}\eta^{\psi}(\rho,u).$
(4.66)

Without loss of generality, we assume $\operatorname{supp}\psi\subset[-L,L]$
for some $L>0$. Then a direct calculation shows that $\eta^{\psi}(\rho,u)=0$
if $|u|\geq k(\rho)+L$. Noticing
$\eta_{u}^{\psi}(\rho,u)=\rho\eta_{m}^{\psi}(\rho,m)$, we have

$\Big{|}\frac{u}{\rho}\partial_{u}\eta^{\psi}(\rho,u)\Big{|}\leq|u|\,|\eta_{m}^{\psi}(\rho,m)|\leq
C_{\psi}(1+\rho^{\theta_{1}})\qquad\mbox{for }0\leq\rho\leq 2\rho^{*}.$ (4.67)

Thus, it suffices to calculate $\partial_{\rho}\eta^{\psi}(\rho,u)$. It
follows from (4.65) that
$\partial_{\rho}\eta^{\psi}(\rho,u)=\partial_{\rho}I_{1}+\partial_{\rho}I_{2}$.
A direct calculation shows that

$\displaystyle\partial_{\rho}I_{1}$
$\displaystyle=\frac{1}{2}\big{(}-\rho^{-2}(k^{\prime}(\rho))^{-1}-\rho^{-1}(k^{\prime}(\rho))^{-2}k^{\prime\prime}(\rho)\big{)}\int_{0}^{\rho}k^{\prime}(s)\,d(s)\,\eta^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s$
$\displaystyle\quad+\frac{1}{2\rho}\int_{0}^{\rho}k^{\prime}(s)\,d(s)\,\eta_{u}^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s+\frac{1}{2\rho}d(\rho)\eta^{\psi}(\rho,u).$
(4.68)

Using (3.6) and Lemma 3.2, we obtain that

$|\partial_{\rho}I_{1}|\leq C+C_{\psi}\rho^{\theta_{1}}+C_{\psi}\leq
C_{\psi}(1+\rho^{\theta_{1}})\qquad\,\,\mbox{for $0\leq\rho\leq 2\rho^{*}$},$

which, with (4.68), yields $|\partial_{\rho}I_{1}|\leq
C_{\psi}(1+\rho^{\theta_{1}})$. Similarly, we obtain that
$|\partial_{\rho}I_{2}|\leq C_{\psi}(1+\rho^{\theta_{1}})$. Thus, we conclude
that
$|\partial_{\rho}\eta^{\psi}(\rho,u)|\leq|\partial_{\rho}I_{1}|+|\partial_{\rho}I_{2}|\leq
C_{\psi}(1+\rho^{\theta_{1}}),$ which, with (4.66)–(4.67), implies that
$|\partial_{\rho}\eta^{\psi}(\rho,m)|\leq C_{\psi}(1+\rho^{\theta_{1}}).$ This
completes the proof. $\hfill\square$

We notice that all the above estimates for the weak entropy pairs in Lemmas
4.2–4.5 hold when the density is bounded. To establish the $L^{p}$-compensated
compactness framework, we need the entropy pair estimates when the density is
large, namely $\rho\geq\rho^{*}$. From now on in this subsection, we use the
representation formula of Lemma 4.4 to estimate $(\eta^{\psi},q^{\psi})$ in
the large density region $\rho\geq\rho^{*}$.

######  Lemma 4.6.

There exists a positive constant $C>0$ depending only on $\rho^{*}$ such that

$\|\chi(\rho,\cdot)\|_{L_{u}^{\infty}}\leq C\rho\qquad\text{for
$\rho\geq\rho^{*}$}.$

Proof. For $\rho\geq\rho^{*}$, $\chi(\rho,u)$ satisfies

$\left\\{\begin{aligned}
&\displaystyle\chi_{\rho\rho}-k^{\prime}(\rho)^{2}\chi_{uu}=0,\\\
&\displaystyle\chi(\rho,u)|_{\rho=\rho^{*}}=\chi(\rho^{*},u),\quad\chi_{\rho}(\rho,u)|_{\rho=\rho^{*}}=\chi_{\rho}(\rho^{*},u).\end{aligned}\right.$

where $\chi(\rho^{*},u)$ and $\chi_{\rho}(\rho^{*},u)$ are given in Lemma 4.2.
Then, applying Lemma 4.4, we obtain that, for $\rho>\rho^{*}$,

$\displaystyle\|k^{\prime}(\rho)\chi(\rho,\cdot)\|_{L_{u}^{\infty}}$
$\displaystyle\leq\frac{1}{\rho-\rho^{*}}\int_{\rho^{*}}^{\rho}d_{*}(s)\|k^{\prime}(s)\chi(s,\cdot)\|_{L_{u}^{\infty}}\,\mathrm{d}s+\frac{1}{2(\rho-\rho^{*})}\int_{-(k(\rho)-k(\rho^{*}))}^{k(\rho)-k(\rho^{*})}|\chi(\rho^{*},u-s)|\,\mathrm{d}s$
$\displaystyle\leq\frac{1}{\rho-\rho^{*}}\int_{\rho^{*}}^{\rho}d_{*}(s)\|k^{\prime}(s)\chi(s,\cdot)\|_{L_{u}^{\infty}}\,\mathrm{d}s+C\rho^{\theta_{2}-1},$

where
$d_{*}(\rho):=2+(\rho-\rho^{*})\frac{k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}$.
By a similar proof to that for Lemma A.3, we have

$\|k^{\prime}(\rho)\chi(\rho,\cdot)\|_{L_{u}^{\infty}}\leq
C\rho^{\theta_{2}}\qquad\text{for $\rho\geq 2\rho^{*}$},$

which, with (3.7), yields that $\|\chi(\rho,\cdot)\|_{L_{u}^{\infty}}\leq
C\rho$ for $\rho\geq 2\rho^{*}$. For $\rho^{*}\leq\rho\leq 2\rho^{*}$, it
follows from Lemma 4.2 that $\|\chi(\rho,\cdot)\|_{L_{u}^{\infty}}\leq C\leq
C\rho$. $\hfill\square$

######  Lemma 4.7.

Let $\rho\geq\rho^{*}$ and $\psi\in C_{0}^{2}(\mathbb{R})$. Then, in the
$(\rho,u)$–coordinates,

$|\eta^{\psi}(\rho,u)|+|\eta_{u}^{\psi}(\rho,u)|+|\eta_{uu}^{\psi}(\rho,u)|\leq
C_{\psi}\rho,\quad|\eta_{\rho}^{\psi}(\rho,u)|+\rho^{1-\theta_{2}}|\eta_{\rho\rho}^{\psi}(\rho,u)|\leq
C_{\psi}\rho^{\theta_{2}}.$

In the $(\rho,m)$–coordinates,

$|\eta^{\psi}_{\rho}(\rho,m)|+\rho^{\theta_{2}}|\eta_{m}^{\psi}(\rho,m)|+\rho^{1+\theta_{2}}|\eta_{mm}^{\psi}(\rho,m)|\leq
C_{\psi}\rho^{\theta_{2}}.$

If we regard $\eta^{\psi}_{m}(\rho,m)$ as a function of $(\rho,u)$, then

$|\eta_{mu}^{\psi}|+\rho^{1-\theta_{2}}|\eta_{m\rho}^{\psi}|\leq C_{\psi}.$

All the above constants $C_{\psi}>0$ depend only on $\|\psi\|_{C^{2}}$ and
$\operatorname{supp}\psi$.

Proof. We divide the proof into five steps.

1\. Using (4.63) and Lemma 4.6, we obtain that, for $\rho\geq\rho^{*}$,

$|{\eta}^{\psi}(\rho,u)|+|{\eta}_{u}^{\psi}(\rho,u)|+|{\eta}_{uu}^{\psi}(\rho,u)|\leq\|{\chi}(\rho,\cdot)\|_{L^{\infty}(\mathbb{R})}\|(\psi,\psi^{\prime},\psi^{\prime\prime})\|_{L^{1}(\mathbb{R})}\leq
C_{\psi}\rho.$ (4.69)

2\. For the estimate of $\eta_{\rho}^{\psi}(\rho,u)$, the proof is similar to
Lemma 4.5. Indeed, $\eta^{\psi}$ satisfies

$\left\\{\begin{aligned}
\displaystyle&\eta_{\rho\rho}^{\psi}-k^{\prime}(\rho)^{2}\eta_{uu}^{\psi}=0,\\\
\displaystyle&\eta^{\psi}(\rho,u)|_{\rho=\rho^{*}}=\eta^{\psi}(\rho^{*},u),\quad\eta_{\rho}^{\psi}(\rho,u)|_{\rho=\rho^{*}}=\eta_{\rho}^{\psi}(\rho^{*},u).\end{aligned}\right.$
(4.70)

It follows from (4.70) and Lemma 4.4 that

$\displaystyle\eta^{\psi}(\rho,u)$
$\displaystyle=\frac{1}{2(\rho-\rho^{*})k^{\prime}(\rho)}\int_{\rho^{*}}^{\rho}d_{*}(s)k^{\prime}(s)\big{(}\eta^{\psi}(s,u+k(\rho)-k(s))-\eta^{\psi}(s,u-k(\rho)+k(s))\big{)}\,\mathrm{d}s$
$\displaystyle\quad-\frac{1}{2(\rho-\rho^{*})k^{\prime}\left(\rho\right)}\int_{-(k(\rho)-k(\rho^{*}))}^{k(\rho)-k(\rho^{*})}\eta^{\psi}(\rho^{*},u-s)\,\mathrm{d}s,$
(4.71)

where
$d_{*}(\rho)=2+(\rho-\rho^{*})\frac{k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}$
and $0<d_{*}(\rho)\leq 3$ for $\rho\geq\rho^{*}$ from (3.11). Then, following
the similar arguments as in the proof of Lemma 4.5, we can obtain that
$|\eta_{\rho}^{\psi}(\rho,u)|\leq C_{\psi}\rho^{\theta_{2}}$ for $\rho\geq
2\rho^{*}$. Moreover, from Lemma 4.5, $|\eta_{\rho}^{\psi}(\rho,u)|\leq
C_{\psi}\leq C_{\psi}\rho^{\theta_{2}}$ for $\rho\in[\rho^{*},2\rho^{*}]$ so
that

$|\eta_{\rho}^{\psi}(\rho,u)|\leq C_{\psi}\rho^{\theta_{2}}\qquad\text{for
}\rho\geq\rho^{*}.$ (4.72)

3\. For $\eta_{\rho\rho}^{\psi}(\rho,u)$, it follows from (4.69)–(4.70) that

$|\eta_{\rho\rho}^{\psi}(\rho,u)|\leq|k^{\prime}(\rho)^{2}|\,|\eta_{uu}^{\psi}(\rho,u)|\leq
C_{\psi}\rho^{2\theta_{2}-1}=C_{\psi}\rho^{\gamma_{2}-2}\qquad\text{ for
}\rho\geq\rho^{*}.$

4\. In the $(\rho,m)$–coordinates, it is clear that

$\displaystyle\eta_{m}^{\psi}(\rho,m)=\rho^{-1}\eta_{u}^{\psi}(\rho,u),\quad\eta_{mm}^{\psi}(\rho,m)=\rho^{-2}\eta_{uu}^{\psi}(\rho,u),\quad\eta_{\rho}^{\psi}(\rho,m)=\eta_{\rho}^{\psi}(\rho,u)-\frac{m}{\rho^{2}}\eta_{u}^{\psi}(\rho,u).$

On the other hand, if $\eta_{m}^{\psi}$ is regarded as a function $(\rho,u)$,
it is direct to obtain

$\eta_{mu}^{\psi}(\rho,u)=\partial_{u}\big{(}\rho^{-1}\eta_{u}^{\psi}(\rho,u)\big{)}=\rho^{-1}\eta_{uu}^{\psi}(\rho,u).$

Thus, using (4.69) and (4.72),

$\displaystyle|\eta_{m}^{\psi}(\rho,m)|+|\eta_{mu}^{\psi}(\rho,u)|+\rho|\eta_{mm}^{\psi}(\rho,m)|\leq
C_{\psi}\rho^{-1},$
$\displaystyle|\eta_{\rho}^{\psi}(\rho,m)|\leq|\eta_{\rho}^{\psi}(\rho,u)|+\frac{|m|}{\rho^{2}}\big{|}\eta_{u}^{\psi}(\rho,u)\big{|}\leq
C_{\psi}\rho^{\theta_{2}}+C_{\psi}(L+k(\rho))\leq C_{\psi}\rho^{\theta_{2}},$

where we have used that $\operatorname{supp}\psi\subset[-L,L]$ and
$\eta_{u}^{\psi}(\rho,u)=\rho\eta_{m}^{\psi}(\rho,m)$.

5\. For the estimates of
$\eta_{m\rho}^{\psi}(\rho,u)=\partial_{\rho}\eta_{m}^{\psi}(\rho,u)$, it
follows from (4.71) that

$\displaystyle\eta_{m}^{\psi}(\rho,m)=\frac{1}{\rho}\eta_{u}^{\psi}(\rho,u)$
$\displaystyle=\frac{1}{2\rho(\rho-\rho^{*})k^{\prime}(\rho)}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{u}^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s$
$\displaystyle\quad+\frac{1}{2\rho(\rho-\rho^{*})k^{\prime}(\rho)}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{u}^{\psi}(s,u-k(\rho)+k(s))\,\mathrm{d}s$
$\displaystyle\quad-\frac{1}{2\rho(\rho-\rho^{*})k^{\prime}(\rho)}\int_{-(k(\rho)-k(\rho^{*}))}^{k(\rho)-k(\rho^{*})}\eta_{u}^{\psi}(\rho^{*},u-s)\,\mathrm{d}s$
$\displaystyle:=J_{1}+J_{2}+J_{3},$ (4.73)

and
$\partial_{\rho}\eta_{m}^{\psi}(\rho,u)=\partial_{\rho}J_{1}+\partial_{\rho}J_{2}+\partial_{\rho}J_{3}$.

A direct calculation shows that

$\displaystyle\partial_{\rho}J_{1}$
$\displaystyle=\partial_{\rho}\Big{(}\frac{1}{2\rho(\rho-\rho^{*})k^{\prime}(\rho)}\Big{)}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{u}^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s$
$\displaystyle\quad+\frac{1}{2\rho(\rho-\rho^{*})}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{uu}^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s+\frac{1}{2\rho(\rho-\rho^{*})}d_{*}(\rho)\,\eta_{u}^{\psi}(\rho,u)$
$\displaystyle:=J_{1,1}+J_{1,2}+J_{1,3},$ (4.74)
$\displaystyle\partial_{\rho}J_{2}$
$\displaystyle=\partial_{\rho}\Big{(}\frac{1}{2\rho(\rho-\rho^{*})k^{\prime}(\rho)}\Big{)}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{u}^{\psi}(s,u-k(\rho)+k(s))\,\mathrm{d}s$
$\displaystyle\quad-\frac{1}{2\rho(\rho-\rho^{*})}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{uu}^{\psi}(s,u-k(\rho)+k(s))\,\mathrm{d}s+\frac{1}{2\rho(\rho-\rho^{*})}d_{*}(\rho)\eta_{u}^{\psi}(\rho,u)$
$\displaystyle:=J_{2,1}+J_{2,2}+J_{2,3}.$ (4.75)

Clearly, we have

$\Big{|}\partial_{\rho}\Big{(}\frac{1}{2\rho(\rho-\rho^{*})k^{\prime}(\rho)}\Big{)}\Big{|}=\frac{1}{2}\Big{|}\frac{\rho-\rho^{*}}{2\rho^{2}\,(\rho-\rho^{*})^{2}\,k^{\prime}(\rho)}+\frac{k^{\prime\prime}(\rho)}{\rho\,(\rho-\rho^{*})\,k^{\prime}(\rho)^{2}}\Big{|}\leq\frac{C}{(\rho-\rho^{*})^{2}\rho^{\theta_{2}}},$

which, with (4.69) and $0<d_{*}(\rho)\leq 3$ for $\rho\geq\rho^{*}$, yields

$\displaystyle|J_{1,1}+J_{2,1}|$
$\displaystyle=\Big{|}\partial_{\rho}\Big{(}\frac{1}{2\rho\,(\rho-\rho^{*})\,k^{\prime}(\rho)}\Big{)}\int_{\rho^{*}}^{\rho}k^{\prime}(s)\,d_{*}(s)\,\eta_{u}^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s\Big{|}$
(4.76)
$\displaystyle\quad+\Big{|}\partial_{\rho}\Big{(}\frac{1}{2\rho\,(\rho-\rho^{*})\,k^{\prime}(\rho)}\Big{)}\int_{\rho^{*}}^{\rho}k^{\prime}(s)\,d_{*}(s)\,\eta_{u}^{\psi}(s,u-k(\rho)+k(s))\,\mathrm{d}s\Big{|}$
$\displaystyle\leq\frac{C_{\psi}}{(\rho-\rho^{*})^{2}\,\rho^{\theta_{2}}}\int_{\rho^{*}}^{\rho}s^{\theta_{2}}\,\mathrm{d}s\leq\frac{C_{\psi}}{\rho-\rho^{*}}.$

It follows from (4.69) and $0<d_{*}(\rho)\leq 3$ for $\rho\geq\rho^{*}$ that

$\displaystyle|J_{1,2}|+|J_{2,2}|$
$\displaystyle=\Big{|}\frac{1}{2\rho(\rho-\rho^{*})}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{uu}^{\psi}(s,u+k(\rho)-k(s))\,\mathrm{d}s\Big{|}$
(4.77)
$\displaystyle\quad+\Big{|}\frac{1}{2\rho(\rho-\rho^{*})}\int_{\rho^{*}}^{\rho}d_{*}(s)\,k^{\prime}(s)\,\eta_{uu}^{\psi}(s,u-k(\rho)+k(s))\,\mathrm{d}s\Big{|}$
$\displaystyle\leq\frac{C_{\psi}}{\rho(\rho-\rho^{*})}\int_{\rho^{*}}^{\rho}s^{\theta_{2}}\,\mathrm{d}s\leq\frac{C_{\psi}}{\rho(\rho-\rho^{*})}\rho^{\theta_{2}}(\rho-\rho^{*})\leq
C_{\psi}\rho^{\theta_{2}-1}.$

For $J_{1,3}+J_{2,3}$, it is direct to see that

$|J_{1,3}+J_{2,3}|\leq\Big{|}\frac{1}{\rho(\rho-\rho^{*})}\,d_{*}(\rho)\,\eta_{u}^{\psi}(\rho,u)\Big{|}\leq\frac{C_{\psi}}{\rho-\rho^{*}}.$
(4.78)

For $\partial_{\rho}J_{3}$, we notice that

$\displaystyle\partial_{\rho}J_{3}$
$\displaystyle=-\partial_{\rho}\Big{(}\frac{1}{2\rho\,(\rho-\rho^{*})\,k^{\prime}(\rho)}\Big{)}\int_{-(k(\rho)-k(\rho^{*}))}^{k(\rho)-k(\rho^{*})}\eta_{u}^{\psi}(\rho^{*},u-s)\,\mathrm{d}s$
$\displaystyle\quad\,-\frac{1}{2\rho\,(\rho-\rho^{*})}\,\big{(}\eta_{u}^{\psi}(\rho^{*},u-k(\rho)+k(\rho^{*}))+\eta_{u}^{\psi}(\rho^{*},u+k(\rho)-k(\rho^{*}))\big{)},$

which, with $0<\theta_{2}\leq 1$, yields

$\displaystyle|\partial_{\rho}J_{3}|\leq\frac{C_{\psi}\rho^{*}}{(\rho-\rho^{*})^{2}}\,|k(\rho)-k(\rho^{*})|+\frac{C_{\psi}\rho^{*}}{\rho\,(\rho-\rho^{*})}\leq\frac{C_{\psi}}{\rho-\rho^{*}}(1+\rho^{\theta_{2}-1})\leq\frac{C_{\psi}}{\rho-\rho^{*}}.$
(4.79)

Combining (4.74)–(4.79) with (4.73) yields that
$|\eta_{m\rho}^{\psi}(\rho,u)|\leq C_{\psi}\rho^{\theta_{2}-1}$ for $\rho\geq
2\rho^{*}$. For $\rho^{*}\leq\rho\leq 2\rho^{*}$, it follows from Lemma 4.5
that $|\eta_{m\rho}^{\psi}(\rho,u)|\leq C_{\psi}\rho^{\theta_{1}-1}\leq
C_{\psi}$ for $\rho^{*}\leq\rho\leq 2\rho^{*}$. Thus, we obtain
$|\eta_{m\rho}^{\psi}(\rho,u)|\leq C_{\psi}\rho^{\theta_{2}-1}$ for
$\rho\geq\rho^{*}$. $\hfill\square$

We now estimate $q^{\psi}$ for $\rho\geq\rho^{*}$. It follows from (4.56) that
$h:=\sigma-u\chi$ satisfies

$\left\\{\begin{aligned}
\displaystyle&h_{\rho\rho}-k^{\prime}(\rho)^{2}h_{uu}=\frac{P^{\prime\prime}(\rho)}{\rho}\chi_{u},\\\
\displaystyle&h(\rho^{*},u)=(\sigma-u\chi)(\rho^{*},u),\quad
h_{\rho}(\rho^{*},u)=(\sigma-u\chi)_{\rho}(\rho^{*},u),\end{aligned}\right.$

where $(\sigma-u\chi)(\rho^{*},u)$ and $(\sigma-u\chi)_{\rho}(\rho^{*},u)$ are
given by Lemma 4.3. Similar to Lemma 4.4, we have the following representation
formula for $h$.

######  Lemma 4.8 (Second representation formula [62, Lemmas 3.4 and 3.9]).

For any $(\rho,u)$ with $|u|\leq k(\rho)$ and $\rho>\rho^{*}$,

$\displaystyle h(\rho,u)=$
$\displaystyle\,\frac{1}{2(\rho-\rho^{*})k^{\prime}(\rho)}\int_{\rho^{*}}^{\rho}k^{\prime}(s)d_{*}(s)\big{(}h(s,u+k(\rho)-k(s))+h(s,u-k(\rho)+k(s))\big{)}\,\mathrm{d}s$
$\displaystyle+\frac{1}{2(\rho-\rho^{*})k^{\prime}(\rho)}\int_{\rho^{*}}^{\rho}(s-\rho^{*})\frac{P^{\prime\prime}(s)}{s}\big{(}\chi(s,u+k(\rho)-k(s))-\chi(s,u-k(\rho)+k(s))\big{)}\,\mathrm{d}s$
$\displaystyle-\frac{1}{2(\rho-\rho^{*})k^{\prime}(\rho)}\int_{-(k(\rho)-k(\rho^{*}))}^{k(\rho)-k(\rho^{*})}h(\rho^{*},u-s)\,\mathrm{d}s,$
(4.80)

where
$d_{*}(\rho)=2+(\rho-\rho^{*})\frac{k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}$.

######  Lemma 4.9.

There exists a constant $C>0$ depending only on $\rho^{*}$ such that

$\|(\sigma-u\chi)(\rho,u)\|_{L_{u}^{\infty}}\leq
C\rho^{1+\theta_{2}}\qquad\,\,\text{for }\rho\geq\rho^{*}.$

Proof. It follows from (3.3), (4.8), and Lemma 4.6 that

$\displaystyle\|k^{\prime}(\rho)h(\rho,\cdot)\|_{L_{u}^{\infty}}$
$\displaystyle\leq\frac{1}{\rho-\rho^{*}}\int_{\rho^{*}}^{\rho}d_{*}(s)\|k^{\prime}(s)h(s,\cdot)\|_{L_{u}^{\infty}}\,\mathrm{d}s+\frac{C}{\rho-\rho^{*}}\int_{\rho^{*}}^{\rho}s^{\gamma_{2}-1}\,\mathrm{d}s+C\rho^{\theta_{2}-1}$
$\displaystyle\leq\frac{1}{\rho-\rho^{*}}\int_{\rho^{*}}^{\rho}d_{*}(s)\|k^{\prime}(s)h(s,\cdot)\|_{L_{u}^{\infty}}\,\mathrm{d}s+C\rho^{2\theta_{2}},$

which, with (3.7) and a similar proof to that for Lemma A.3, yields

$\|k^{\prime}(\rho)h(\rho,\cdot)\|_{L_{u}^{\infty}}\leq
C\rho^{2\theta_{2}}\,\Longrightarrow\,\|h(\rho,\cdot)\|_{L_{u}^{\infty}}\leq
C\rho^{1+\theta_{2}}\qquad\text{for }\rho\geq 2\rho^{*}.$

For $\rho^{*}\leq\rho\leq 2\rho^{*}$, it follow from Lemma 4.3 that
$\|h(\rho,\cdot)\|_{L_{u}^{\infty}}\leq C\leq C\rho^{1+\theta_{2}}$.
$\hfill\square$

######  Lemma 4.10.

For $\rho\geq\rho^{*}$ and $\psi\in C_{0}^{2}(\mathbb{R})$,

$|q^{\psi}(\rho,u)|\leq C_{\psi}\rho^{1+\theta_{2}}.$ (4.81)

Proof. Recall that

$\displaystyle
q^{\psi}(\rho,u)=\int_{\mathbb{R}}\big{(}\sigma(\rho,u,s)-u\chi(\rho,u-s)\big{)}\psi(s)\,\mathrm{d}s+u\int_{\mathbb{R}}\chi(\rho,u-s)\psi(s)\,\mathrm{d}s:=h^{\psi}(\rho,u)+u\,\eta^{\psi}(\rho,u).$
(4.82)

It follows from Lemma 4.9 that

$|h^{\psi}(\rho,u)|\leq
C\|(\sigma-u\chi)(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\|\psi\|_{L^{1}(\mathbb{R})}\leq
C_{\psi}\rho^{1+\theta_{2}}.$ (4.83)

Since there exists $L>0$ such that $\operatorname{supp}\psi\subset[-L,L]$,
then it follows from Lemma 4.7 that
$|u\eta^{\psi}(\rho,u)|\leq(k(\rho)+L)|\eta^{\psi}(\rho,u)|\leq
C_{\psi}\rho^{1+\theta_{2}}$ for $\rho\geq\rho^{*}$, which, with
(4.82)–(4.83), yields (4.81). $\hfill\square$

###  4.3. Singularities of the entropy kernel and the entropy flux kernel

As indicated in [10, 61, 62], understanding the singularities of the entropy
kernel and the entropy flux kernel is essential for the reduction of the Young
measure. Thus, it requires some detailed estimates of the singularities of the
entropy kernel and the entropy flux kernel. The arguments in this subsection
are similar to [62, §6], the main difference is that a more subtle Grönwall
inequality (see Lemma A.3) is needed to obtain the desired estimates of the
singularities.

######  Lemma 4.11.

For $\rho\geq\rho^{*}$, the coefficient functions $a_{1}(\rho)$ and
$a_{2}(\rho)$ and the remainder term $g_{1}(\rho,u)$ in Lemma 4.2 satisfy

$\displaystyle|a_{1}(\rho)|+\rho^{\theta_{2}}|a_{2}(\rho)|\leq
C\rho^{\frac{1}{2}-\frac{\theta_{2}}{2\theta_{1}}},\qquad\,\,\|g_{1}(\rho,u)\|_{L_{u}^{\infty}(\mathbb{R})}\leq\begin{cases}C\rho\quad&\text{if
$\theta_{2}<\theta_{1}$},\\\ C\rho\ln\rho\,\,\,&\text{if
$\theta_{2}=\theta_{1}$},\end{cases}$
$\displaystyle\|\partial_{u}g_{1}(\rho,u)\|_{L_{u}^{\infty}(\mathbb{R})}+\|\partial_{u}^{\lambda_{1}+1}g_{1}(\rho,u)\|_{L_{u}^{\infty}(\mathbb{R})}+\|\partial_{u}^{\lambda_{1}+1+\alpha_{0}}g_{1}(\rho,u)\|_{L_{u}^{\infty}(\mathbb{R})}\leq
C\rho,$

where $\alpha_{0}\in(0,1)$ is the Hölder exponent.

Proof. We divide the proof into five steps.

1\. It follows from (4.60) that

$|a_{1}(\rho)|+\rho^{1-2\theta_{1}}|a_{1}^{\prime}(\rho)|+\rho^{2-2\theta_{1}}|a_{1}^{\prime\prime}(\rho)|+|a_{2}(\rho)|+\rho|a_{2}^{\prime}(\rho)|+\rho^{2}|a_{2}^{\prime\prime}(\rho)|\leq
C\quad\,\text{for }\rho\in[0,\rho^{*}].$ (4.84)

For $\rho\geq\rho^{*}$, it follows from (3.7), (4.58), and a direct
calculation that

$\displaystyle|a_{1}(\rho)|+\rho|a_{1}^{\prime}(\rho)|+\rho^{2}|a_{1}^{\prime\prime}(\rho)|\leq
C\rho^{\frac{1}{2}-\frac{\theta_{2}}{2\theta_{1}}},$ (4.85)
$\displaystyle|a_{2}(\rho)|=\Big{|}\frac{1}{4\lambda_{1}+1}k(\rho)^{-\lambda_{1}-1}k^{\prime}(\rho)^{-\frac{1}{2}}\int_{0}^{\rho}k(s)^{\lambda_{1}}k^{\prime}(s)^{-\frac{1}{2}}a_{1}^{\prime\prime}(s)\,\mathrm{d}s\Big{|}$
$\displaystyle\qquad\quad\leq
C\rho^{\frac{1}{2}-\frac{\theta_{2}}{2\theta_{1}}-\theta_{2}}\Big{(}\rho_{*}^{\theta_{1}}+1+\int_{\rho^{*}}^{\rho}s^{-1-\theta_{2}}\,\mathrm{d}s\Big{)}\leq
C\rho^{\frac{1}{2}-\theta_{2}-\frac{\theta_{2}}{2\theta_{1}}}.$ (4.86)

Moreover, calculating the derivatives explicitly, we conclude

$|a_{2}(\rho)|+\rho|a_{2}^{\prime}(\rho)|+\rho^{2}|a_{2}^{\prime\prime}(\rho)|\leq
C\rho^{\frac{1}{2}-\theta_{2}-\frac{\theta_{2}}{2\theta_{1}}}\qquad\,\text{for
}\rho\geq\rho^{*}.$

2\. For the remainder term $g_{1}(\rho,u)$, it follows from [10, Proof of
Theorem 2.1]) that

$\left\\{\begin{aligned}
\displaystyle&\partial_{\rho\rho}g_{1}(\rho,u)-k^{\prime}(\rho)^{2}\partial_{uu}g_{1}(\rho,u)=A(\rho)k(\rho)^{-1}f_{\lambda_{1}+1}(\frac{u}{k(\rho)}),\\\
\displaystyle&g_{1}(0,\cdot)=0,\quad\partial_{\rho}g_{1}(0,\cdot)=0,\end{aligned}\right.$

where $f_{\lambda_{1}}(y)=[1-y^{2}]_{+}^{\lambda_{1}}$ and
$A(\rho)=-a_{2}^{\prime\prime}(\rho)k(\rho)^{2\lambda_{1}+3}$. By (4.84),
$A(\rho)\sim O(\rho^{-1+2\theta_{1}})$ as $\rho\to 0$ and
$|A(\rho)|\leq\rho^{-\frac{3}{2}+\theta_{2}(1+\frac{1}{2\theta_{1}})}$ for
$\rho\geq\rho^{*}$. Similar to Lemma 4.4, we have following representation
formula for $g_{1}(\rho,u)$:

$\displaystyle k^{\prime}(\rho)g_{1}(\rho,u)$
$\displaystyle=\frac{1}{2\rho}\int_{0}^{\rho}d(s)\,k^{\prime}(s)\,\big{(}g_{1}(s,u+k(\rho)-k(s))+g_{1}(s,u-k(\rho)+k(s))\big{)}\,\mathrm{d}s$
$\displaystyle\quad\,+\frac{1}{2\rho}\int_{0}^{\rho}s\,A(s)\,k(s)^{-1}\,\Big{(}\int_{u-k(\rho)+k(s)}^{u+k(\rho)-k(s)}f_{\lambda_{1}+1}(\frac{y}{k(s)})\,\mathrm{d}y\Big{)}\,\mathrm{d}s,$
(4.87)

where $d(\rho)=2+\rho\frac{k^{\prime\prime}(\rho)}{k^{\prime}(\rho)}$
satisfying (3.12). Since $\rho A(\rho)k(\rho)^{-1}\sim O(\rho^{\theta_{1}})$
as $\rho\to 0$, the second integral is well-defined. Then it follows directly
from (3.6)–(3.7) and (4.87) that

$\displaystyle\|k^{\prime}(\rho)g_{1}(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}$
$\displaystyle\leq\frac{1}{\rho}\int_{0}^{\rho}d(s)\|k^{\prime}(s)g_{1}(s,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\,\mathrm{d}s+\frac{2k(\rho)}{\rho}\Big{(}\int_{0}^{\rho_{*}}+\int_{\rho_{*}}^{\rho^{*}}+\int_{\rho^{*}}^{\rho}\Big{)}|sA(s)k(s)^{-1}|\,\mathrm{d}s$
$\displaystyle\leq\frac{1}{\rho}\int_{0}^{\rho}d(s)\|k^{\prime}(s)g_{1}(s,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\,\mathrm{d}s+C\rho^{\theta_{2}-1}\big{(}\rho_{*}^{1+\theta_{1}}+1+\rho^{\frac{1}{2}+\frac{\theta_{2}}{2\theta_{1}}}\big{)}$
$\displaystyle\leq\frac{1}{\rho}\int_{0}^{\rho}d(s)\|k^{\prime}(s)g_{1}(s,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\,\mathrm{d}s+C\rho^{-\frac{1}{2}+\frac{\theta_{2}}{2\theta_{1}}+\theta_{2}}\qquad\,\,\text{for
}\rho\geq\rho^{*}.$ (4.88)

Since $g_{1}(\rho,u)$ is Hölder continuous and
$\operatorname{supp}g_{1}(\rho,\cdot)\subset[-k(\rho),k(\rho)]$, it follows
from (3.6)–(3.7) that
$\|k^{\prime}(\rho)g_{1}(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}$ is
locally integrable with respect to $\rho\in[0,\infty)$. Applying Lemma A.3 to
(4.88), we obtain that, for $\rho\geq\rho^{*}$,

$\|k^{\prime}(\rho)g_{1}(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\leq\left\\{\begin{aligned}
&C\rho^{\theta_{2}}\qquad\quad\;\;\text{if $\theta_{2}<\theta_{1}$},\\\
&C\rho^{\theta_{2}}\ln\rho\qquad\text{if
$\theta_{2}=\theta_{1}$},\end{aligned}\right.$

which, with (3.7), yields that, for $\rho\geq\rho^{*}$,

$\|g_{1}(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\leq\left\\{\begin{aligned}
&C\rho\qquad\quad\;\;\text{if }\theta_{2}<\theta_{1},\\\
&C\rho\ln\rho\qquad\text{if }\theta_{2}=\theta_{1}.\end{aligned}\right.$

3\. Applying $\partial_{u}$ to (4.87), we have

$\displaystyle k^{\prime}(\rho)\partial_{u}g_{1}(\rho,u)$
$\displaystyle=\frac{1}{2\rho}\int_{0}^{\rho}d(s)k^{\prime}(s)\Big{(}\partial_{u}g_{1}(s,u+k(\rho)-k(s))+\partial_{u}g_{1}(s,u-k(\rho)+k(s))\Big{)}\,\mathrm{d}s$
$\displaystyle\quad\,+\frac{1}{2\rho}\int_{0}^{\rho}sA(s)k(s)^{-1}\Big{(}f_{\lambda_{1}+1}(\frac{u+k(\rho)-k(s)}{k(s)})-f_{\lambda_{1}+1}(\frac{u-k(\rho)+k(s)}{k(s)})\Big{)}\,\mathrm{d}s.$
(4.89)

Since $|f_{\lambda_{1}+1}(s)|\leq 1$, by similar arguments as in Step 2, we
can obtain

$\|\partial_{u}g_{1}(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\leq
C\rho\qquad\,\,\text{for }\rho\geq\rho^{*}.$

4\. Applying the fractional derivative $\partial_{u}^{\lambda_{1}}$ to (4.3),
we have

$\displaystyle k^{\prime}(\rho)\partial_{u}^{\lambda_{1}+1}g_{1}(\rho,u)$
$\displaystyle=\frac{1}{2\rho}\int_{0}^{\rho}d(s)k^{\prime}(s)\Big{(}(\partial_{u}^{\lambda_{1}+1}g_{1})(s,u+k(\rho)-k(s))+(\partial_{u}^{\lambda_{1}+1}g_{1})(s,u-k(\rho)+k(s))\Big{)}\,\mathrm{d}s$
$\displaystyle\quad+\frac{1}{2\rho}\int_{0}^{\rho}sA(s)k(s)^{-1-\lambda_{1}}\Big{(}(\partial_{u}^{\lambda_{1}}f_{\lambda_{1}+1})(\frac{u+k(\rho)-k(s)}{k(s)})-(\partial_{u}^{\lambda_{1}}f_{\lambda_{1}+1})(\frac{u-k(\rho)+k(s)}{k(s)})\Big{)}\,\mathrm{d}s,$

where we have taken into account the homogeneity of the factional derivative
in the last term. Using the Fourier transform relation as in [46,
(I.26)–(I.27)], we can obtain

$\big{|}\mathscr{F}\big{(}(\partial_{u}^{\lambda_{1}}f_{\lambda_{1}+1})(u)\big{)}(\xi)\big{|}=C_{\lambda_{1}+1}|\xi|^{-\frac{3}{2}}\,|J_{\lambda_{1}+\frac{3}{2}}(|\xi|)|\leq\frac{\tilde{C}_{\lambda_{1}+1}}{1+\xi^{2}},$

for some positive constants $C_{\lambda_{1}+1}$ and
$\tilde{C}_{\lambda_{1}+1}$ depending only on $\lambda_{1}+1$, where we have
used the asymptotic relations for the first kind of Bessel functions
$J_{\lambda_{1}+\frac{3}{2}}(|\xi|)$ to obtain the final inequality. Since
$(1+|\xi|^{2})^{-1}$ is integrable, applying the Fourier inversion theorem, we
see that $(\partial_{u}^{\lambda_{1}}f_{\lambda_{1}+1})(u)$ is uniformly
bounded. Hence, by similar arguments as in Step 2, we have

$\|\big{(}\partial_{u}^{\lambda_{1}+1}g_{1}\big{)}(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\leq
C\rho\qquad\,\,\text{for $\rho\geq\rho^{*}$}.$

5\. By Lemma 4.2, we assume that $\alpha_{0}\in(0,1)$ is the Hölder exponent
of $(\partial_{u}^{\lambda_{1}+1})g_{1}(\rho,u)$. Then, applying the
fractional derivative $\partial_{u}^{\lambda_{1}}$ to (4.3), we have

$\displaystyle
k^{\prime}(\rho)\partial_{u}^{\lambda_{1}+1+\alpha_{0}}g_{1}(\rho,u)$
$\displaystyle=\frac{1}{2\rho}\int_{0}^{\rho}d(s)k^{\prime}(s)\Big{(}(\partial_{u}^{\lambda_{1}+1+\alpha_{0}}g_{1})(s,u+k(\rho)-k(s))+(\partial_{u}^{\lambda_{1}+1+\alpha_{0}}g_{1})(s,u-k(\rho)+k(s))\Big{)}\,\mathrm{d}s$
$\displaystyle\quad+\frac{1}{2\rho}\int_{0}^{\rho}sA(s)k(s)^{-1-\lambda_{1}-\alpha_{0}}(\partial_{u}^{\lambda_{1}+\alpha_{0}}f_{\lambda_{1}+1})(\frac{u+k(\rho)-k(s)}{k(s)})\,\mathrm{d}s$
$\displaystyle\quad-\frac{1}{2\rho}\int_{0}^{\rho}sA(s)k(s)^{-1-\lambda_{1}-\alpha_{0}}(\partial_{u}^{\lambda_{1}+\alpha_{0}}f_{\lambda_{1}+1})(\frac{u-k(\rho)+k(s)}{k(s)})\,\mathrm{d}s.$

Noting

$\big{|}\mathscr{F}\big{(}(\partial_{u}^{\lambda_{1}+\alpha_{0}}f_{\lambda_{1}+1})(u)\big{)}(\xi)\big{|}=C_{\lambda_{1}+1}|\xi|^{-\frac{3}{2}+\alpha_{0}}\big{|}J_{\lambda_{1}+\frac{3}{2}}(|\xi|)\big{|}\leq\frac{\tilde{C}_{\lambda_{1}+1}}{1+\xi^{2-\alpha_{0}}},$

and using the Fourier inversion theorem, we find that
$(\partial_{u}^{\lambda_{1}+\alpha_{0}}f_{\lambda_{1}+1})(u)$ is uniformly
bounded. By similar arguments as in Step 2, we obtain that
$\|\big{(}\partial_{u}^{\lambda_{1}+1+\alpha_{0}}g_{1}\big{)}(\rho,\cdot)\|_{L_{u}^{\infty}(\mathbb{R})}\leq
C\rho$ for $\rho\geq\rho^{*}$. This completes the proof of Lemma 4.11.
$\hfill\square$

From Lemmas 4.2 and 4.11, we conclude

######  Corollary 4.12.

$\chi(\rho,\cdot)$ is Hölder continuous and