# Existence of global entropy solution for Eulerian droplet models and two-
phase flow model with non-constant air velocity

Abhrojyoti Sen1 1Department of Mathematics, Indian Institute of Science
Education and Research, Dr. Homi Bhabha Road, Pune 411008, India. Email:
<EMAIL_ADDRESS> and  Anupam Sen2 2Centre for Applicable
Mathematics, Tata Institute of Fundamental Research, Post Bag No 6503,
Sharadanagar, Bangalore - 560065, India. Email<EMAIL_ADDRESS>
###### Abstract.

This article addresses the question concerning the existence of global entropy
solution for generalized Eulerian droplet models with air velocity depending
on both space and time variables. When $f(u)=u,$ $\kappa(t)=const.$ and
$u_{a}(x,t)=const.$ in (1.1), the study of the Riemann problem has been
carried out by Keita and Bourgault [42] $\&$ Zhang et al. [38]. We show the
global existence of the entropy solution to (1.1) for any strictly increasing
function $f(\cdot)$ and $u_{a}(x,t)$ depending only on time with mild
regularity assumptions on the initial data via shadow wave tracking approach.
This represents a significant improvement over the findings of Yang [26].
Next, by using the generalized variational principle, we prove the existence
of an explicit entropy solution to (1.1) with $f(u)=u,$ for all time $t>0$ and
initial mass $v_{0}>0,$ where $u_{a}(x,t)$ depends on both space and time
variables, and also has an algebraic decay in the time variable. This improves
the results of many authors such as Ha et al. [40], Cheng and Yang [27] $\&$
Ding and Wang [50] in various ways. Furthermore, by employing the shadow wave
tracking procedure, we discuss the existence of global entropy solution to the
generalized two-phase flow model with time-dependent air velocity that extends
the recent results of Shen and Sun [9].

###### Key words and phrases:

Eulerian droplet model, pressureless gas dynamics system, two-phase flow
model, shadow wave tracking, non-constant air velocity, entropy solution,
generalized variational principle

###### 2020 Mathematics Subject Classification:

Primary: 35L65, 35L67, 76N15, 35Q35

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Overview
    2. 1.2 Main results
    3. 1.3 Plan of the paper
  2. 2 Riemann problem and interactions
    1. 2.1 Riemann problem to the system Eq. 1.9
    2. 2.2 Riemann problem to the system Eq. 1.12
  3. 3 Entropy inequality
  4. 4 Global existence of entropy solution
    1. 4.1 The wave tracking algorithm
    2. 4.2 Proof of Theorem 1.1
    3. 4.3 Proof of Theorem 1.2
  5. 5 Some examples
    1. 5.1 System of nonlinear geometric optics
    2. 5.2 A model with $f(u)=u^{k},$ $k$ odd.
    3. 5.3 Pressureless hydrodynamic model
  6. 6 Eulerian droplet model involving space-time dependent source
    1. 6.1 Derivation of the formula Eq. 6.6
    2. 6.2 Verification of weak formulation, entropy criterion, and initial condition
      1. 6.2.1 Verification of weak formulation Eq. 1.13-(1.14)
      2. 6.2.2 Entropy criterion
      3. 6.2.3 Verification of initial condition

##  1\. Introduction and main results

###  1.1. Overview

In this article, we consider the $1$D Eulerian droplet model

$\displaystyle\begin{aligned}
v_{t}+(vf(u))_{x}&=0,\,\,\,&x\in\mathds{R},t>0,\\\
(vu)_{t}+(vuf(u))_{x}&=\kappa(t)(u_{a}(x,t)-u)v,\,\,\,&x\in\mathds{R},t>0,\end{aligned}$
(1.1)

adjoined with the initial data

$(v,u)(x,0)=(v_{0}(x),u_{0}(x)),\,\,\,\,\,\,x\in\mathds{R}.$ (1.2)

The precise conditions on the initial data will be specified later depending
on different situations. Here $v,u$ in (1.1) denotes the volume friction and
the velocity of the droplets, respectively. Moreover, we take the following
assumption on $f$ in (1.1):

$f:\mathds{R}\to\mathds{R}$ is a $C^{1}$ strictly increasing function.

Furthermore, $u_{a}(x,t)$ is a locally bounded function that indicates the
velocity of the carrier fluid or air, which depends on the position of the
particle and time, and $\kappa(t)\in L^{\infty}([0,T);\mathds{R})$ is the drag
coefficients between the carrier fluid and the droplets. The above system
(1.1) can be derived from a more general system

$\displaystyle\begin{aligned} v_{t}+(vf(u))_{x}&=0,\\\
(vu)_{t}+(vuf(u))_{x}&=\kappa(t)(u_{a}(x,t)-u)v+\left(1-\frac{\rho}{\rho_{l}}\right)\frac{1}{Fr^{2}}\textbf{g},\end{aligned}$
(1.3)

by neglecting the source term involving the gravitational force g. The system
(1.3), for $f(u)=u,$ $\kappa(t)=\frac{C_{D}Re_{d}}{24K}$ where
$K=\frac{\rho_{l}d^{2}U_{\infty}}{18L\mu}$ is an inertia parameter,
$Fr=\frac{U_{\infty}}{\sqrt{Lg_{0}}}$ is the Froude number, $U_{\infty}$ is
the speed of air at infinity; $g_{0}$ is a characteristic external field; and
$L$ is a characteristic length, and $u_{a}(x,t)=\bf{u_{a}}$ was introduced by
Bourgault et al. [55]. For a detailed physical description of (1.3),
applications, and numerical experiments, see [55, 43, 42].

In order to understand the system (1.1) from a more analytical point of view,
we consider different cases depending on the function $f(\cdot),$ the air
velocity $u_{a}(x,t)$ and the drag coefficient $\kappa(t)$ as follows:

Case I: $f(u)=u.$ This case can be split into two subcases depending on the
contributions from the drag coefficient $\kappa(t).$

Subcase I. $\kappa(t)=0.$ In this case the system (1.1) turns out to be the
usual system of pressureless gas dynamics and the initial value problem has
been extensively studied in the last few decades. As it is well known now,
among others, one of the main issues is that, $v$ is no longer a function, but
a measure. So the natural space where one should search for a weak solution to
(1.1) is the space of Radon measures. The existing results consist of
different notions of weak solutions, for example, measure-valued solutions
[17], duality solutions, and solutions via vanishing viscosity approach [18,
19, 29]. The global existence of weak solutions via mass and momentum
potentials was established in [53, 49]. An explicit formula using generalized
potentials and variational principles was obtained in [16, 21, 57]. A new
perspective to the global existence of weak solutions for $1$D pressureless
gas dynamics equation is due to Natile and Savaŕe [32] by constructing sticky
particle solutions using a suitable metric projection onto the cone of
monotone maps. Later, Cavalletti et al. [24] gave a more direct proof by using
the notion of differentiability of metric projections introduced by Haraux. On
the other hand, Nguyen and Tudorascu [47, 48] gave a general global existence
result for (1.1) with or without viscosity by constructing an entropy solution
for appropriate scalar conservation laws. They also obtained the uniqueness of
the solution via the contraction principle in the Wasserstein metric. Other
uniqueness results are due to Wang and Ding [58] $\&$ Haung and Wang [21]
where they used generalized characteristics introduced by Dafermos [12]. On
the contrary, Bressan and Nguyen [5] showed the non-uniqueness and non-
existence of solutions in the multi-D case by constructing different initial
data. Regarding numerical methods employed to the pressureless gas model, we
refer to [30, 20].

Subcase II. Now we consider $\kappa(t)\neq 0.$ The work of Ha et al. [40] is
the first useful result for us to consider in this scenario. They took
$\kappa(t)=1$ and $u_{a}(x,t)=0,$ i.e., the system (1.1) takes the form

$\displaystyle\begin{aligned} v_{t}+(vu)_{x}&=0,\\\
(vu)_{t}+(vu^{2})_{x}&=-uv.\end{aligned}$ (1.4)

The system (1.4) is strongly related to the pressureless Euler system with
flocking dissipation

$\displaystyle\begin{aligned}
\partial_{t}v+\nabla_{x}\cdot(vu)&=0,\,\,\,\,x\in{\mathds{R}^{n}},\,\,t>0,\\\
\partial_{t}(uv)+\nabla_{x}\cdot(vu\otimes
u)&=-Kv\int_{{\mathds{R}^{n}}}\psi(|x-y|)(u(x)-u(y))v(y)dy,\end{aligned}$
(1.5)

where $K$ is the positive coupling strength and $\psi$ is a Lipschitz
continuous function that denotes the communication weight. The system (1.4)
can be obtained from (1.5) by setting the following quantities:

$\displaystyle n=1,K\equiv 1,\psi\equiv
1,\int_{-\infty}^{\infty}vdx=1,\,\,\,\text{and}\,\,\,\int_{-\infty}^{\infty}vudx=0.$

To study the initial value problem for (1.4), the authors used a variational
approach. Furthermore, they showed the uniqueness of the entropy solution by
adopting the arguments of [58] in their setting. As it is mentioned earlier,
in the case when $\kappa(t)$ and $u_{a}(x,t)$ both are constants, the Riemann
problem for (1.1) is studied by Bourgault and Keita in [42]. More recently,
Cheng and Yang [27] studied the Riemann problem for the system

$\displaystyle\begin{aligned} v_{t}+(vu)_{x}&=0,\\\
(vu)_{t}+(vu^{2})_{x}&=(kx-\alpha u)v.\end{aligned}$ (1.6)

The nonhomogeneous term in (1.6) can easily be obtained by setting
$\kappa(t)=\alpha>0$ and $u_{a}(x,t)=kx/\alpha$ in (1.1). For other related
studies, we refer to [10, 50] and the references cited therein where the
authors considered the pressureless Euler system with a coulomb-like friction
term $\beta v,$ $\beta>0$ and a source term of the form $vx,$ respectively.
Recently, Leslie and Tan [46] developed a global wellposedness theory and
long-time behavior for weak solutions of the 1D Euler-alignment system
(similar to the system (1.5)) with measure-valued density, and bounded
velocity which is an example of a nonlocal system [6].

Case II: $f:\mathds{R}\to\mathds{R}$ be any function satisfying
$f^{\prime}(u)>0.$ Similar to the above, we consider two subcases below.
Subcase I. $\kappa(t)=0.$ This case corresponds to the generalized
pressureless Euler system. To the best of our knowledge, Yang [26] first
considered the Riemann problem for the homogeneous version of (1.1), where
$f(u)$ is assumed to be a smooth and strictly monotone function. He used the
characteristics method to obtain the Riemann solution and showed the existence
of a non-classical measure-valued solution. In fact, he proved there are only
two kinds of solution: one that involves vacuum and the other one contains a
delta measure in the component $v.$ Furthermore, he proposed a generalized
Rankine-Hugoniot relation for delta shock solutions to the system (1.1).

The next result on the homogeneous version of system (1.1) is due to Huang
[22], where he established the existence of a global weak solution with the
initial data $v_{0}(x)\geq 0,u_{0}(x)\in L^{\infty}(\mathds{R}).$ When
$f(u)\neq u,$ the key difference between this system and the usual
pressureless gas dynamics is that, here one has to deal with two kinds of
speeds: one is the characteristics speed $f(u)$ and the other one is the
physical velocity $u.$ Due to this, several tools, including duality
solutions, sticky particle approaches, and most importantly the generalized
variational principle (GVP) are inapplicable in this situation. Huang [22]
established the global weak solution by developing an approach that can be
seen as a combination of the front-tracking method and generalized
characteristics. First, the initial data is approximated by piecewise constant
functions and a sequence of Riemann solution $(v_{n},u_{n})$ is obtained up to
a finite time when the first interactions of waves occur. Then a set of
Riemann problems with $\delta$-initial data are solved to continue the
process.

In [14], Mitrović and Nedeljkov showed that the Riemann solutions to the
generalized pressureless gas dynamics equation (which is a non-strictly
hyperbolic system) can be obtained as a vanishing pressure limit of the
strictly hyperbolic system

$\displaystyle\begin{aligned} v_{t}+(vf(u))_{x}&=0,\\\
(vu)_{t}+(vuf(u)+\varepsilon p(v))_{x}&=0,\end{aligned}$ (1.7)

where $\varepsilon>0$ is a small parameter and the pressure term $p$ is a non-
negative $C^{2}$-function satisfying: $p^{\prime}\geq 0$ and
$p^{\prime\prime}>0.$ The system (1.7) is strictly hyperbolic [2] and can be
solved for arbitrary Riemann data. The distributional limit as $\varepsilon\to
0$ of the $BV$ solutions to the system (1.7) converges to the delta shock
solution of the homogeneous version of (1.1).

######  Remark 1.1.

In this context, it is important to note that when $f^{\prime}(u)$ changes
sign, system (1.1) can be associated with the general system of Keyfitz-
Kranzer [7] or Aw-Rascle type [1]. The global existence results to such
systems have been obtained by Lu (see [51, 52] and the references cited
therein) using compensated compactness arguments for the homogeneous case,
i.e., when $\kappa(t)=0.$

Subcase II. $\kappa(t)\neq 0.$ For $f(u)\neq u,$ there are very few papers in
the literature that considers this case. Recently, Zhang et al. [38] studied
the Riemann problem for the system

$\displaystyle\begin{aligned} v_{t}+(vf(u))_{x}&=0,\\\
(vu)_{t}+(vuf(u))_{x}&=(\beta-\alpha u)v,\end{aligned}$ (1.8)

where the constants $\alpha$ and $\beta$ denote the dissipation coefficient
and the friction coefficient, respectively. Note that (1.8) can be derived
from (1.1) by simply setting $u_{a}(x,t)=\beta/\alpha$ and $\kappa(t)=\alpha.$
Also, for $\alpha=0,$ the Riemann solutions are obtained by Zhang and Zhang in
[54].

###  1.2. Main results

In this section, we state our main results. Keeping the above literature in
mind, we ask the following question:

Q. Depending on the function $f$ and the source term, what are the possible
cases for which the system (1.1)-(1.2) admits a global entropy solution?

Our answer is two-fold:

  * •

When $f$ is any $C^{1}$ strictly increasing function, we establish the
existence of global entropy solution for the following system

$\displaystyle\begin{aligned} v_{t}+(vf(u))_{x}&=0,\\\
(vu)_{t}+(vuf(u))_{x}&=\kappa(t)(u_{a}(t)-u)v.\end{aligned}$ (1.9)

To achieve our objective, we utilize shadow wave tracking method [41]. Note
that here the drag coefficient $\kappa$ and the air velocity $u_{a}$ are
locally bounded functions of $t.$

  * •

When $f(u)=u,$ using the generalized variational principle, we obtain an
explicit representation of the entropy solution for the system

$\displaystyle\begin{aligned} v_{t}+(vu)_{x}&=0,\\\
(vu)_{t}+(vu^{2})_{x}&=\frac{1}{t+\upkappa}\left(\frac{x}{t+\upkappa}-u\right)v.\end{aligned}$
(1.10)

We derive the above system (1.10) by setting $\kappa(t)=\frac{1}{t+\upkappa}$
and $u_{a}(x,t)=\frac{x}{t+\upkappa}$ in (1.1) where
$\upkappa\in\mathds{R}^{+}.$ Note that, here the air velocity $u_{a}(x,t)$
depends both on the time and space variables. Also, $\kappa(t)$ is a function
that decays algebraically in time.

The first part of the article is devoted to the results related to the system
(1.9). We start with the concept of shadow waves [34, 35, 36]. Shadow waves
(in short SDW) are constructed as a net of piecewise constant (more precisely,
piecewise constant for each time $t$) functions that approximate delta shocks
in a small neighborhood of the shock location. Let a delta shock is supported
by a curve $x=c(t)$ with speed $c^{\prime}(t).$ We perturb the curve from both
sides by a small parameter $\varepsilon>0,$ and replace the delta shock with a
fan of shocks that depend on $\varepsilon.$ Next we give a formal definition
of shadow wave solution.

######  Definition 1.1.

A shadow wave is a piecewise constant (for each time $t$) function of the form

$\displaystyle
U^{\varepsilon}(x,t)=\begin{cases}(v_{l}(t),u_{l}(t)),\,\,\,\,\,x<c(t)-a_{\varepsilon}(t)-x_{l,\varepsilon},\\\
(v_{l,\varepsilon}(t),u_{l,\varepsilon}(t)),\,\,\,\,\,c(t)-a_{\varepsilon}(t)-x_{l,\varepsilon}<x<c(t),\\\
(v_{r,\varepsilon}(t),u_{r,\varepsilon}(t)),\,\,\,\,\,c(t)<x<c(t)+b_{\varepsilon}(t)+x_{r,\varepsilon},\\\
(v_{r}(t),u_{r}(t)),\,\,\,\,\,x>c(t)+b_{\varepsilon}(t)+x_{r,\varepsilon},\end{cases}$
(1.11)

where
$a_{\varepsilon}(t),b_{\varepsilon}(t),x_{l,\varepsilon},x_{r,\varepsilon}$
are $\mathcal{O}(\varepsilon)$ for each $t>0.$ We say that the SDW (1.11)
solves the system (1.9) if its substitution in the RHS and LHS of (1.9) gives
the same limit as $\varepsilon\to 0$ in the sense of distributions.

The idea of replacing delta (or singular) shocks with a fan of shocks reminds
us of the method of front tracking for conservation laws (see [2, 3, 11, 12,
25, 37, 4] ). As a first step, the initial data (1.2) is approximated by
piecewise constant functions and finitely many Riemann problems are solved at
the initial level $t=0.$ The solution can be continued until $t=t_{1},$ when
the first interaction of waves occurs. Since the interaction of two waves
produces only a single delta wave, the number of shock fronts decreases in
time. At the time level $t=t_{1},$ one needs to solve a finite number (less
than the initial case) of Riemann problems with delta initial data and the
process can be continued further.

In [22], Huang started with a similar method but later on, he defined
generalized characteristics and mass-momentum-energy potentials by using the
approximate solution to produce a complete solution. We take a different route
of using shadow wave solution at each stage of interaction (including the
initial stage where no interaction happens) and obtain a complete solution in
an approximated sense. One of the advantages of this approach is that it can
be implemented to study $3\times 3$ systems (see Section 2) of having
unbounded solutions whereas it seems that Huang’s method is restrictive in
such cases. However, the solution constructed by Huang can be seen as an
actual solution that satisfies the weak formulation.

Next, we present the global existence result for the system (1.9).

######  Theorem 1.1.

Let $v(x)\in L^{\infty}([R,\infty))$ be positive, $u(x)\in
L^{\infty}([R,\infty))\cap C([R,\infty))$ and $u(x)$ be a function having
finitely many extremes. Take a partition
$\\{Y_{i}\\}_{i\in\mathds{N}\cup\\{0\\}}$ of $[R,\infty)$ such that $Y_{0}=R$
and $C_{1}\varepsilon^{\alpha}<Y_{i}-Y_{i+1}<C_{2}\rho(\varepsilon)$ for every
$i=0,1,2,\cdots\cdots$ where $C_{1},C_{2}\geq 1,\alpha\in(0,1)$ and
$\rho(\varepsilon)\to 0$ as $\varepsilon\to 0.$ Then there exists a global
admissible solution to (1.9) and (4.1). More precisely, there exists a
function $U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})$ that satisfies

$\displaystyle\lim_{\varepsilon\to
0}\begin{cases}\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}(v^{\varepsilon}f(u^{\varepsilon})),\varphi\right\rangle=0,\\\
\left\langle\frac{\partial}{\partial
t}(v^{\varepsilon}u^{\varepsilon}),\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}(v^{\varepsilon}u^{\varepsilon}f(u^{\varepsilon})),\varphi\right\rangle=\left\langle\kappa(t)(u_{a}(t)-u^{\varepsilon})v^{\varepsilon},\varphi\right\rangle,\end{cases}$

for every test function $\varphi\in
C^{\infty}_{c}\left(\mathds{R}\times[0,\infty)\right)$ and the admissibility
condition.

Theorem 1.1 can be extended to a $3\times 3$ system of pressureless drift-flux
equations of two-phase flow model

$\displaystyle\begin{aligned} v_{t}+(vf(u))_{x}&=0,\\\
w_{t}+(wf(u))_{x}&=0,\\\
((v+w)u)_{t}+((v+w)uf(u))_{x}&=\kappa(t)(u_{a}(t)-u)(v+w),\end{aligned}$
(1.12)

in which $v$ and $w$ represent the masses of gas and liquid, respectively. For
more on the drift-flux model, we refer to [23, 44, 45]. Recently, considering
$f(u)=u$ and the source term to be $-\mu(v+w),$ Shen and Sun [9] studied the
Riemann problem for (1.12) and showed the existence of delta shock wave
invoking the vanishing pressure limit approach. We can prove an analogous
result of Theorem 1.1 for (1.12) as our method only requires the existence of
a unique solution to the Riemann problem and the interactions consisting of
shadow waves or elementary waves.

Next, we prove that a sequence of solutions constructed in Theorem 1.1 has a
weak limit in the space of Radon measures.

######  Theorem 1.2.

Grant the assumptions of Theorem 1.1 on the initial data (4.1). Take a
partition $\\{Y^{\nu}_{i}\\}_{i\in\mathds{N}\cup\\{0\\}}$ of $[R,\infty)$ such
that
$Y_{0}=R,C_{1}\varepsilon_{\nu}^{\alpha}<Y_{i}-Y_{i+1}<C_{2}\rho(\varepsilon_{\nu})$
for every $i=0,1,2,\cdot\cdots$ where $C_{1},C_{2}\geq 1,\alpha\in(0,1)$ and
$\rho(\varepsilon_{\nu})\to 0$ as $\varepsilon_{\nu}\to 0$ for any sequence
$\\{\varepsilon_{\nu}\\}_{\nu\in\mathds{N}\cup\\{0\\}}.$ Let
$\\{U^{\nu}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$ be a sequence of approximated
solution obtained in Theorem 1.1. Then there exists a subsequence still
denoted as $\\{U^{\nu}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$ and a Radon measure
$U^{*}$ such that
$U^{\nu}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}U^{*}$ as
$\nu\to\infty.$

In the second part of the paper, we obtain the explicit formula for (1.10) and
show that it satisfies the weak formulation (see Definition 1.2). We use the
method of generalized variational principle (GVP). The next paragraphs are
dedicated to briefly discussing the method and stating this part’s main
result.

As mentioned earlier, Rykov et al. [16] introduced the generalized variational
principle for pressureless gas dynamics equation by generalizing the
variational principle due to Lax and Oleinik for scalar conservation laws, in
particular for Burger’s equation. Huang and Wang [21] $\&$ Ding et al. [57]
extended the method of generalized potentials when the initial data $u_{0}$ is
not continuous and $v_{0}\geq 0$ is a Radon measure, respectively. In this
setting, the solution concept is the following: we show that $(v,u)$ is
actually a weak solution to the system (1.10). First, we construct locally
bounded measurable functions $m(x,t)$ and $u(x,t)$ such that $m(x,t)$ is of
locally $BV$ in $x$ for a.e $t.$ Therefore $m$ defines a Lebesgue-Stieltjes
measure $dm$ and its derivative in the sense of distribution defines a Radon
measure $v=m_{x}.$ These two objects are the same through the identification

$\displaystyle-\left\langle
m,\varphi_{x}\right\rangle=-\int_{-\infty}^{\infty}\varphi_{x}m\mathrm{d}{x}=\int_{-\infty}^{\infty}\varphi
m_{x}\mathrm{d}{x}=\int_{-\infty}^{\infty}\varphi\mathrm{d}{m}=\left\langle
v,\varphi\right\rangle\,\,\,\,\text{for all}\,\,\,\,\varphi\in
C^{\infty}_{c}(\mathds{R}).$

Furthermore, similar identification allows us to define

$\displaystyle\left\langle
vu,\varphi\right\rangle=\int_{-\infty}^{\infty}\varphi u\mathrm{d}{m}.$

These identifications lead to the notion of generalized solution to (1.10).
The first equation can be written in the distributional sense as

$\displaystyle 0=\left\langle v,\varphi_{t}\right\rangle+\left\langle
vu,\varphi_{x}\right\rangle=-\iint\varphi_{xt}m\mathrm{d}{x}\mathrm{d}{t}+\iint\phi_{x}u\mathrm{d}{m}\mathrm{d}{t}.$

Similarly, the second equation of (1.10) can be written as

$\displaystyle 0$ $\displaystyle=\left\langle
vu,\varphi_{t}\right\rangle+\left\langle
vu^{2},\varphi_{x}\right\rangle+\left\langle\frac{1}{t+\upkappa}\left(\frac{x}{t+\upkappa}-u\right)v,\varphi\right\rangle$
$\displaystyle=\iint u\varphi_{t}\mathrm{d}{m}\mathrm{d}{t}+\iint
u^{2}\phi_{x}\mathrm{d}{m}\mathrm{d}{t}+\iint\frac{1}{t+\upkappa}\left(\frac{x}{t+\upkappa}-u\right)\varphi\mathrm{d}{m}\mathrm{d}{t}.$

Therefore the weak formulation to the system (1.10) is the following:

######  Definition 1.2.

The pair $(v,u)$ is said to be a generalized solution to the system (6.1) if
the following integral identities

$\displaystyle\iint\varphi_{t}m\mathrm{d}{x}\mathrm{d}{t}-\iint\varphi
u\mathrm{d}{m}\mathrm{d}{t}=0,$ (1.13)
$\displaystyle\iint\varphi_{t}u+\varphi_{x}u^{2}+\frac{1}{(t+\upkappa)}\left(\frac{x}{t+\upkappa}-u\right)\varphi\mathrm{d}{m}\mathrm{d}{t}=0,$
(1.14)

hold for all test functions $\varphi\in
C^{\infty}_{c}(\mathds{R}\times\mathds{R}^{+}),$ where the distributional
derivative $m_{x}$ defines the Radon measure $v.$

The construction of the generalized solution $(m,u)$ is done in two levels.
First, by introducing generalized potential $F(y,x,t)$ we construct $u$ and
then we introduce the momentum and energy potentials $q(x,t)$ and $E(x,t),$
respectively and some auxiliary functionals
$H(\cdot,x,t),I(\cdot,x,t),J(\cdot,x,t).$ Moreover, by establishing relations
between the measures $\mathrm{d}{q},\mathrm{d}{E},$ $\mathrm{d}{m},$ and
$\mathrm{d}{J}$ we show that $q,E,m$ satisfy Definition 1.2.

Now we state the main result of this part.

######  Theorem 1.3.

Let $v_{0}(x)>0,u_{0}(x)$ are locally bounded measurable functions, then the
pair $(m,u)$ given by (6.5)-(6.6) is a global weak solution to the system
(1.10)-(1.2) in the sense of Definition 1.2.

######  Remark 1.2.

We want to point out that the system (1.10) can be associated with the system
(1.5) if we consider a more general communication weight $\psi(|x-y|,t)$ which
is of the form $\kappa(t):=1/(t+\upkappa).$ Also, if we take further
assumptions $\int_{-\infty}^{\infty}v(y)\mathrm{d}{y}=1$ and
$\int_{-\infty}^{\infty}v(y)u(y)\mathrm{d}{y}=1,$ the nonhomogeneous term in
(1.5) would take a form $\frac{1}{t+\upkappa}\left(1-u\right)v.$ The source
term considered in (1.10) is even more general involving the space variable.
Note that, in the situation described above the communication weight is a
decaying function of time. Therefore, Theorem 1.3 essentially gives an answer
to the question of Ha et al. [40, Section 7] where they made a query: whether
the generalized variational principle would apply for non-constant
communication weights, for instance, algebraically decaying communication
weights.

###  1.3. Plan of the paper

The article is organized into two separate parts. The first part of the
article consists of Section 2, Section 3, Section 4 and Section 5. In Section
2, we study the Riemann problem and the interactions for (1.9) and (1.12). In
Section 3, using the entropy-entropy flux pair, we introduce the notion of
dissipative shadow waves for the system (1.9) and show its equivalence to the
overcompressibility condition. Section 4 is devoted to prove Theorem 1.1 and
Theorem 1.2. In Section 5, we provide some examples of physically relevant
models that are included in (1.9). The second part of the paper consists of
Section 6 where we give the proof of Theorem 1.3.

Part I: Global existence results for (1.9) and (1.12).

##  2\. Riemann problem and interactions

In this section, we study the shadow wave solution for the Riemann type
initial data and initial data containing $\delta$-measure for the systems
(1.9) and (1.12). We start with the system (1.9).

Notation. Let $g$ be any function that depends “only” on time $t,$ i.e.
$g:[0,\infty)\to\mathds{R},$ then to denote the derivative of $g$ with respect
to $t,$ we use $\frac{\partial}{\partial
t},\frac{\mathrm{d}}{\mathrm{d}{t}},\cdot,\prime$ interchangeably throughout
the article which convey the same meaning.

###  2.1. Riemann problem to the system Eq. 1.9

First, we observe that for a smooth solution the system (1.9) reduces to

$u_{t}+f(u)u_{x}=\kappa(t)(u_{a}(t)-u).$

From the characteristic equation we have the system of ODEs as follows:

$\displaystyle\begin{cases}&\frac{\mathrm{d}{x(t)}}{\mathrm{d}{t}}=f(u(x(t),t)),\\\
&\frac{\mathrm{d}{u(x(t),t)}}{\mathrm{d}{t}}=\kappa(t)(u_{a}(t)-u(x(t),t)),\\\
&x(0)=x_{0}.\end{cases}$ (2.1)

Solving (2.1), we obtain

$\displaystyle
u(x,t)=e^{-\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}\left(\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+u_{0}(x_{0})\right),\,\,\,\text{and}\,\,\,$
$\displaystyle
x(t)=x_{0}+\int_{0}^{t}f\left(e^{-\int_{0}^{s}\kappa(\theta)\mathrm{d}{\theta}}\left(\int_{0}^{s}F(\theta)\mathrm{d}{\theta}+u_{0}(x_{0})\right)\right)\mathrm{d}{s},$

where $F$ is given by $F(t)=\kappa(t)\cdot u_{a}(t)\cdot
e^{\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}$. This motivates us to
consider the shadow wave in the following form:

$\displaystyle
U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})(x,t)=\begin{cases}\left(v_{l},U_{l}(t)\right),\,\,\,\,\,\,\,\,\,\,\,\,&x<c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},\\\
\left(v_{\varepsilon}(t),u_{\varepsilon}(t)\right),\,\,\,\,\,\,&c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}<x<c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},\\\
\left(v_{r},U_{r}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},\end{cases}$
(2.2)

where
$U_{l,r}(t):=e^{-\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}\cdot\left(\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+u_{l,r}\right)$
and $x_{\varepsilon},v_{\varepsilon}(t)$ are $\mathcal{O}(\varepsilon)$ and
$\mathcal{O}(1/\varepsilon),$ and $\lim\limits_{\varepsilon\to
0}u_{\varepsilon}(t)=\chi(t).$ First, we study the above system (1.9) when
initial data contains a $\delta$-measure and is of the following form

$\displaystyle(v,u)(x,0)=\begin{cases}(v_{l},u_{l}),\,\,\,\,\,\,&x<0,\\\
(\bar{m}\delta(x),\bar{u}),\,\,\,\,\,\,&x=0,\\\
(v_{r},u_{r}),\,\,\,\,\,\,&x>0,\end{cases}$ (2.3)

where $v_{l,r}\geq 0$, $\bar{m}>0$ and $u_{l}>\bar{u}>u_{r}$. This situation
arises when two approaching shock waves interact. Suppose we are given a
piecewise constant data: $(v_{l},u_{l}),(v_{m},u_{m})$ and $(v_{r},u_{r})$
with $u_{l}>u_{m}>u_{r}.$ The delta shock curve joining $(v_{l},u_{l})$ to
$(v_{m},u_{m})$ interacts with another delta shock curve connecting the states
$(v_{m},u_{m})$ to $(v_{r},u_{r})$ at some point $(X,T)$ and at this level, we
need to solve a Riemann problem with a $\delta$-initial data. When
$\bar{m}=0,$ then the data is purely of Riemann type and can be seen as a
particular case of (2.3). Without loss of any generality, we may assume
$(X,T)=(x,0).$

Substituting the shadow wave solution (2.2) into the system (1.9), from the
definition of shadow wave we have

$\displaystyle\lim_{\varepsilon\to
0}\left[\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}\left(v^{\varepsilon}f(u^{\varepsilon})\right),\varphi\right\rangle\right]=0,$
(2.4) $\displaystyle\lim_{\varepsilon\to
0}\left[\left\langle\frac{\partial}{\partial
t}\left(v^{\varepsilon}u^{\varepsilon}\right),\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}\left(v^{\varepsilon}u^{\varepsilon}f(u^{\varepsilon})\right),\varphi\right\rangle-\left\langle\kappa(t)(u_{a}(t)-u^{\varepsilon})v^{\varepsilon},\varphi\right\rangle\right]=0,$
(2.5)

for all $\varphi\in C^{\infty}_{c}(\mathds{R}\times[0,\infty))$.
Now for a fixed $\varepsilon>0,$ by using integration by parts in the term
involving time derivative of (2.4), we get

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle$
$\displaystyle=\int_{0}^{\infty}\int_{-\infty}^{\infty}v^{\varepsilon}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}+\int_{-\infty}^{\infty}v^{\varepsilon}(x,0)\varphi(x,0)\mathrm{d}{x}$
$\displaystyle=\int_{0}^{\infty}\int_{-\infty}^{c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}}v_{l}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}+\int_{-\infty}^{-x_{\varepsilon}}v^{\varepsilon}(x,0)\varphi(x,0)\mathrm{d}{x}$
$\displaystyle+\int_{0}^{\infty}\int_{c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}}^{c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon}}v_{\varepsilon}(t)\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}+\int_{-x_{\varepsilon}}^{x_{\varepsilon}}v^{\varepsilon}(x,0)\varphi(x,0)\mathrm{d}{x}$
$\displaystyle+\int_{0}^{\infty}\int_{c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon}}^{\infty}v_{r}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}+\int_{x_{\varepsilon}}^{\infty}v^{\varepsilon}(x,0)\varphi(x,0)\mathrm{d}{x}.$

Simplifying the above expression, we obtain

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle$
$\displaystyle=\int_{0}^{\infty}\left(v_{\varepsilon}(t)-v_{l}\right)\varphi\left(c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},t\right)\left(c^{\cdot}(t)-\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{\infty}\left(v_{r}-v_{\varepsilon}(t)\right)\varphi\left(c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},t\right)\left(c^{\cdot}(t)+\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle-\int_{0}^{\infty}\int_{c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}}^{c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon}}\frac{\partial}{\partial
t}v_{\varepsilon}(t)\varphi(x,t)\mathrm{d}{t}+\int_{-x_{\varepsilon}}^{x_{\varepsilon}}v_{\varepsilon}(0)\varphi(x,0)\mathrm{d}{x}.$
(2.6)

Similarly, the term involving spatial derivatives of (2.4) gives

$\displaystyle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle$
$\displaystyle=\int_{0}^{\infty}\left[v_{l}f(U_{l}(t))-v_{\varepsilon}(t)f(u_{\varepsilon}(t))\right]\varphi\left(c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},t\right)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{\infty}\left[v_{\varepsilon}(t)f(u_{\varepsilon}(t))-v_{r}f(U_{r}(t))\right]\varphi\left(c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},t\right)\mathrm{d}{t}.$
(2.7)

Next, we use the following Taylor series expansion for $\varphi$ with respect
to $x=c(t)$ to evaluate the above integrals, we have

$\displaystyle\varphi\left(c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},t\right)=\varphi(c(t),t)-\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\frac{\partial}{\partial
x}\varphi(c(t),t)+\mathcal{O}(\varepsilon^{2}),$
$\displaystyle\varphi\left(c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},t\right)=\varphi(c(t),t)+\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\frac{\partial}{\partial
x}\varphi(c(t),t)+\mathcal{O}(\varepsilon^{2}),$ (2.8)
$\displaystyle\varphi(x,t)=\varphi(c(t),t)+\mathcal{O}(\varepsilon),\,\,\,\textnormal{for}\,\,\,c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}<x<c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon}.$

By employing the Taylor expansions into the equations (2.1)-(2.1) and
simplifying, we obtain

$\displaystyle\begin{aligned} &-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle=\int_{0}^{\infty}\left[c^{\cdot}(t)(v_{r}-v_{l})-\frac{\partial}{\partial
t}\left(2\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)v_{\varepsilon}(t)\right)\right]\varphi(c(t),t)\mathrm{d}{t}\\\
&+\int_{0}^{\infty}\left[\left(v_{l}f(U_{l}(t))-v_{r}f(U_{r}(t))\right)+\frac{\varepsilon}{2}\left(v_{l}+v_{r}\right)\right]\varphi(c(t),t)\mathrm{d}{t}\\\
&+\int_{0}^{\infty}c^{\cdot}(t)\left[v_{r}-2v_{\varepsilon}(t)+v_{r}\right]\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\frac{\partial}{\partial
x}\varphi(c(t),t)\mathrm{d}{t}+\int_{-x_{\varepsilon}}^{x_{\varepsilon}}v_{\varepsilon}(0)\varphi(x,0)\mathrm{d}{x}\\\
&+\int_{0}^{\infty}\left[2v_{\varepsilon}(t)f(u_{\varepsilon}(t))-v_{l}f\left(U_{l}(t)\right)-v_{r}f(U_{r}(t))\right]\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\frac{\partial}{\partial
x}\varphi(c(t),t)\mathrm{d}{t}+\mathcal{O}(\varepsilon).\end{aligned}$ (2.9)

In a similar way as above, we can calculate (2.5). However, we have to
consider the contribution of the nonhomogeneous term. Following the same
calculations, simplifying the terms for time derivative and the source, we get

$\displaystyle\begin{aligned} &-\left\langle\frac{\partial}{\partial
t}\left(v^{\varepsilon}u^{\varepsilon}\right),\varphi\right\rangle+\left\langle\kappa(t)(u_{a}(t)-u^{\varepsilon})v^{\varepsilon},\varphi\right\rangle\\\
&=\int_{0}^{\infty}\left[v_{\varepsilon}(t)u_{\varepsilon}(t)-v_{l}U_{l}(t)\right]\varphi\left(c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},t\right)\left(\dot{c}(t)-\frac{\varepsilon}{2}t\right)\mathrm{d}{t}\\\
&+\int_{0}^{\infty}\left[v_{r}U_{r}(t)-v_{\varepsilon}(t)u_{\varepsilon}(t)\right]\varphi\left(c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},t\right)\left(\dot{c}(t)+\frac{\varepsilon}{2}t\right)\mathrm{d}{t}\\\
&+\int_{0}^{\infty}\int_{c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}}^{c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon}}\left[\kappa(t)(u_{a}(t)-u_{\varepsilon}(t))v_{\varepsilon}(t)-\frac{\partial}{\partial
t}\left(v_{\varepsilon}(t)u_{\varepsilon}(t)\right)\right]\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}+\int_{-x_{\varepsilon}}^{x_{\varepsilon}}v_{\varepsilon}(0)u_{\varepsilon}(0)\varphi(x,0)\mathrm{d}{x},\end{aligned}$
(2.10)

and for spatial derivatives, we have

$\displaystyle\begin{aligned} -\left\langle\frac{\partial}{\partial
x}\left(v^{\varepsilon}u^{\varepsilon}f(u^{\varepsilon})\right),\varphi\right\rangle&=\int_{0}^{\infty}\left[v_{l}U_{l}(t)f(U_{l}(t))-v_{\varepsilon}(t)u_{\varepsilon}(t)f(u_{\varepsilon}(t))\right]\varphi\left(c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},t\right)\mathrm{d}{t}\\\
&+\int_{0}^{\infty}\left[v_{\varepsilon}(t)u_{\varepsilon}(t)f(u_{\varepsilon}(t))-v_{r}U_{r}(t)f(U_{r}(t))\right]\varphi\left(c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},t\right)\mathrm{d}{t}.\end{aligned}$
(2.11)

Again using the Taylor expansion of the test function $\varphi$ in
(2.10)-(2.11), we obtain

$\displaystyle-\left\langle\frac{\partial}{\partial
t}\left(v^{\varepsilon}u^{\varepsilon}\right),\varphi\right\rangle-\left\langle\frac{\partial}{\partial
x}\left(v^{\varepsilon}u^{\varepsilon}f(u^{\varepsilon})\right),\varphi\right\rangle+\left\langle\kappa(t)(u_{a}(t)-u^{\varepsilon})v^{\varepsilon},\varphi\right\rangle$
$\displaystyle=\int_{0}^{\infty}\left[\dot{c}(t)\left(v_{r}U_{r}(t)-v_{l}U_{l}(t)\right)+\frac{\varepsilon}{2}\left(v_{r}U_{r}(t)+v_{l}U_{l}(t)\right)+v_{l}U_{l}(t)f(U_{l}(t))-v_{r}U_{r}(t)f(U_{r}(t))\right]\varphi(c(t),t)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{\infty}2\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\left[\kappa(t)(u_{a}(t)-u_{\varepsilon}(t))v_{\varepsilon}(t)-\partial_{t}\left(2\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)v_{\varepsilon}(t)u_{\varepsilon}(t)\right)\right]\varphi(c(t),t)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{\infty}\left[\dot{c}(t)\left(v_{r}U_{r}(t)-2v_{\varepsilon}(t)u_{\varepsilon}(t)+v_{l}U_{l}(t)\right)\right]\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\frac{\partial}{\partial
x}\varphi(x,t)\mathrm{d}{t}$
$\displaystyle+2\int_{0}^{\infty}\left[v_{\varepsilon}(t)u_{\varepsilon}f(u_{\varepsilon}(t))-v_{l}U_{l}(t)f(U_{l}(t))-v_{r}U_{r}(t)f(U_{r}(t))\right]\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\varphi_{x}(x,t)\mathrm{d}{t}.$
(2.12)

Now passing to the limit as $\varepsilon$ tends to zero in the equations (2.9)
and (2.1), we obtain the following relations

$\displaystyle\frac{\partial}{\partial
t}(\xi(t))=\dot{c}(t)[v]-[vf(U(t))],\,\,\,\xi(0)=\bar{m},$ (2.13)
$\displaystyle\frac{\partial}{\partial
t}\left(\xi(t)\chi(t)\right)+\kappa(t)(\chi(t)-u_{a}(t))\xi(t)=\dot{c}(t)[vU(t)]-[vU(t)f(U(t))],\,\,\,\xi(0)\chi(0)=\bar{m}\bar{u},$
$\displaystyle f(\chi(t))=\dot{c}(t),$

where $\lim\limits_{\varepsilon\to
0}2\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)v_{\varepsilon}(t)=\xi(t)$,
$\lim\limits_{\varepsilon\to 0}u_{\varepsilon}(t)=\chi(t)$ and
$[\cdot]:=\cdot_{r}-\cdot_{l}$ denotes the jump across the discontinuity
curve. From the first equation of (2.13), we have

$\xi(t)=c(t)[v]+v_{l}\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}-v_{r}\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}+\bar{m}.$

Setting $\Theta(t)=\xi(t)\chi(t),$ second equation of (2.13) can be written as

$\displaystyle
e^{\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}\left[\frac{\partial}{\partial
t}\Theta(t)+\kappa(t)\Theta(t)\right]=$ $\displaystyle
F(t)\left[c(t)(v_{r}-v_{l})+v_{l}\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}-v_{r}\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}+\bar{m}\right]$
$\displaystyle+\dot{c}(t)\Big{[}v_{r}\Big{(}\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+u_{r}\Big{)}-v_{l}\Big{(}\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+u_{l}\Big{)}\Big{]}$
$\displaystyle-\Big{[}v_{r}\Big{(}\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+u_{r}\Big{)}f(U_{r}(t))-v_{l}\Big{(}\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+u_{l}\Big{)}f(U_{l}(t))\Big{]},$

which implies the following identity.

$\displaystyle\begin{aligned} \frac{\partial}{\partial
t}\left(e^{\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}\Theta(t)\right)&=\dot{c}(t)[vu]+[v]\frac{\partial}{\partial
t}\left(c(t)\int_{0}^{t}F(\theta)\mathrm{d}{\theta}\right)+[vuf(U(t))]+\bar{m}F(t)\\\
&+v_{l}\frac{\partial}{\partial
t}\left(\int_{0}^{t}F(\theta)\mathrm{d}{\theta}\cdot\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}\right)-v_{r}\frac{\partial}{\partial
t}\left(\int_{0}^{t}F(\theta)\mathrm{d}{\theta}\cdot\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}\right).\end{aligned}$
(2.14)

The ODE presented in (2.14) directly yields,

$\Theta(t):=\xi(t)\chi(t)=c(t)[vU(t)]+v_{l}U_{l}(t)\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}-v_{r}U_{r}(t)\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}+\bar{m}\bar{U}(t)$

where
$\bar{U}(t)=e^{-\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}\left(\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+\bar{u}\right)$.
Thus $\chi(t)$ is of the following form

$\displaystyle\chi(t)=\frac{c(t)[vU(t)]+v_{l}(U_{l}(t))\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}-v_{r}U_{r}(t)\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}+\bar{m}(\bar{U}(t))}{c(t)[v]+v_{l}\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}-v_{r}\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}+\bar{m}}.$

Now, we need to consider the region
$\mathscr{D}:=\\{(c(t),t)|\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}\leq
c(t)\leq\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}\\}$ to prove the
overcompressibility condition as $u_{l}>\bar{u}>u_{r}$. In the above region
$\mathscr{D}$, we find $\xi(t)>0,$

$\displaystyle\chi(t)=U_{l}(t)+\frac{v_{r}(U_{l}(t)-U_{r}(t))\left(\int_{0}^{t}f(U_{r}(\theta))d\theta-c(t)\right)+\bar{m}\left(\bar{U}(t)-U_{l}(t)\right)}{\xi(t)}<U_{l}(t),$

and

$\displaystyle\chi(t)=U_{r}(t)+\frac{v_{l}(U_{l}(t)-U_{r}(t))\left(\int_{0}^{t}f(U_{l}(\theta))d\theta-c(t)\right)+\bar{m}\left(\bar{U}(t)-U_{r}(t)\right)}{\xi(t)}>U_{r}(t).$

Furthermore, since $f$ is $C^{1}$ the following ODE

$\displaystyle\dot{c}(t)=f(\chi(t))={\mathcal{H}}(c(t),t),$ $\displaystyle
c(0)=0,$

has a unique solution [15, 28] in the region $\mathscr{D}$. Finally, by using
the increasing property of $f,$ we have $f(U_{r}(t))<f(\chi(t))<f(U_{l}(t)).$
Next, we turn our attention to the case of Riemann-type initial data, i.e.,

$\displaystyle(v,u)(x,0)=\begin{cases}(v_{l},u_{l}),\,\,\,\,\,\,\ x<0,\\\
(v_{r},u_{r}),\,\,\,\,\,\,x>0,\end{cases}$ (2.15)

where $v_{l,r}>0$. When $u_{l}>u_{r}$ in (2.15), substituting the shadow wave

$\displaystyle
U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})(x,t)=\begin{cases}\left(v_{l},U_{l}(t)\right),\,\,\,\,\,\,\,\,\,\,\,\,&x<c(t)-\frac{\varepsilon}{2}t,\\\
\left(v_{\varepsilon}(t),u_{\varepsilon}(t)\right),\,\,\,\,\,\,&c(t)-\frac{\varepsilon}{2}t<x<c(t)+\frac{\varepsilon}{2}t,\\\
\left(v_{r},U_{r}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>c(t)+\frac{\varepsilon}{2}t,\end{cases}$
(2.16)

we obtain the system of ODE’s (2.13) where $\lim\limits_{\varepsilon\to
0}\varepsilon tv_{\varepsilon}(t)=\xi(t),\lim\limits_{\varepsilon\to
0}u_{\varepsilon}(t)=\chi(t)$ with the initial conditions $\xi(0)=0$ and
$\xi(0)\chi(0)=0$. We take the particular form of
$u_{\varepsilon}(t)=e^{-\int_{0}^{t}\kappa(\theta)d\theta}\left(\int_{0}^{t}F(\theta)d\theta+u_{\varepsilon}\right)$,
in which $u_{\varepsilon}$’s are constants and independent of $t$. Hence,

$\lim\limits_{\varepsilon\to
0}e^{-\int_{0}^{t}\kappa(\theta)d\theta}\left(\int_{0}^{t}F(\theta)d\theta+u_{\varepsilon}\right)=e^{-\int_{0}^{t}\kappa(\theta)d\theta}\left(\int_{0}^{t}F(\theta)d\theta+\chi_{0}\right)=\chi(t),$

where $\chi_{0}$ is constant. Thus the second equation of (2.13) takes the
simple form

$\displaystyle\frac{\partial\xi(t)}{\partial
t}\cdot\chi(t)=c^{\cdot}(t)[v(U(t))]-[v(U(t))f(U(t))].$

The first equation of (2.13) yields

$f(\chi(t))\left([v](\chi(t))-[v(U(t))]\right)-(\chi(t))\left[vf(U(t))\right]+\left[v(U(t))f(U(t))\right]=0.$

Let us consider the function ${\mathcal{K}}(x)$ as

${\mathcal{K}}(x)=f(x)\Big{(}[v]x-[v(U(t))]\Big{)}-x[vf(U(t))]+[v(U(t))f(U(t))].$

One can observe that
${\mathcal{K}}(U_{l}(t))=v_{r}(u_{l}-u_{r})\left(f(U_{l}(t))-f(U_{r}(t))\right)>0$,
since $f$ is increasing. Similarly, we have ${\mathcal{K}}(U_{r}(t))<0.$ Also,
a simple calculation shows

${\mathcal{K}}^{\prime}(x)=v_{r}\left(x-(U_{r}(t))\right)f^{\prime}(x)+v_{l}\left((U_{l}(t))-x\right)f^{\prime}(x)+v_{r}\left(f(x)-f(U_{r}(t))\right)+v_{l}\left(f(U_{l}(t))-f(x)\right).$

Since $f$ is increasing, we find ${\mathcal{K}}^{\prime}(x)>0$ and therefore
${\mathcal{K}}(x)=0$ has a unique solution in $(U_{r}(t),U_{l}(t))$. Now using
${\mathcal{K}}(\chi(t))=0$, we obtain the overcompressibility condition:

$U_{r}(t)<\chi(t)<U_{l}(t)\,\,\,\,\text{implies}\,\,\,\,f(U_{r}(t))<f(\chi(t))<f(U_{l}(t)).$

When $u_{l}<u_{r}$ the solution consists of contact discontinuity and vacuum,
i.e.,

$\displaystyle(v,u)(x,t)=\begin{cases}\left(v_{l},U_{l}(t)\right),\,\,\,\,\,\,\,\,\,\,\,\,&x<\int_{0}^{t}f(U_{l}(s))ds,\\\
\left(0,z(x,t)\right),\,\,\,\,\,\,&\int_{0}^{t}f(U_{l}(s))ds<x<\int_{0}^{t}f(U_{r}(s))ds,\\\
\left(v_{r},U_{r}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>\int_{0}^{t}f(U_{r}(s))ds,\end{cases}$
(2.17)

where $z(x,t)$ is a continuous function that satisfies
$z\left(\int_{0}^{t}f(U_{l}(s))ds,t\right)=U_{l}(t)$ and
$z\left(\int_{0}^{t}f(U_{r}(s))ds,t\right)=U_{r}(t).$ Summarizing the above
discussion we prove the following:

######  Lemma 2.1 (Riemann solution).

The system (1.9) with initial data (2.15) has a unique shadow wave solution of
the form (2.16) in the case $u_{l}>u_{r}.$ If $u_{l}\leq u_{r},$ then the
solution is a combination of contact discontinuities and vacuum of the form
(2.17).

######  Lemma 2.2 (Interaction of shadow waves).

If $u_{l}>\bar{u}>u_{r},$ then the system (1.9) with initial data (2.3) has a
unique shadow wave solution of the form (2.2).

######  Remark 2.1.

We observe that conservation of mass holds true due to the first equation in
the system (1.9). For the quantity

$\displaystyle{\mathcal{M}}_{0}(t):=\xi(t)+\int_{-\infty}^{c(t)}v(x,t)\mathrm{d}{x}+\int_{c(t)}^{\infty}v(x,t)\mathrm{d}{x},$

we have ${\mathcal{M}}_{0}(0)={\mathcal{M}}_{0}(t)$ for all $t>0.$ Indeed,
using the first equation of (2.13), we get

$\displaystyle\dot{{\mathcal{M}}_{0}}(t)$
$\displaystyle=\dot{\xi}(t)+v(c(t)-,t)\dot{c}(t)+\int_{-\infty}^{c(t)}\frac{\partial}{\partial
t}v(x,t)\mathrm{d}{x}-v(c(t)+,t)\dot{c}(t)+\int_{c(t)}^{\infty}\frac{\partial}{\partial
t}v(x,t)\mathrm{d}{x}$ $\displaystyle=\dot{\xi}(t)-\dot{c}(t)[v]+[vf(U)]=0.$

However, the momentum satisfies a differential equation. Define

$\displaystyle{\mathcal{M}}_{1}(t):=\xi(t)\chi(t)+\int_{-\infty}^{c(t)}v(x,t)u(x,t)\mathrm{d}{x}+\int_{c(t)}^{\infty}v(x,t)u(x,t)\mathrm{d}{x}.$

A similar calculation as above leads to

$\displaystyle\dot{{\mathcal{M}}_{1}}(t)=$
$\displaystyle\frac{\partial}{\partial
t}(\xi(t)\chi(t))-\dot{c}(t)[v]+[vUf(U)]+\int_{-\infty}^{c(t)}\kappa(t)(u_{a}(t)-u(x,t))v(x,t)\mathrm{d}{x}$
$\displaystyle+\int_{c(t)}^{\infty}\kappa(t)(u_{a}(t)-u(x,t))v(x,t)\mathrm{d}{x}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
t}(\xi(t)\chi(t))-\dot{c}(t)[v]+[vUf(U)]-\kappa(t)\left[\int_{-\infty}^{c(t)}v(x,t)u(x,t)\mathrm{d}{x}+\int_{c(t)}^{\infty}v(x,t)u(x,t)\mathrm{d}{x}\right]$
$\displaystyle+\kappa(t)u_{a}(t)\left[\int_{-\infty}^{c(t)}v(x,t)\mathrm{d}{x}+\int_{c(t)}^{\infty}v(x,t)\mathrm{d}{x}\right]$
$\displaystyle=$
$\displaystyle\kappa(t)(u_{a}(t)-\chi(t))\xi(t)-\kappa(t)({\mathcal{M}}_{1}(t)-\xi(t)\chi(t))+\kappa(t)u_{a}(t)({\mathcal{M}}_{0}(t)-\xi(t))$
$\displaystyle=$
$\displaystyle\kappa(t)(u_{a}(t){\mathcal{M}}_{0}(t)-{\mathcal{M}}_{1}(t)).$

Therefore momentum ${\mathcal{M}}_{1}$ satisfies the ODE

$\displaystyle\dot{{\mathcal{M}}_{1}}(t)+\kappa(t){\mathcal{M}}_{1}(t)=\kappa(t)u_{a}(t){\mathcal{M}}_{0}(t).$
(2.18)

Solving (2.18) explicitly, we get

$\displaystyle{\mathcal{M}}_{1}(t)=e^{-\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}\left[\int_{0}^{t}e^{\int_{0}^{s}\kappa(\theta)\mathrm{d}{\theta}}\kappa(s)u_{a}(s){\mathcal{M}}_{0}(s)\mathrm{d}{s}+\bar{m}\bar{u}\right].$
(2.19)

When $\kappa(t):=\kappa$ and $u_{a}(t):=u_{a}$ are constants, using
$\dot{{\mathcal{M}}_{0}}(t)=0$ we get a simplified form of (2.18)

$\displaystyle\ddot{{\mathcal{M}}_{1}}(t)+\kappa\dot{{\mathcal{M}}_{1}}(t)=0,$

which gives

$\displaystyle{\mathcal{M}}_{1}(t)=\left(\bar{m}\bar{u}-C\right)+Ce^{-\kappa
t}$ (2.20)

for some constant $C.$ From the expressions (2.19) and (2.20) it can be easily
seen that the momentum is conserved as $t\to 0$.

###  2.2. Riemann problem to the system Eq. 1.12

Now we extend the Lemma 2.1 and Lemma 2.2 for the drift flux equation of two-
phase flow. We start with the construction of shadow wave solution for (1.12)
adjoined with the $\delta$ initial data. Similar to Section 2.1, we consider
the shadow waves solution

$\displaystyle
U^{\varepsilon}=(v^{\varepsilon},w^{\varepsilon},u^{\varepsilon})(x,t)=\begin{cases}\left(v_{l},w_{l},U_{l}(t)\right),\,\,\,\,\,\,\,\,\,\,\,\,&x<c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},\\\
\left(v_{\varepsilon}(t),w_{\varepsilon}(t),u_{\varepsilon}(t)\right),\,\,\,\,\,\,&c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}<x<c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},\\\
\left(v_{r},w_{r},U_{r}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},\end{cases}$
(2.21)

where
$U_{l,r}(t):=e^{-\int_{0}^{t}\kappa(\theta)\mathrm{d}{\theta}}\cdot\left(\int_{0}^{t}F(\theta)\mathrm{d}{\theta}+u_{l,r}\right)$
and $x_{\varepsilon},v_{\varepsilon}(t)$ are $\mathcal{O}(\varepsilon)$ and
$\mathcal{O}(1/\epsilon),$ and $\lim\limits_{\varepsilon\to
0}u_{\varepsilon}(t)=\chi(t).$ We study the system (1.12) when initial data
contains a $\delta$-measure and is of the following form

$\displaystyle(v,w,u)(x,0)=\begin{cases}(v_{l},v_{l},u_{l}),\,\,\,\,\,\,&x<0,\\\
(\bar{m}\delta(x),\bar{n}\delta(x),\bar{u}),\,\,\,\,\,\,&x=0,\\\
(v_{r},w_{r},u_{r}),\,\,\,\,\,\,&x>0,\end{cases}$

where $v_{l,r},w_{l,r}\geq 0$, $\bar{m},\bar{n}>0$ and $u_{l}>\bar{u}>u_{r}$.

Substituting the above shadow wave solution (2.21) into the system (1.12),
from Definition 1.1 we have

$\displaystyle\lim_{\varepsilon\to
0}\left[\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}\left(v^{\varepsilon}f(u^{\varepsilon})\right),\varphi\right\rangle\right]=0,$
(2.22) $\displaystyle\lim_{\varepsilon\to
0}\left[\left\langle\frac{\partial}{\partial
t}w^{\varepsilon},\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}\left(w^{\varepsilon}f(u^{\varepsilon})\right),\varphi\right\rangle\right]=0,$
(2.23) $\displaystyle\lim_{\varepsilon\to
0}\left[\left\langle\frac{\partial}{\partial
t}\left((v^{\varepsilon}+w^{\varepsilon})u^{\varepsilon}\right),\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}\left((v^{\varepsilon}+w^{\varepsilon})u^{\varepsilon}f(u^{\varepsilon})\right),\varphi\right\rangle-\left\langle\kappa(t)(u_{a}(t)-u^{\varepsilon})(v^{\varepsilon}+w^{\varepsilon}),\varphi\right\rangle\right]=0,$
(2.24)

for all $\varphi\in C^{\infty}_{c}(\mathds{R}\times[0,\infty))$.

Now passing to the limit as $\varepsilon$ tends to zero in the equations
(2.22)-(2.24), we obtain the following relations

$\displaystyle\frac{\partial}{\partial
t}(\xi_{v}(t))=\dot{c}(t)[v]-[vf(U(t))],\,\,\,\xi_{v}(0)=\bar{m},$ (2.25)
$\displaystyle\frac{\partial}{\partial
t}(\xi_{w}(t))=\dot{c}(t)[w]-[wf(U(t))],\,\,\,\xi_{w}(0)=\bar{n},$ (2.26)
$\displaystyle\frac{\partial}{\partial
t}\left((\xi_{v}(t)+\xi_{w}(t))\chi(t)\right)+\kappa(t)(\chi(t)-u_{a}(t))(\xi_{v}(t)+\xi_{w}(t))=\dot{c}(t)[(v+w)U(t)]-[(v+w)U(t)f(U(t))],$
$\displaystyle(\xi_{v}(0)+\xi_{w}(0))\chi(0)=(\bar{m}+\bar{n})\bar{u},$ (2.27)
$\displaystyle f(\chi(t))=\dot{c}(t),$ (2.28)

where $\lim\limits_{\varepsilon\to
0}2\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)v_{\varepsilon}(t)=\xi_{v}(t)$,
$\lim\limits_{\varepsilon\to
0}2\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)w_{\varepsilon}(t)=\xi_{w}(t)$,
$\lim\limits_{\varepsilon\to 0}u_{\varepsilon}(t)=\chi(t)$ and
$[\cdot]:=\cdot_{r}-\cdot_{l}$ denotes the jump across the discontinuity
curve.

Following exactly the same calculations of Section 2.1 from (2.25)-(2.28), we
obtain

$\displaystyle\Theta(t)+\Phi(t):$
$\displaystyle=\chi(t)(\xi_{v}(t)+\xi_{w}(t))=c(t)[(v+w)U(t)]$
$\displaystyle+(v_{l}+w_{l})U_{l}(t)\int_{0}^{t}f(U_{l}(\theta))\mathrm{d}{\theta}-(v_{r}+w_{r})U_{r}(t)\int_{0}^{t}f(U_{r}(\theta))\mathrm{d}{\theta}+(\bar{m}+\bar{n})\bar{U}(t),$

where $\Theta(t):=\xi_{v}(t)\chi(t)$ and $\Phi(t):=\xi_{w}(t)\chi(t)$ and
$\bar{U}$ denotes the same from Section 2.1. Again, the overcompressibility
for shadow wave solution (2.21) follows from the arguments of Section 2.1.

Now we will consider the case of Riemann-type initial data, i.e.,

$\displaystyle(v,w,u)(x,0)=\begin{cases}(v_{l},w_{l},u_{l}),\,\,\,\,\,\,\
x<0,\\\ (v_{r},w_{r},u_{r}),\,\,\,\,\,\,x>0,\end{cases}$ (2.29)

where $v_{l,r},w_{l,r}>0$. For Riemann type of initial data (2.29) when
$u_{l}>u_{r},$ substituting the shadow wave

$\displaystyle
U^{\varepsilon}=(v^{\varepsilon},w^{\varepsilon},u^{\varepsilon})(x,t)=\begin{cases}\left(v_{l},w_{l},U_{l}(t)\right),\,\,\,\,\,\,\,\,\,\,\,\,&x<c(t)-\frac{\varepsilon}{2}t,\\\
\left(v_{\varepsilon}(t),w_{\varepsilon}(t),u_{\varepsilon}(t)\right),\,\,\,\,\,\,&c(t)-\frac{\varepsilon}{2}t<x<c(t)+\frac{\varepsilon}{2}t,\\\
\left(v_{r},w_{r},U_{r}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>c(t)+\frac{\varepsilon}{2}t,\end{cases}$

we have the ODE’s (2.25)-(2.27) with $\lim\limits_{\varepsilon\to
0}\varepsilon tv_{\varepsilon}(t)=\xi_{v}(t),\lim\limits_{\varepsilon\to
0}\varepsilon
tw_{\varepsilon}(t)=\xi_{w}(t)~{}\text{and}~{}\lim\limits_{\varepsilon\to
0}u_{\varepsilon}(t)=\chi(t)$ with the initial conditions $\xi_{v}(0)=0$,
$\xi_{w}(0)=0$ and $(\xi_{v}(0)+\xi_{w}(0))\chi(0)=0$. As in the Section 2, we
take the particular form of
$u_{\varepsilon}(t)=e^{-\int_{0}^{t}\kappa(\theta)d\theta}\left(\int_{0}^{t}F(\theta)d\theta+u_{\varepsilon}\right)$,
where $u_{\varepsilon}$’s are constants and independent of $t$. Hence

$\lim\limits_{\varepsilon\to
0}e^{-\int_{0}^{t}\kappa(\theta)d\theta}\left(\int_{0}^{t}F(\theta)d\theta+u_{\varepsilon}\right)=e^{-\int_{0}^{t}\kappa(\theta)d\theta}\left(\int_{0}^{t}F(\theta)d\theta+\chi_{0}\right),$

where $\chi_{0}$ is constant. Thus the third equation of (2.25) takes the
simple form

$\displaystyle\frac{\partial}{\partial
t}\left((\xi_{v}(t)+\xi_{w}(t)\right)\cdot\chi(t)=c^{\cdot}(t)[(v+w)U(t)]-[(v+w)U(t)f(U(t))].$

The first equation of (2.25) implies

$f(\chi(t))\left([v+w](\chi(t))-[(v+w)(U(t))]\right)-(\chi(t))\left[(v+w)f(U(t))\right]+\left[(v+w)(U(t))f(U(t))\right]=0.$

Now considering the function ${\mathcal{K}}(x)$ as

${\mathcal{K}}(x)=f(x)\Big{(}[v+w]x-[(v+w)(U(t))]\Big{)}-x[(v+w)f(U(t))]+[(v+w)(U(t))f(U(t))].$

and following the arguments of Section 2, we show that the overcompressibility
condition

$U_{r}(t)<\chi(t)<U_{l}(t)\,\,\,\,\text{implies}\,\,\,\,f(U_{r}(t))<f(\chi(t))<f(U_{l}(t))$

holds.

In the case of $u_{l}<u_{r}$ the solution consists of contact discontinuity
and vacuum, i.e.,

$\displaystyle(v,w,u)(x,t)=\begin{cases}\left(v_{l},w_{l},U_{l}(t)\right),\,\,\,\,\,\,\,\,\,&x<\int_{0}^{t}f(U_{l}(s))ds,\\\
\left(0,0,z(x,t)\right),\,\,\,\,\,\,&\int_{0}^{t}f(U_{l}(s))ds<x<\int_{0}^{t}f(U_{r}(s))ds,\\\
\left(v_{r},w_{r},U_{r}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>\int_{0}^{t}f(U_{r}(s))ds,\end{cases}$

where $z(x,t)$ is a continuous function that satisfies
$z\left(\int_{0}^{t}f(U_{l}(s))ds,t\right)=U_{l}(t)$ and
$z\left(\int_{0}^{t}f(U_{r}(s))ds,t\right)=U_{r}(t).$ Summarizing the above
calculations, we proved the analogous results of Lemma 2.1 and Lemma 2.2 for
system (1.12).

##  3\. Entropy inequality

In this section, we introduce the dissipative shadow wave solution for the
system (1.9) and show the equivalence of dissipative shadow waves and
overcompressibility condition. We can express the entropy-entropy flux pair as
follows:

$\displaystyle\eta(v,u)=\frac{1}{2}vf^{2}(u)~{}~{}\text{and}~{}~{}q(v,u)=\frac{1}{2}vf^{3}(u),$

where $\eta$ and $q$ represent entropy and entropy flux, respectively.

######  Definition 3.1.

A shadow wave $U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})$ is called a
dissipative shadow wave solution if it satisfies the following entropy
inequality

$\displaystyle\lim_{\varepsilon\to
0}\left(\left\langle\frac{\partial}{\partial
t}\eta(v^{\varepsilon},u^{\varepsilon}),\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}q(v^{\varepsilon},u^{\varepsilon}),\varphi\right\rangle-\left\langle\kappa(t)(u_{a}(t)-u^{\varepsilon})v^{\varepsilon}f(u^{\varepsilon})f^{\prime}(u^{\varepsilon}),\varphi\right\rangle\right)\leq
0,$ (3.1)

for all non-negative $\varphi\in C^{\infty}_{c}(\mathds{R}\times(0,\infty))$.

######  Theorem 3.1.

A shadow wave solution $U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})$ is
dissipative if and only if

$\displaystyle
2\xi(t)f(\chi(t))f^{\prime}(\chi(t))\left\\{\frac{\partial}{\partial
t}\chi(t)+\kappa(t)\left(\chi(t)-u_{a}(t)\right)\right\\}+\frac{\partial}{\partial
t}\xi(t)f^{2}(\chi(t))$ $\displaystyle\leq
f(\chi(t))[vf^{2}(U(t))]-[vf^{3}(U(t))],$ (3.2)

where $\xi(t)$ and $\chi(t)$ are defined as before.

###### Proof.

Inserting the the shadow wave solution (2.2) and following the similar
calculation as previous, we obtain

$\displaystyle-\left\langle\frac{\partial}{\partial
t}\eta(v^{\varepsilon},u^{\varepsilon}),\varphi\right\rangle$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\left(v_{\varepsilon}(t)f^{2}(u_{\varepsilon}(t))-v_{l}f^{2}(U_{l}(t))\right)\left(\dot{c}(t)-\frac{\varepsilon}{2}\right)\varphi\left(c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},t\right)\mathrm{d}{t}$
$\displaystyle+\frac{1}{2}\int_{0}^{\infty}\left(v_{r}f^{2}(U_{r}(t))-v_{\varepsilon}(t)f^{2}(u_{\varepsilon}(t))\right)\left(\dot{c}(t)+\frac{\varepsilon}{2}\right)\varphi\left(c(t)+\frac{\varepsilon}{2}t+x_{\epsilon},t\right)\mathrm{d}{t}$
$\displaystyle-\int_{0}^{\infty}\int_{-\infty}^{c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}}\kappa(t)(u_{a}(t)-u_{l})v_{l}f(U_{l}(t))f^{\prime}(U_{l}(t))\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle-\frac{1}{2}\int_{0}^{\infty}\int_{c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon}}^{c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon}}\frac{\partial}{\partial
t}\Big{(}v_{\epsilon}(t)f^{2}(u_{\epsilon}(t))\Big{)}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle-\int_{0}^{\infty}\int_{c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon}}^{\infty}\kappa(t)(u_{a}(t)-u_{r})v_{r}f(U_{r}(t))f^{\prime}(U_{r}(t))\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t},$
(3.3)

where in the third and fifth lines we used

$\dot{U}_{l,r}(t)=\kappa(t)\left(u_{a}(t)-U_{l,r}(t)\right).$

Similarly,

$\displaystyle-\left\langle\frac{\partial}{\partial
x}q(v^{\varepsilon},u^{\varepsilon}),\varphi\right\rangle$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\left(v_{l}f^{3}(U_{l}(t))-v_{\varepsilon}(t)f^{3}(u_{\varepsilon}(t))\right)\varphi\left(c(t)-\frac{\varepsilon}{2}t-x_{\varepsilon},t\right)\mathrm{d}{t}$
$\displaystyle+\frac{1}{2}\int_{0}^{\infty}\left(v_{\varepsilon}(t)f^{3}(u_{\varepsilon}(t))-v_{r}f^{3}(U_{r}(t))\right)\varphi\big{(}c(t)+\frac{\varepsilon}{2}t+x_{\varepsilon},t\big{)}\mathrm{d}{t}.$
(3.4)

Now using the Taylor expansions (2.1) in (3)-(3) and considering the
contributions from the source term of (3.1), we find

$\displaystyle\frac{1}{2}\int_{0}^{\infty}\Bigg{[}\left(\dot{c}(t)[vf^{2}(U(t))]+\frac{\varepsilon}{2}\left(v_{r}f(U_{r}(t))+v_{l}f^{2}(U_{l}(t))\right)\right)-\frac{\partial}{\partial
t}\left(2\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)v_{\varepsilon}(t)f^{2}(u_{\varepsilon}(t))\right)$
$\displaystyle-[vf^{3}(U(t))]+4\kappa(t)\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)(u_{a}(t)-u_{\varepsilon}(t))v_{\varepsilon}(t)f(u_{\varepsilon}(t))f^{\prime}(u_{\varepsilon}(t))\Bigg{]}\varphi(c(t),t)\mathrm{d}{t}$
$\displaystyle+\frac{1}{2}\int_{0}^{\infty}\Bigg{[}c^{\cdot}(t)\left(v_{r}f^{2}(U_{r}(t))+v_{l}f(U_{l}(t))-2v_{\varepsilon}(t)f^{2}(U_{\varepsilon}(t))\right)$
$\displaystyle+\left(2v_{\varepsilon}(t)f^{3}(u_{\varepsilon}(t))-v_{l}f^{3}(U_{l}(t))-v_{r}f^{3}(U_{r}(t))\right)\left(\frac{\varepsilon}{2}t+x_{\varepsilon}\right)\Bigg{]}\frac{\partial}{\partial
x}\varphi(c(t),t)\mathrm{d}{t}.$ (3.5)

From the equation (3.1), we have

$\displaystyle\lim_{\varepsilon\to
0}\int_{0}^{\infty}\int_{-\infty}^{\infty}\left(\eta(v^{\varepsilon},u^{\varepsilon})\frac{\partial}{\partial
t}\varphi(x,t)+q(v^{\varepsilon},u^{\varepsilon})\frac{\partial}{\partial
t}\varphi(x,t)\right)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\int_{0}^{\infty}\int_{-\infty}^{\infty}\kappa(t)(u_{a}(t)-u^{\varepsilon})f(u^{\varepsilon})f^{\prime}(u^{\varepsilon})v^{\varepsilon}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}\geq
0.$ (3.6)

Now passing to the limit as $\varepsilon\to 0$, from (3) and (3), we obtain

$\frac{\partial}{\partial
t}\left(\xi(t)f^{2}(\chi(t))\right)\leq\dot{c}(t)[vf^{2}(U(t))]-[vf^{3}(U(t))]+2\kappa(t)\left(u_{a}(t)-\chi(t)\right)\xi(t)f(\chi(t))f^{\prime}(\chi(t)),$
(3.7)

and simplifying we get the inequality (3.1). ∎

######  Lemma 3.1.

Let $f^{2}$ be a convex function, then a shadow wave solution
$U^{\varepsilon}$ is dissipative if and only if it satisfies the
overcompressibility condition.

###### Proof.

Using convexity of $f^{2}$, we have a function $g$ such that

$\displaystyle f^{2}(x)=f^{2}(x_{0})+2(x-x_{0})f(x_{0})f^{\prime}(x_{0})+g(x)$
(3.8)

with $g\geq 0$ and $g(x_{0})=g^{\prime}(x_{0})=0.$ Inserting (3.8) into the
inequality (3.7) and taking $x=\chi(t),x_{0}=\chi(t_{0})$ we have

$\displaystyle\frac{\partial}{\partial
t}\left(\xi(t)\left[2f(\chi(t_{0}))f^{\prime}(\chi(t_{0}))\chi(t)+f^{2}(\chi(t_{0}))-2f(\chi(t_{0}))f^{\prime}(\chi(t_{0}))\chi(t_{0})+g(\chi(t))\right]\right)\Bigg{|}_{t=t_{0}}$
$\displaystyle\leq
f(\chi(t_{0}))[vf^{2}(U(t_{0}))]-[vf^{3}(U(t_{0}))]+2\kappa(t_{0})\left(u_{a}(t_{0})-\chi(t_{0})\right)\xi(t_{0})f(\chi(t_{0}))f^{\prime}(\chi(t_{0})).$

Since $g(\chi(t_{0}))=g^{\prime}(\chi(t_{0}))=0$, we find
$\frac{\partial}{\partial t}(\xi(t)g(\chi(t)))\Big{|}_{t=t_{0}}=0$ and the
above inequality takes the form

$\displaystyle 2f(\chi(t_{0}))f^{\prime}(\chi(t_{0}))\frac{\partial}{\partial
t}\left(\xi(t)\chi(t)\right)\Big{|}_{t=t_{0}}+\left(f^{2}(\chi(t_{0}))-2f(\chi(t_{0}))f^{\prime}(\chi(t_{0}))\chi(t_{0})\right)\frac{\partial}{\partial
t}\xi(t)\Big{|}_{t=t_{0}}$ $\displaystyle\leq
f(\chi(t_{0}))[vf^{2}(U(t_{0}))]-[vf^{3}(U(t_{0}))]+2\kappa(t_{0})(u_{a}(t_{0})-\chi(t_{0}))\xi(t_{0})f(\chi(t_{0}))f^{\prime}(\chi(t_{0})).$

Now using the ODE (2.13) and simplifying the above inequality, we find

$v_{r}\Big{(}f(\chi(t_{0}))-f(U_{r}(t_{0}))\Big{)}g(U_{r}(t_{0}))-v_{l}\Big{(}f(\chi(t_{0}))-f(U_{l}(t_{0}))\Big{)}g(U_{l}(t_{0}))\geq
0.$ (3.9)

If the shadow wave is overcompressive, non-negativity of $v_{l,r}$ and
positivity of $g$ implies that the above inequality (3.9) holds, i.e., the
shadow wave is dissipative for any $t_{0}$. The other way is not difficult to
see and follows from the arguments of [33]. ∎

######  Remark 3.1.

In the above proof, we assumed that $f\in C^{1}$ and $f^{2}$ be convex, but
there is no assumption on the sign of $f^{\prime}$, which is assumed to be
positive in the previous section. $f^{\prime}>0$ becomes an admissible
assumption if we consider the relation of the system (1.9) with scalar
conservation laws. As we mentioned earlier for the smooth solution the system
(1.9) reduced to

$u_{t}+f(u)u_{x}=\kappa(t)(u_{a}(t)-u)$

which is of the form $u_{t}+F(u)_{x}=h(x,t,u)$ with $F^{\prime}(u)=f(u).$ For
scalar conservation laws convexity condition
$F^{\prime\prime}(u)=f^{\prime}(u)>0$ on flux function is useful to obtain
explicit formulas. Let us define the sets
$S_{1}:=\left\\{f\,\,|\,\,f\,\,\,\text{is}\,\,\,C^{1}\,\,\,\text{and}\,\,\,f^{\prime}>0\right\\}$
and $S_{2}:=\left\\{f\,\,|\,\,f^{2}\,\,\,\text{is convex}\right\\}$. We
observe that there is no containment relation between $S_{1}$ and $S_{2}.$ For
instance, $S_{1}\not\subset S_{2}:$ let $f(x)=\log x,$
$f^{\prime}=\frac{1}{x}>0$ for $x>0$ and
$(f^{2})^{\prime\prime}(x)=\frac{2}{x^{2}}(1-\log x)<0$ for some $x>0.$ Again
$S_{2}\not\subset S_{1}:$ let $f(x)=-x,$ then $f^{2}=x^{2}$ which is convex
but $f^{\prime}=-1<0.$ Hence in this section we are working with the class of
functions $S_{1}\cap S_{2}.$

##  4\. Global existence of entropy solution

In this section, we begin with a concise description of the shadow wave
tracking procedure. Subsequently, we give the proofs for Theorem 1.1 and
Theorem 1.2.

###  4.1. The wave tracking algorithm

Let us introduce the shadow wave tracking algorithm. To begin, we establish
the notations primarily borrowed from [41].

  1. 1.

For any two given states $(v_{i},u_{i})$ and $(v_{j},u_{j})$ with
$i<j,v_{i},v_{j}>0,$ $SDW_{i,j}$ denotes a shadow wave solution joining
$(v_{i},u_{i})$ on the left and $(v_{j},u_{j})$ on the right. Note that
$SDW_{i,j}$ exists if $u_{i}>u_{j}.$

  2. 2.

For any given three states $(v_{i},u_{i}),(v_{m},u_{m}),(v_{j},u_{j})$ with
$i<m<j,v_{i},v_{m},v_{j}>0.$ Suppose the states $(v_{i},u_{i})$ and
$(v_{m},u_{m})$ are connected by contact discontinuities
$CD^{i}_{1},CD^{m}_{2}$ and a vacuum $Vac_{i,m}=(0,u_{i}(x,t))$ and the states
$(v_{m},u_{m})$ and $(v_{j},u_{j})$ are connected by a shadow wave
$SDW_{m,j}.$ Now at time $t=T,$ if the states $(0,u_{i}(x,T))$ on the left and
$(v_{j},u_{j})$ on the right are connected by a shadow wave, we denote it as
$~{}^{i}\\!{SDW}_{j}.$ Note that $~{}^{i}\\!{SDW}_{m}=CD^{m}_{2}.$

  3. 3.

In a similar situation described above, $SDW^{j}_{i}$ denotes the shadow wave
joining the states $(v_{i},u_{i})$ on the left to the state $(0,u_{j}(x,t))$
on the right.

  4. 4.

Finally $~{}^{i}\\!{SDW}^{j}$ denotes a shadow wave joining the state
$(0,u_{i}(x,t))$ on the left to a state $(0,u_{j}(x,t))$ on the right.

Now we describe the wavefront tracking algorithm. We take the initial data for
the system (1.9) as

$\displaystyle(v,u)(x,0)=\begin{cases}(v_{0},u_{0}),\,\,\,\,\,\,&x\leq R,\\\
(v(x),u(x)),\,\,\,\,\,\,&x>R.\end{cases}$ (4.1)

Here $v_{0},u_{0}\in\mathds{R},$ $v_{0}>0$ and $v(x)>0,u(x)\in
C_{b}\left([R,\infty)\right).$ Let $\varepsilon$ be given sufficiently small
positive real number. Define
$((v^{\varepsilon}(x),u^{\varepsilon}(x)))_{\varepsilon}$ as a piecewise
constant approximation for $(v(x),u(x)).$ For a fixed $\varepsilon>0,$ take a
corresponding partition $R:=Y_{0}<Y_{1}<Y_{2}\cdot\cdot\cdots$ of the interval
$[R,\infty)$ that satisfies
$Y_{i}-Y_{i+1}\leq\rho(\varepsilon),i=0,1,2,\cdot\cdot\cdots$ where
$\rho(\varepsilon)\to 0$ as $\varepsilon\to 0.$ Now choose
$v_{i+1}:=v^{\varepsilon}(x)=v(Y_{i+1})$ and
$u_{i+1}:=u^{\varepsilon}(x)=u(Y_{i+1})$ for
$x\in(Y_{i},Y_{i+1}],i\in\mathds{N}\cup\\{0\\}.$

Algorithm:

Step 0: Let $v_{0},u_{0}$ be given as in (4.1) and
$\left(\\{v_{i}\\},\\{u_{i}\\}\right)_{i\in\mathds{N}}$ be a piecewise
constant approximation obtained by the procedure describes as above. Therefore
an approximation of the initial data (4.1) can be expressed as follows:

$\displaystyle U_{i}=(v,u)(x,0)=\begin{cases}(v_{i},u_{i}),\,\,\,\,\,\,&x\leq
Y_{i},\\\ (v_{i+1},u_{i+1}),\,\,\,\,\,\,&x>Y_{i},\end{cases}$ (4.2)

for $i=0,1,2,\cdot\cdot\cdots$.

Step 1: Let $S_{0}$ denotes the collection of all states at time $t=T_{0},$
i.e., $S_{0}=\\{U_{i}\,\,\,\,|\,\,\,\,i\in I_{0}\\}$ where $U_{i}$’s are given
by (4.2) and $I_{0}=\\{0,1,2,3,\cdot\cdots\\}$ be an index set. A shadow wave
solution of the Riemann problem consisting of the states in $S_{0}$ gives rise
to two possibilities: either there is no further interaction of waves or two
(possibly more) waves interact at time $t=T_{1}.$ If there is no further
interaction then the procedure has to be stopped at $t=T_{0}.$ In the event of
wave interaction, there are four possible ways in which the interaction can
occur, as described above. Each interaction leads to a single resulting shadow
wave. The new resulting waves and the non-interacted waves together give a new
collection of initial states $S_{1}=\\{U_{i}\,\,\,\,|\,\,\,\,i\in I_{1}\\}$
where $I_{1}\subset I_{0}$ is a new index set for $t\geq T_{1}.$

Step j to j+1: Suppose that $j$-th interaction happens at a time $t=T_{j}$.
Then eliminating all the middle states from $S_{j-1},$ we obtain a new
collection of states $S_{j}=\\{U_{i}\,\,\,\,|\,\,\,\,i\in I_{j}\\}$ where
$I_{j}=\\{0,j_{1},j_{2},\cdots\cdots\\}\subseteq I_{j-1},1\leq
j_{1}<j_{2}<\cdots$ denotes the index set at $j$-th level. All non-interacting
waves continue to propagate after $t>T_{j}$. We can repeat the procedure by
substituting $j+1$ in place of $j$ after a new interaction at $t=T_{j+1}.$ The
algorithm finishes if there is no such $T_{j+1}$.

The above-stated procedure will help us to prove the global existence of an
admissible solution to the problem (1.9) and (4.1).

###  4.2. Proof of Theorem 1.1

Let us consider the initial data

$\displaystyle(v,u)(x,0)=\begin{cases}(v_{l},u_{l}),\,\,\,\,\,\,&x<X_{1},\\\
(v_{m},u_{m}),\,\,\,\,\,\,&X_{1}<x<X_{2},\\\
(v_{r},u_{r}),\,\,\,\,\,\,&x>X_{2},\end{cases}$

with $u_{l}>u_{m}>u_{r}$. Then two shadow waves are emanating from $X_{1}$ and
$X_{2}$ with the central shadow wave line $x(t)=X_{1}+c(t)$ and
$x(t)=X_{2}+c(t)$. Let
$x_{1}(t)=X_{1}+c(t)+\frac{\varepsilon}{2}t+x_{1,\varepsilon}$ and
$x_{2}(t)=X_{2}+c(t)-\frac{\varepsilon}{2}t-x_{2,\varepsilon}$ are the right
external shadow wave line and left external shadow wave line, respectively.
Suppose $T_{\varepsilon}$ is the time when two external shadow wave lines
interact and $T$ is the time when two central shadow wave lines interact. Then
$T-T_{\varepsilon}=\mathcal{O}(\varepsilon)$. In summary of the aforementioned
fact, we can state the following lemma.

######  Lemma 4.1 ([41], Lemma 3.2).

Let two approaching shadow waves with the central lines given by $x=c_{l}(t)$
and $x=c_{r}(t)$ interact at time $t=\tilde{T}$. The value of $\tilde{T}$ is
obtained by solving the equation

$\displaystyle
c_{l}(t)+\frac{\varepsilon}{2}(t-T_{l})+x_{l,\varepsilon}=c_{r}(t)-\frac{\varepsilon}{2}(t-T_{r})+x_{r,\varepsilon}$

where $c_{l}(t)+\frac{\varepsilon}{2}(t-T_{l})+x_{l,\varepsilon}$ is the right
external SDW line of the first approaching shadow wave, while
$c_{r}(t)-\frac{\varepsilon}{2}(t-T_{r})+x_{r,\varepsilon}$ is the left
external SDW line of the second approaching shadow wave. Also, let
$x_{l,\varepsilon},x_{r,\varepsilon}\sim\varepsilon$. A solution T to
$c_{l}(t)=c_{r}(t)$ will be called the interaction time since the area bounded
by two external shadow wave lines, and the lines $t=T$ and $t=\tilde{T}$ is of
order $\varepsilon^{2}$ and all terms of order $\varepsilon^{\alpha},\alpha>1$
are neglected. Note that $T=\tilde{T}+\mathcal{O}(\varepsilon)$. The assertion
stays true if one of the shadow waves is substituted by a contact
discontinuity.

The next lemma is the first step towards proving Theorem 1.1. We will be using
Lemma 4.1 in the following proofs without mentioning it repeatedly.

######  Lemma 4.2.

Let $v(x)>0,u(x)\in L^{\infty}([R,\infty))\cap C([R,\infty))$ and $u(x)$ be an
increasing function on $[R,\infty)$. Take a partition
$\\{Y_{i}\\}_{i\in\mathds{N}\cup\\{0\\}}$ of $[R,\infty)$ such that
$Y_{0}=R,C_{1}\varepsilon^{\alpha}<Y_{i}-Y_{i+1}<C_{2}\rho(\varepsilon)$ for
every $i=0,1,2,\cdot\cdots$ where $C_{1},C_{2}\geq 1,\alpha\in(0,1)$ and
$\rho(\varepsilon)\to 0$ as $\varepsilon\to 0.$ Then there exists a global
admissible solution to (1.9) and (4.1). More precisely, there exists a
function $U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})$ that satisfies

$\displaystyle\lim_{\varepsilon\to
0}\begin{cases}\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}(v^{\varepsilon}f(u^{\varepsilon})),\varphi\right\rangle=0,\\\
\left\langle\frac{\partial}{\partial
t}(v^{\varepsilon}u^{\varepsilon}),\varphi\right\rangle+\left\langle\frac{\partial}{\partial
x}(v^{\varepsilon}u^{\varepsilon}f(u^{\varepsilon})),\varphi\right\rangle=\left\langle\kappa(t)(u_{a}(t)-u^{\varepsilon})v^{\varepsilon},\varphi\right\rangle,\end{cases}$
(4.3)

for every test function $\varphi\in
C^{\infty}_{c}\left(\mathds{R}\times[0,\infty)\right)$ and the admissibility
condition.

###### Proof.

To prove this lemma, we consider two cases.

Case 1. When $u_{0}\leq u(Y_{1}).$ Since $u$ is an increasing function on
$[R,\infty),$ we get $u_{0}<u_{1}<u_{2},\cdot\cdot\cdots$ and at each
$\\{Y_{i}\\}_{i\in\mathds{N}\cup\\{0\\}}$ we have a solution as a combination
of contact discontinuities and vacuum, i.e.,
$CD^{i}_{1}+Vac_{i,i+1}+CD^{i+1}_{2}$ for $i=0,1,2,\cdot\cdot\cdots.$ As the
speeds of the fronts do not overlap each other, no interaction occurs in this
case.

Case 2. When $u_{0}>u(Y_{1}).$ Since $u(x)$ is bounded for $x>R,$ we have
$\lim\limits_{i\to\infty}u_{i}=\tilde{u}.$ Then two cases arise.

Subcase 1. Let $\tilde{u}\leq u_{0}.$ Let $T_{01}$ be the time when first
interaction occurs, i.e., $SDW_{0,1}$ meets
$CD^{1}_{1}=Y_{1}+\int_{0}^{t}f(U_{1}(\xi))d\xi.$ Therefore the first step is
to study $-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle$ in the interval $[0,T_{01}].$ In the
interval $[0,T_{0,1}],$ we have

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle$
$\displaystyle=\int_{0}^{T_{01}}\int_{-\infty}^{\infty}v^{\varepsilon}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}-\int_{-\infty}^{\infty}\left[(v^{\varepsilon}\varphi)(x,T_{01}-)-(v^{\varepsilon}\varphi)(x,0)\right]\mathrm{d}{x}$
$\displaystyle=I_{0}-\int_{-\infty}^{\infty}\left[(v^{\varepsilon}\varphi)(x,T_{01}-)-(v^{\varepsilon}\varphi)(x,0)\right]\mathrm{d}{x}.$

Inserting the shadow wave

$\displaystyle
U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})=\begin{cases}\left(v_{0},U_{0}(t)\right),\,\,\,\,\,\,\,\,\,\,\,\,&x<R+c(t)-\frac{\varepsilon}{2}t,\\\
\left(v_{0,\varepsilon}(t),u_{0,\varepsilon}(t)\right),\,\,\,\,\,\,&R+c(t)-\frac{\varepsilon}{2}t<x<R+c(t)+\frac{\varepsilon}{2}t,\\\
\left(v_{1},U_{1}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>R+c(t)+\frac{\varepsilon}{2}t,\end{cases}$

in the above equation, we obtain

$\displaystyle I_{0}$
$\displaystyle=\int_{0}^{T_{01}}\int_{-\infty}^{R+c(t)-\frac{\varepsilon}{2}t}v_{0}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}+\int_{0}^{T_{01}}\int_{R+c(t)-\frac{\varepsilon}{2}t}^{R+c(t)+\frac{\varepsilon}{2}t}v_{0,\varepsilon}(t)\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}\int_{R+c(t)+\frac{\varepsilon}{2}t}^{Y_{1}+\int_{0}^{t}f(U_{1}(\xi))d\xi}v_{1}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\sum_{i=1}^{\infty}\int_{0}^{T_{01}}\int_{Y_{i}+\int_{0}^{t}f(U_{i}(\xi))d\xi}^{Y_{i+1}+\int_{0}^{t}f(U_{i+1}(\xi))d\xi}v_{i+1}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}=I_{01}+I_{02}+I_{03}+I_{04}.$

Simplifying the above expressions, we get

$\displaystyle
I_{01}=-\int_{0}^{T_{01}}v_{0}\varphi\left(R+c(t)-\frac{\varepsilon}{2}t,t\right)\left(\dot{c}(t)-\frac{\varepsilon}{2}\right)\mathrm{d}{t}+\int_{-\infty}^{R+c(T_{01})-\frac{\varepsilon}{2}T_{01}}v_{0}\varphi(x,T_{01})\mathrm{d}{x}-\int_{-\infty}^{R}v_{0}\varphi(x,0)\mathrm{d}{x},$
$\displaystyle
I_{02}=\int_{R+c(T_{01})-\frac{\varepsilon}{2}T_{01}}^{R+c(T_{01})+\frac{\varepsilon}{2}T_{01}}v_{0,\varepsilon}(T_{01})\varphi\left(x,T_{01}\right)\mathrm{d}{x}-\int_{0}^{T_{01}}v_{0,\varepsilon}(t)\varphi\left(R+c(t)+\frac{\varepsilon}{2}t,t\right)\left(\dot{c}(t)+\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}v_{0,\varepsilon}(t)\varphi\left(R+c(t)-\frac{\varepsilon}{2}t,t\right)\left(\dot{c}(t)-\frac{\varepsilon}{2}\right)\mathrm{d}{t}-\int_{R+c(t)-\frac{\varepsilon}{2}t}^{R+c(t)+\frac{\varepsilon}{2}t}\frac{\partial}{\partial
t}v_{0,\varepsilon}(t)\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t},$ $\displaystyle
I_{03}$
$\displaystyle=\int_{R+c(T_{01})+\frac{\varepsilon}{2}T_{01}}^{Y_{1}+\int_{0}^{T_{01}}f(U_{1}(\xi))d\xi}v_{1}\varphi(x,T_{01})\mathrm{d}{x}-\int_{R}^{Y_{1}}v_{1}\varphi(x,0)\mathrm{d}{x}+\int_{0}^{T_{01}}v_{1}\varphi\left(R+c(t)+\frac{\varepsilon}{2}t\right)\left(\dot{c}(t)+\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle-\int_{0}^{T_{01}}v_{1}\varphi\left(Y_{1}+\int_{0}^{t}f(U_{1}(\xi))d\xi,t\right)f\left(U_{1}(t)\right)\mathrm{d}{t},$
$\displaystyle
I_{04}=\sum_{i=1}^{\infty}\int_{Y_{i}+\int_{0}^{T_{01}}f(U_{i}(\xi))d\xi}^{Y_{i+1}+\int_{0}^{T_{01}}f(U_{i+1}(\xi))d\xi}v_{i+1}\varphi(x,T_{01})\mathrm{d}{x}-\sum_{i=1}^{\infty}\int_{Y_{i}}^{Y_{i+1}}v_{i+1}\varphi(x,0)\mathrm{d}{x}$
$\displaystyle-\sum_{i=1}^{\infty}\int_{0}^{T_{01}}v_{i+1}\varphi\left(Y_{i+1}+\int_{0}^{t}f(U_{i+1}(\xi))d\xi,t\right)f\left(U_{i+1}(t)\right)\mathrm{d}{t}$
$\displaystyle+\sum_{i=1}^{\infty}\int_{0}^{T_{01}}v_{i+1}\varphi\left(Y_{i}+\int_{0}^{t}f(U_{i+1}(\xi))d\xi,t\right)f\left(U_{i+1}(t)\right)\mathrm{d}{t}.$

Finally adding up $I_{01},I_{02},I_{03}$ and $I_{04},$ we obtain

$\displaystyle I_{0}=$
$\displaystyle\int_{0}^{T_{01}}(v_{0,\varepsilon}(t)-v_{0})\varphi\left(R+c(t)-\frac{\varepsilon}{2}t,t\right)\left(\dot{c}(t)-\frac{\varepsilon}{2}\right)\mathrm{d}{t}-\int_{R+c(t)-\frac{\varepsilon}{2}t}^{R+c(t)+\frac{\varepsilon}{2}t}\frac{\partial}{\partial
t}v_{0,\varepsilon}(t)\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}(v_{1}-v_{0,\varepsilon}(t))\varphi\left(R+c(t)+\frac{\varepsilon}{2}t,t\right)\left(\dot{c}(t)+\frac{\varepsilon}{2}\right)\mathrm{d}{t}+\int_{-\infty}^{\infty}v^{\varepsilon}\varphi(x,T_{01})\mathrm{d}{x}$
$\displaystyle-\int_{-\infty}^{\infty}v^{\varepsilon}\varphi(x,0)\mathrm{d}{x}-\int_{0}^{T_{01}}v_{1}\varphi\left(Y_{1}+\int_{0}^{t}f(U_{1}(\xi))d\xi,t\right)f\left(U_{1}(t)\right)\mathrm{d}{t}$
$\displaystyle-\sum_{i=1}^{\infty}\int_{0}^{T_{01}}v_{i+1}\varphi\left(Y_{i+1}+\int_{0}^{t}f(U_{i+1}(\xi))d\xi,t\right)f\left(U_{i+1}(t)\right)\mathrm{d}{t}$
$\displaystyle+\sum_{i=1}^{\infty}\int_{0}^{T_{01}}v_{i+1}\varphi\left(Y_{i}+\int_{0}^{t}f(U_{i+1}(\xi))d\xi,t\right)f\left(U_{i+1}(t)\right)\mathrm{d}{t}.$

A similar calculation for the flux term in $[0,T_{01}]$ gives

$\displaystyle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle=$
$\displaystyle\int_{0}^{T_{01}}\left(v_{0}f(U_{0}(t))-v_{0,\varepsilon}(t)f(u_{0,\varepsilon}(t))\right)\varphi\left(R+c(t)-\frac{\varepsilon}{2}t,t\right)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}\left(v_{0,\varepsilon}(t)f(u_{0,\varepsilon}(t))-v_{1}f(U_{1}(t))\right)\varphi\left(R+c(t)+\frac{\varepsilon}{2}t,t\right)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}v_{1}f(U_{1}(t))\varphi\left(Y_{1}+\int_{0}^{t}f(U_{1}(\xi))d\xi,t\right)\mathrm{d}{t}$
$\displaystyle+\sum_{i=1}^{\infty}\int_{0}^{T_{01}}v_{i+1}f(U_{i+1}(t))\varphi\left(Y_{i+1}+\int_{0}^{t}f(U_{i+1}(\xi))d\xi,t\right)\mathrm{d}{t}$
$\displaystyle-\sum_{i=1}^{\infty}\int_{0}^{T_{01}}v_{i+1}f(U_{i+1}(t))\varphi\left(Y_{i}+\int_{0}^{t}f(U_{i+1}(\xi))d\xi,t\right)\mathrm{d}{t}$

and hence we have

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle=$
$\displaystyle\int_{0}^{T_{01}}(v_{0,\varepsilon}(t)-v_{0})\varphi\left(R+c(t)-\frac{\varepsilon}{2}t,t\right)\left(\dot{c}(t)-\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}(v_{1}-v_{0,\varepsilon}(t))\varphi\left(R+c(t)+\frac{\varepsilon}{2}t,t\right)\left(\dot{c}(t)+\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle-\int_{R+c(t)-\frac{\varepsilon}{2}t}^{R+c(t)+\frac{\varepsilon}{2}t}\frac{\partial}{\partial
t}v_{0,\varepsilon}(t)\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}\left(v_{0}f(U_{0}(t))-v_{0,\varepsilon}(t)f(u_{0,\varepsilon}(t))\right)\varphi\left(R+c(t)-\frac{\varepsilon}{2}t,t\right)\mathrm{d}{t}$
$\displaystyle+\int_{0}^{T_{01}}\left(v_{0,\varepsilon}(t)f(u_{0,\varepsilon}(t))-v_{1}f(U_{1}(t))\right)\varphi\left(R+c(t)+\frac{\varepsilon}{2}t,t\right)\mathrm{d}{t}.$
(4.4)

Now using the Taylor series expansion (2.1) in (4.4) and following the
calculations of Section 2 (cf. Lemma 2.1), we find

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle=\mathcal{O}(\varepsilon)\quad\text{in}\quad[0,T_{01}].$
(4.5)

Next by replacing $v^{\varepsilon}$ with $v^{\varepsilon}u^{\varepsilon}$ in
(4.2) and performing calculations similar to the ones described above, Lemma
2.1 gives

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon}u^{\varepsilon},\varphi\right\rangle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}u^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle+\left\langle\kappa(t)\left(u_{a}(t)-u^{\varepsilon}\right)v^{\varepsilon}\right\rangle=\mathcal{O}(\varepsilon)\quad\text{in}\quad[0,T_{01}].$
(4.6)

Now we consider the interval $[T_{0i},T_{1i}].$ Note that at $T_{0i}$-th level
for each $i\in\\{1,2,\cdot\cdot\cdots\\},$ we have initial data that contains
delta function as follows

$\displaystyle(v,u)(x,T_{0i})=\begin{cases}(v_{0},u_{0}),\,\,\,\,\,&\text{if}\,\,\,\,x<X_{i,i},\\\
(\xi(T_{0i})\delta_{T_{0i}},\chi(T_{0i})),\,\,\,\,\,&\text{if}\,\,\,\,x=X_{i,i},\\\
(0,u_{j}),\,\,\,\,\,&\text{if}\,\,\,\,X_{j,j}<x<X_{j,j+1},\\\
(v_{j+1},u_{j+1}),\,\,\,\,\,&\text{if}\,\,\,\,X_{j,j+1}<x<X_{j+1,j+1},\end{cases}$

where

$\displaystyle
X_{j,j}=Y_{j}+\int_{0}^{T_{0i}}f(U_{j}(\xi))d\xi,~{}~{}X_{j,j+1}=Y_{j}+\int_{0}^{T_{0i}}f(U_{j+1}(\xi))\mathrm{d}{\xi},$
$\displaystyle
X_{j+1,j+1}=Y_{j+1}+\int_{0}^{T_{0i}}f(U_{j+1}(\xi))\mathrm{d}{\xi}\,\,\,\,\text{for}\,\,\,\,j=i,i+1,i+2\cdot\cdot\cdots.$

Therefore we use the shadow wave (2.2), i.e.,

$\displaystyle
U^{\varepsilon}=(v^{\varepsilon},u^{\varepsilon})(x,t)=\begin{cases}\left(v_{0},U_{0}(t)\right),\,\,\,\,\,\,\,\,&x<X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon},\\\
\left(v_{T_{0i},\varepsilon}(t),u_{T_{0i},\varepsilon}(t)\right),\,\,\,\,\,\,&X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon}\\\
&<x<X_{i,i}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon},\\\
\left(0,U_{i}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&X_{ii}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon}\\\
&<x<Y_{i}+\int_{0}^{t}f(U_{i+1}(\xi))\mathrm{d}{\xi},\\\
(v_{j+1},U_{j+1}(t)),\,\,\,\,&Y_{j}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi}<x<Y_{j+1}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi},\end{cases}$
(4.7)

for $j=i,i+1,i+2,\cdots\cdot\cdots.$ In the intervals $[T_{0i},T_{1i}]$, we
have

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle$
$\displaystyle=\int_{T_{0i}}^{T_{1i}}\int_{-\infty}^{\infty}v^{\varepsilon}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}-\int_{-\infty}^{\infty}\left[(v^{\varepsilon}\varphi)(x,T_{1i}-)-(v^{\varepsilon}\varphi)(x,T_{0i}+)\right]\mathrm{d}{x}$
$\displaystyle=I_{1}-\int_{-\infty}^{\infty}\left[(v^{\varepsilon}\varphi)(x,T_{1i}-)-(v^{\varepsilon}\varphi)(x,T_{0i}+)\right]\mathrm{d}{x}.$

Substituting the shadow wave (4.7) in the above equation, we get

$\displaystyle I_{1}$
$\displaystyle=\int_{T_{0i}}^{T_{1i}}\int_{-\infty}^{X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon}}v_{0}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\int_{T_{0i}}^{T_{1i}}\int_{{X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon}}}^{{X_{i,i}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon}}}v_{T_{0i},\varepsilon}(t)\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\sum_{j=i}^{\infty}\int_{T_{0i}}^{T_{1i}}\int_{Y_{j}+\int_{0}^{t}f(U_{j+1}(\xi))d\xi}^{Y_{j+1}+\int_{0}^{t}f(U_{j+1}(\xi))d\xi}v_{j+1}\frac{\partial}{\partial
t}\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}=I_{11}+I_{12}+I_{13}.$

Simplifying the above, we have

$\displaystyle I_{11}$
$\displaystyle=-\int_{T_{0i}}^{T_{1i}}v_{0}\varphi\left(X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon},t\right)\left(\dot{c}(t-T_{0i})-\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle+\int_{-\infty}^{X_{i,i}+c(T_{1i}-T_{0i})-\frac{\varepsilon}{2}(T_{1i}-T_{0i})-x^{T_{0i}}_{\varepsilon}}v_{0}\varphi(x,T_{1i})\mathrm{d}{x}-\int_{-\infty}^{X_{i,i}-x^{T_{0i}}_{\varepsilon}}v_{0}\varphi(x,T_{0i})\mathrm{d}{x},$
$\displaystyle I_{12}$
$\displaystyle=\int_{X_{i,i}+c(T_{1i}-T_{0i})-\frac{\varepsilon}{2}(T_{1i}-T_{0i})-x^{T_{0i}}_{\varepsilon}}^{X_{i,i}+c(T_{1i}-T_{0i})+\frac{\varepsilon}{2}(T_{1i}-T_{0i})+x^{T_{0i}}_{\varepsilon}}v_{T_{0i},\varepsilon}(T_{1i})\varphi(x,T_{1i})\mathrm{d}{x}-\int_{X_{i,i}-x^{T_{0i}}_{\varepsilon}}^{X_{i,i}+x^{T_{0i}}_{\varepsilon}}v_{T_{0i},\varepsilon}(T_{0i})\varphi(x,T_{0i})\mathrm{d}{x}$
$\displaystyle+\int_{T_{0i}}^{T_{1i}}v_{T_{0i},\varepsilon}(t)\varphi\left(X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon},t\right)\left(\dot{c}(t-T_{0i})-\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle-\int_{T_{0i}}^{T_{1i}}v_{T_{0i},\varepsilon}(t)\varphi\left(X_{i,i}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon},t\right)\left(\dot{c}(t-T_{0i})+\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle-\int_{T_{0i}}^{T_{1i}}\int_{X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon}}^{X_{i,i}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon}}\frac{\partial}{\partial
t}v_{T_{0i},\varepsilon}(t)\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t},$

and

$\displaystyle I_{13}$
$\displaystyle=\sum_{j=i}^{\infty}\int_{Y_{j}+\int_{0}^{T_{1i}}f(U_{j+1}(\xi))\mathrm{d}{\xi}}^{Y_{j+1}+\int_{0}^{T_{1i}}f(U_{j+1}(\xi))\mathrm{d}{\xi}}v_{j+1}\varphi(x,T_{1i})\mathrm{d}{x}\mathrm{d}{t}-\sum_{j=i}^{\infty}\int_{Y_{j}+\int_{0}^{T_{0i}}f(U_{j+1}(\xi))\mathrm{d}{\xi}}^{Y_{j+1}+\int_{0}^{T_{0i}}f(U_{j+1}(\xi))\mathrm{d}{\xi}}v_{j+1}\varphi(x,T_{0i})\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\sum_{j=i}^{\infty}\int_{T_{0i}}^{T_{1i}}v_{j+1}f(U_{j+1}(t))\left[\varphi\left(Y_{j}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi},t\right)-\varphi\left(Y_{j+1}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi},t\right)\right]\mathrm{d}{t}.$

Summing up $I_{11},I_{12},I_{13}$ and inserting the expression of $I_{1}$ in
above, we obtain

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle$
$\displaystyle=\int_{T_{0i}}^{T_{1i}}\left(v_{T_{0i},\varepsilon}(t)-v_{0}\right)\varphi\left(X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon},t\right)\left(\dot{c}(t-T_{0i})-\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle-\int_{T_{0i}}^{T_{1i}}v_{T_{0i},\varepsilon}(t)\varphi\left(X_{i,i}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon},t\right)\left(\dot{c}(t-T_{0i})+\frac{\varepsilon}{2}\right)\mathrm{d}{t}$
$\displaystyle-\int_{T_{0i}}^{T_{1i}}\int_{X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon}}^{X_{i,i}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon}}\frac{\partial}{\partial
t}v_{T_{0i},\varepsilon}(t)\varphi(x,t)\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle+\sum_{j=i}^{\infty}\int_{T_{0i}}^{T_{1i}}v_{j+1}f(U_{j+1}(t))\left[\varphi\left(Y_{j}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi},t\right)-\varphi\left(Y_{j+1}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi},t\right)\right]\mathrm{d}{t}.$

Again,

$\displaystyle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle$
$\displaystyle=\int_{T_{0i}}^{T_{1i}}\left(v_{0}f(U_{0}(t))-v_{T_{0i},\varepsilon}(t)f(u_{T_{0i},\varepsilon}(t))\right)\varphi\left(X_{i,i}+c(t-T_{0i})-\frac{\varepsilon}{2}(t-T_{0i})-x^{T_{0i}}_{\varepsilon},t\right)\mathrm{d}{t}$
$\displaystyle+\int_{T_{0i}}^{T_{1i}}v_{T_{0i},\varepsilon}(t)f(u_{T_{0i},\varepsilon}(t))\varphi\left(X_{i,i}+c(t-T_{0i})+\frac{\varepsilon}{2}(t-T_{0i})+x^{T_{0i}}_{\varepsilon},t\right)\mathrm{d}{t}$
$\displaystyle+\sum_{j=i}^{\infty}\int_{T_{0i}}^{T_{1i}}v_{j+1}f(U_{j+1}(t))\left[\varphi\left(Y_{j+1}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi},t\right)-\varphi\left(Y_{j}+\int_{0}^{t}f(U_{j+1}(\xi))\mathrm{d}{\xi},t\right)\right]\mathrm{d}{t}.$

Hence, we have

$\displaystyle-\left\langle\frac{\partial}{\partial
t}v^{\varepsilon},\varphi\right\rangle-\left\langle\frac{\partial}{\partial
x}v^{\varepsilon}f(u^{\varepsilon}),\varphi\right\rangle=\mathcal{O}(\varepsilon)\quad\text{in}\quad[T_{0i},T_{1i}]$

follows from Lemma 2.2, and replacing $v^{\varepsilon}$ by
$v^{\varepsilon}u^{\varepsilon}$ we conclude (4.6) in $[T_{0i},T_{1i}].$
Finally $[T_{1i},T_{0(i+1)}]$ can be treated in the same way by using an
appropriate shadow wave. The interaction procedure terminates at a finite
stage due to the compact support of $\varphi$ and the fact $Y_{i+1}-Y_{i}\geq
C_{1}\varepsilon^{\alpha}$ for $0<\alpha<1.$ It can be seen that we have at
most $\frac{C_{1}C_{\varphi}}{\varepsilon^{\alpha}}$ number of interactions
where $C_{\varphi}$ denotes the constant depending on $\varphi.$ Thus we
conclude (4.3) as both the equations in (4.2) and (4.6) are of order
$\mathcal{O}(\varepsilon^{1-\alpha}).$

Subcase 2. Let $\tilde{u}>u_{0}.$ Since $\tilde{u}$ is the limit of $u_{i}$ as
$i\to\infty,$ there exists a large $n_{0}\in\mathds{N}$ such that
$u_{{n_{0}}+1}\geq u_{0}$ and $u_{n_{0}}<u_{0}.$ In this case the shock curve
$x=c(t)$ will stay between $CD^{0}_{1}$ and $CD^{{n_{0}}+1}_{2}$ emanating
from $X_{n_{0},n_{0}}$ and the interaction stops after the time level
$T_{0n_{0}}.$ This completes the proof.

∎

######  Lemma 4.3.

Let $u(x)\in L^{\infty}([R,\infty))\cap C([R,\infty))$ be a decreasing
function and grant all other assumptions of Lemma 4.2. Then (4.3) holds.

###### Proof.

The proof is similar to the arguments of Lemma 4.2 and hence we give a sketch
of the proof by omitting the detailed analysis. We consider two cases.

Case 1. When $u_{0}\leq u(Y_{1}).$ In this case we have $u_{0}\leq u_{1}$ and
$u_{i}>u_{i+1}$ for $i=1,2,\cdot\cdot\cdots.$ By Lemma 2.1, the solution
consists of contact discontinuities and vacuum of the following form

$\displaystyle(v,u)(x,t)=\begin{cases}\left(v_{0},U_{0}(t)\right),\,\,\,\,\,\,\,\,\,\,\,\,&x<R+\int_{0}^{t}f(U_{0}(\xi))\mathrm{d}{\xi},\\\
\left(0,w(x,t)\right),\,\,\,\,\,\,&R+\int_{0}^{t}f(U_{0}(\xi))\mathrm{d}{\xi}<x<R+\int_{0}^{t}f(U_{1}(\xi))\mathrm{d}{\xi},\\\
\left(v_{1},U_{1}(t)\right),\,\,\,\,\,\,\,\,\,\,\,&x>R+\int_{0}^{t}f(U_{1}(\xi))\mathrm{d}{\xi},\end{cases}$

and for each $Y_{i},$ $i=1,2,3,\cdot\cdot\cdots$ a unique shadow wave
$SDW_{i,i+1}$ emanates. Due to the overcompressibility
$CD^{1}_{2}:=R+\int_{0}^{1}f(U_{1}(\xi))\mathrm{d}{\xi}$ meets $SDW_{1,k}$ at
a time level $T_{k(k+1)}$ for $k=2,3,\cdot\cdot\cdots.$ At this time level, we
have a set of new initial data with delta-function and (4.3) can be concluded
by combining the arguments of Lemma 2.2 and Lemma 4.2.

Case 2. When $u_{0}>u(Y_{1}).$ Since $u(x)$ is decreasing, in this case we
have $u_{i}>u_{i+1}$ for $i=0,1,2,\cdot\cdot\cdots.$ By Lemma 2.1 a unique
shadow wave solution $SDW_{i,i+1}$ emerges from each $Y_{i},$
$i=0,1,2,\cdot\cdot\cdots.$ Consider the initial data

$(v,u)(x,0)=\begin{cases}(v_{i},u_{i}),\,\,\,\,x<Y_{i},\\\
(v_{i+1},u_{i+1}),\,\,\,\,x>Y_{i},\end{cases}$

and

$(v,u)(x,0)=\begin{cases}(v_{i+1},u_{i+1}),\,\,\,\,x<Y_{i+1},\\\
(v_{i+2},u_{i+2}),\,\,\,\,x>Y_{i+1}.\end{cases}.$

Then $SDW_{i,i+1}$ and $SDW_{i+1,i+2}$ emerges from $Y_{i}$ and $Y_{i+1}$,
respectively with the speed $f(\chi_{i}(t))$ and $f(\chi_{i+1}(t)).$ Since
$SDW_{i,i+1}$ and $SDW_{i+1,i+2}$ are overcompressive, it follows that
$f(\chi_{i+1}(t))\leq f(\chi_{i}(t))$ and therefore these two shadow waves
meet at some time say $T_{i(i+1)}.$ It is difficult to determine for which $i$
first interaction happens, since the comparison of speed between any
consecutive shadow waves $SDW_{i,i+1}$ and $SDW_{i+1,i+2}$ is not precisely
known. Then at this level, we have a new set of initial data that contains a
$\delta$-function. Again, since $u(x)$ is decreasing, applying Lemma 2.2 we
conclude (4.3). The interaction procedure terminates after a finite number of
steps because of the same reason given in Lemma 4.2. ∎

Now we are ready to prove the Theorem 1.1.

###### Proof of Theorem 1.1.

Let $u(x)$ be a function having finitely many local extremes. For the sake of
concreteness, we only consider that $u(x)$ has a local maximum and local
minimum at $Y_{M},Y_{m}\in\\{Y_{i}\\},$ $M<m,$ respectively for
$i=0,1,2,\cdot\cdot\cdots.$ So, $u(x)$ is increasing on $[R,Y_{M}],$
decreasing on $(Y_{M},Y_{m}]$ and again increasing on $(Y_{m},\infty).$

Now if $u_{0}\leq u(Y_{1})$ then no interactions occur up to the state
$Y_{M-1}$ and the solution is given by $CD^{i-1}_{1}+Vac_{i-1,i}+CD^{i}_{2},$
$i=1,2,3\cdot\cdot\cdots M.$ By Lemma 2.1, there exists a unique shadow wave
$SDW_{M,M+1}$ from $Y_{M}.$ The waves continued to propagate until the first
interaction occurs and this case is dealt with in Lemma 4.2. Further, a
sequence of shadow waves $SDW_{i,i+1},$ $i=M,\cdot\cdot\cdots m-1$ emanates
from each $\\{Y_{i}\\}^{m-1}_{i=M}$ and they interact due to the
overcompressibility and this case is studied in Lemma 4.3. Finally, since
$Y_{m}$ is the local minimum, there exists a solution consisting
$CD^{m}_{1}+Vac_{m,m+1}+CD^{m+1}_{1}$ starting from $Y_{m}.$ Again the
solution propagates until the time when $SDW_{m-1,m}$ meets $CD^{m}_{1}$ and
this interaction occur as
$CD^{m}_{1}=f(U_{m}(t))<f(\chi_{m-1}(t))<f(U_{m-1}(t))$ where
$f(\chi_{m-1}(t))$ denotes the speed of the shadow wave $SDW_{m-1,m}$ emerging
from $Y_{m-1}.$ This case falls under Lemma 4.2.

If $u_{0}>u(Y_{1}),$ then $SDW_{0,1}$ emanates from $Y_{0}=R$ and interacts
with $CD^{1}_{1}$ which is considered in Lemma 4.2. The other cases are also
similar and an application of Lemma 4.2 and Lemma 4.3 depending on the
situation concludes the proof. ∎

An analogue of Theorem 1.1 can be proved for the $3\times 3$ system (1.12)
following the above steps and using the Riemann problem of Section 2.

###  4.3. Proof of Theorem 1.2

The aim of this section is to prove the Theorem 1.2. We start with the well-
known Riesz’ representation theorem.

######  Theorem 4.1 (Riesz’ representation theorem).

Let $I_{\mu}:C_{c}(\mathds{R}^{d})\to\mathds{R}$ be a nonnegative linear
functional. Then there exists a unique signed Radon measure $\mu$ such that

$I_{\mu}(\varphi):=\int_{K}\varphi\,\,\mathrm{d}{\mu}$

for any compact set $K\subset\mathds{R}^{d}$ with
$\operatorname*{support}(\varphi)\subset K.$

We denote ${\mathcal{M}}(\mathds{R}^{d})$ as the space of all signed Radon
measures endowed with the topology of the dual space of
$C_{c}(\mathds{R}^{d}).$

######  Definition 4.1 (Weak* convergence of measures).

Let $\\{\mu_{\nu}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$ be a sequence of
nonnegative locally finite measures on $\mathds{R}^{d}.$ The sequence
$\\{\mu_{\nu}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$ is said to be converges
weakly* to $\mu$ (denoted as
$\mu_{\nu}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\mu$) if

$\lim_{\nu\to\infty}\int\varphi\,\,\mathrm{d}{\mu_{\nu}}=\int\varphi\,\,\mathrm{d}{\mu}$

for every $\varphi\in C_{c}(\mathds{R}^{d}).$

######  Lemma 4.4 ([13], Proposition 2.5).

Let $\\{\mu_{\nu}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$ be a sequence of uniformly
locally bounded measures. Then there exists a subsequence still denoted as
$\\{\mu_{\nu}\\}$ and a Radon measure $\mu$ such that
$\mu_{\nu}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\mu.$

To prove Theorem 1.2, we need to show that $|U^{\nu}|=(v^{\nu},|u^{\nu}|)$ is
locally uniformly bounded for each $\nu\in\mathds{N}\cup\\{0\\}.$ This is the
content of the next lemma.

######  Lemma 4.5.

Suppose $v(x)>0$ is bounded and $u(x)\in L^{\infty}([R,\infty))\cap
C([R,\infty))$ and $u(x)$ has finitely many local extremes. Also assume
$U^{\nu}=(v^{\nu},u^{\nu})$ be the admissible solution to (1.9) and (4.1).
Then $\\{U^{\nu}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$ is locally uniformly
bounded for each $\nu.$

###### Proof.

The proof follows from overcompressibility and conservation of mass principle.

Step 1. For $t\in(t_{0},T),$ using

$\left|\int_{0}^{t}F(\xi)\mathrm{d}{\xi}\right|\leq||u_{a}||_{L^{\infty}([t_{0},T])}\left(e^{||\kappa||_{L^{\infty}([t_{0},T])}T}-1\right),$

we get

$\chi(t)\leq\left|U_{l,r}(t)\right|\leq||u_{a}||_{L^{\infty}([t_{0},T])}\left(e^{||\kappa||_{L^{\infty}([t_{0},T])}T}-1\right)+|u_{l,r}|,$

and hence $\dot{c}(t)$ is also locally uniformly bounded. This, combined with
the fact that $u(x)$ in (4.1) is bounded gives $|u^{\nu}|$ is locally
uniformly bounded for each $\nu\in\mathds{N}\cup\\{0\\}.$

Step 2. To prove $v^{\nu}\geq 0$ is locally uniformly bounded, observe that
for any compact set $K\subset\mathds{R}\times[0,\infty)$ we have

$\displaystyle 0\leq\int_{K}v^{\nu}(x,t)\mathrm{d}{x}\mathrm{d}{t}\leq
C_{K}\max\left\\{v_{0},\sup_{x>R}v(x)\right\\}<\infty.$

Therefore, $|U^{\nu}|=(v^{\nu},|u^{\nu}|)$ is bounded in $L^{1}(K)$ for each
compact set $K\subset\mathds{R}\times[0,\infty).$ This completes the proof. ∎

Now we complete the proof of Theorem 1.2. We will use the Jordan Decomposition
of signed measures. If $\mu$ is a signed measure, then there exist unique
positive mutually singular measures $\mu^{+}$ and $\mu^{-}$ such that
$\mu=\mu^{+}-\mu^{-}.$ The measures $\mu^{+}$ and $\mu^{-}$ are called
positive and negative variations of $\mu$ and $|\mu|=\mu^{+}+\mu^{-}$ is
defined as the total variation of the measure $\mu.$

###### Proof of Theorem 1.2.

The proof can be completed following the same lines given in [41, Theorem
7.1]. We repeat it here for the sake of completeness. Using Lemma 4.5 and
decomposition of measures, $\\{U^{\nu}_{+}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$
and $\\{U^{\nu}_{-}\\}_{\nu\in\mathds{N}\cup\\{0\\}}$ are locally uniformly
bounded where $U^{\nu}=U^{\nu}_{+}-U^{\nu}_{-}.$ Thus by Lemma 4.4,
$U^{\nu}_{+}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}U^{*}_{+}$ and
$U^{\nu}_{-}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}U^{*}_{-}$ where
$U^{*}_{\pm}$ are Radon measures. Therefore $U^{\nu}$ converges weakly* to a
Radon measure $U^{*}=U^{*}_{+}-U^{*}_{-}.$ Also a direct use of Lemma 4.4
gives $|U^{\nu}|$ converges weakly* to $|U^{*}|.$ ∎

##  5\. Some examples

In this section, we present some examples of physically relevant models which
can be derived from (1.9). The first obvious example is the pressureless gas
dynamics model with flocking dissipation that can be derived by setting
$f(u)=u,$ $\kappa(t)=1,$ and $u_{a}(t)=0.$ We provide some more examples
below.

###  5.1. System of nonlinear geometric optics

The following examples can be constructed by setting
$f:U\mapsto\frac{U}{\sqrt{1+U^{2}}}$ in (1.9), i.e., consider the system of
the following form

$\displaystyle\begin{cases}v_{t}+\left(\frac{vU}{\sqrt{1+U^{2}}}\right)_{x}=0,\\\
(vU)_{t}+\left(\frac{vU^{2}}{\sqrt{1+U^{2}}}\right)_{x}=\kappa(t)(u_{a}(t)-U)v,\end{cases}$

with the initial data $(v,U)(x,0)=(v_{0}(x),U_{0}(x)).$ One can easily check
that $f^{\prime}(U)>0.$

The motivation for considering the above system comes from the system of
nonlinear geometric optics with a damping source. We use the transformation
$U=\frac{u}{v}$, i.e., $u=vU$ to obtain the following:

$\displaystyle\begin{cases}v_{t}+\left(\frac{u^{2}}{\sqrt{u^{2}+v^{2}}}\right)_{x}=0,\\\
u_{t}+\left(\frac{uv}{\sqrt{u^{2}+v^{2}}}\right)_{x}=\kappa(t)(u_{a}(t)v-u).\end{cases}$

For the homogeneous version of the above example, see [54, 26].

###  5.2. A model with $f(u)=u^{k},$ $k$ odd.

In their work [39], Cruz and Juajibioy investigated the Riemann problem
associated with the pressureless model featuring linear damping, described by
the following equations:

$\displaystyle\begin{cases}v_{t}+\left(vu^{k}\right)_{x}=0,\\\
(vu)_{t}+\left(vu^{k+1}\right)_{x}=-\alpha uv,\end{cases}$

where $k$ is an odd natural number, $\alpha>0$ is a constant. This system can
immediately be obtained by plugging $f(u)=u^{k},$ $\kappa(t)=\alpha,$ and
$u_{a}(t)=0.$ Since $k$ is odd, clearly $f^{\prime}(u)=ku^{k-1}>0.$

###  5.3. Pressureless hydrodynamic model

Recently, Piccoli et al. [8] derived a pressureless hydrodynamic model from
the second-order macroscopic traffic flow. The model reads

$\displaystyle\begin{cases}v_{t}+\left(\frac{vu}{a+u}\right)_{x}=0,\\\
(vu)_{t}+\left(\frac{vu^{2}}{a+u}\right)_{x}=\frac{\mu
pv^{2}(s_{d}(v)-u)}{2(\nu+1)},\end{cases}$

where $v$ and $u$ denote the traffic density and the local mean headway (i.e.,
the distance from their leading vehicles), respectively. Furthermore, $a>0$
denotes the magnitude of the driver sensitivity, $0<\mu<1$ corresponds to the
equilibrium coefficient in order to control the deviation, whereas $\nu>0$ is
related to the cost of control. The function $s_{d}(v)$ is used to express the
safe distance from the preceding vehicle and $0\leq p\leq 1$ denotes the
percentage of vehicles in a traffic flow. More recently, the Riemann problem
for the non-homogeneous version of the above system is studied by Wang and Sun
[56]. Setting $f(u)=\frac{u}{a+u}$ in (1.9), we propose a pressureless
hydrodynamic model with a damping source that depends on time. For this $f,$
it is easy to check $f^{\prime}(u)=\frac{a}{(a+u)^{2}}>0.$

Part II: Explicit formula for (1.10).

##  6\. Eulerian droplet model involving space-time dependent source

In this section, we deal with the case $f(u)=u$ and $u_{a}(x,t)$ depending on
both space and time variables. We employ generalized variational techniques
and obtain an explicit representation of the solution to the system (6.1) with
air velocity that has an algebraic decay in time. We start by introducing the
generalized potential for the initial value problem as follows

$\displaystyle v_{t}+(vu)_{x}$ $\displaystyle=0,$ (6.1)
$\displaystyle(vu)_{t}+(vu^{2})_{x}$ $\displaystyle=\kappa(t)(u_{a}(x,t)-u)v,$

adjoined with the initial data $(v(x,0),u(x,0))=(v_{0}(x),u_{0}(x)).$ We
recall $\kappa(t)=\frac{1}{t+\upkappa}$ and $u_{a}(x,t)=\frac{x}{t+\upkappa}$
for any $\upkappa\in\mathds{R}^{+}.$ The characteristics equation for the
above system is reduced to

$\displaystyle\begin{cases}&\frac{\mathrm{d}{x(t)}}{\mathrm{d}{t}}=u(x(t),t),\\\
&\frac{\mathrm{d}{u(x(t),t)}}{\mathrm{d}{t}}=\frac{1}{t+\upkappa}(\frac{x}{t+\upkappa}-u(x(t),t)),\\\
&x(0)=x_{0}.\end{cases}$ (6.2)

From (6.2), we obtain second-order ODE as follows:

$\displaystyle\frac{\mathrm{d}^{2}{x}}{\mathrm{d}{t^{2}}}+\frac{1}{t+\upkappa}\frac{\mathrm{d}{x}}{\mathrm{d}{t}}-\frac{1}{(t+\upkappa)^{2}}x=0,$
(6.3) $\displaystyle x(0)=x_{0}.$

Solving the above ODE (6.3), we obtain

$\displaystyle
x(t)=\left(\frac{x_{0}}{2\upkappa}+\frac{u_{0}(x_{0})}{2}\right)(t+\upkappa)+\left(\frac{x_{0}}{2}-\frac{u_{0}(x_{0})\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)},$
$\displaystyle
u(x(t),t)=\left(\frac{x_{0}}{2\upkappa}+\frac{u_{0}(x_{0})}{2}\right)-\left(\frac{x_{0}}{2}-\frac{u_{0}(x_{0})\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}.$

Now we introduce our generalized potential as the following.

$\displaystyle
F(y,x,t)=\int_{0}^{y}\left(\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)(t+\upkappa)+\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)}-x\right)v_{0}(\eta)d\eta.$
(6.4)

Given a point $(x,t),$ let $y_{*}(x,t)$ and $y^{*}(x,t)$ be the leftmost and
the rightmost points on the $x$\- axis such that

$\displaystyle\min_{y\in\mathds{R}}F(y,x,t)=F(y_{*}(x,t),x,t)=F(y^{*}(x,t),x,t).$

Below, we give the explicit representation for $m(x,t)$ and $u(x,t).$ The
formula for the pair $(m,u)$ is given by

$\displaystyle m(x,t)=\int_{0}^{y_{*}(x,t)}v_{0}(\eta)\mathrm{d}{\eta},$ (6.5)
$\displaystyle
u(x,t)=\begin{cases}&\frac{1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}}{(t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}}\left[x-\frac{1}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)y_{*}(x,t)\right]+\frac{y_{*}(x,t)}{2}\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right),\,\,\,\,\text{if}\,\,\,\,y_{*}(x,t)=y^{*}(x,t),\\\
&\frac{1}{\int_{y_{*}}^{y^{*}}v_{0}(\eta)\mathrm{d}{\eta}}\int_{y_{*}}^{y^{*}}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\mathrm{d}{\eta},\,\,\,\,\text{if}\,\,\,\,y_{*}(x,t)\neq
y^{*}(x,t).\end{cases}$ (6.6)

###  6.1. Derivation of the formula Eq. 6.6

In this section we derive the above formula. Derivation of the formula (6.6)
consists of several steps that are formulated as lemmas. We Define the left
backward characteristics and right backward characteristics for $0\leq
t<t_{0}$ as

$\displaystyle
X_{l}(x_{0},t_{0},t)=\frac{y_{*}(x_{0},t_{0})}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)+\left((t+\upkappa)-\frac{\upkappa^{2}}{t+\upkappa}\right)\frac{x_{0}-\frac{y_{*}(x_{0},t_{0})}{2}\left(\frac{t_{0}+\upkappa}{\upkappa}+\frac{\upkappa}{t_{0}+\upkappa}\right)}{\left((t_{0}+\upkappa)-\frac{\upkappa^{2}}{t_{0}+\upkappa}\right)},$
(6.7) $\displaystyle
X_{r}(x_{0},t_{0},t)=\frac{y^{*}(x_{0},t_{0})}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)+\left((t+\upkappa)-\frac{\upkappa^{2}}{t+\upkappa}\right)\frac{x_{0}-\frac{y^{*}(x_{0},t_{0})}{2}\left(\frac{t_{0}+\upkappa}{\upkappa}+\frac{\upkappa}{t_{0}+\upkappa}\right)}{\left((t_{0}+\upkappa)-\frac{\upkappa^{2}}{t_{0}+\upkappa}\right)}.$

Also, let us denote

$\displaystyle X^{+}_{l,r}(x_{0},t_{0},t):=\left\\{(x,t)\big{|}x\geq
X_{l,r}(x_{0},t_{0},t)\right\\},~{}~{}X^{-}_{l,r}(x_{0},t_{0},t):=\left\\{(x,t)\big{|}x\leq
X_{l,r}(x_{0},t_{0},t)\right\\}.$

Next, we define the characteristic triangle.

######  Definition 6.1.

The area $\\{(x,t)\,\,|\,\,x\in\mathds{R},0\leq t\leq t_{0}\\}\cap
X^{+}_{l}(x_{0},t_{0},t)\cap X^{-}_{r}(x_{0},t_{0},t)$ is said to be a
characteristic triangle associated to $(x_{0},t_{0})$ and it is denoted as
$\Delta(x_{0},t_{0}).$

Next, we collect some basic properties of the functional $F(y,x,t),$
minimizers $y^{*}(x,t),y_{*}(x,t)$ and the characteristic triangles which can
be proved easily following [21, 57, 31].

######  Lemma 6.1.

Grant the assumptions on the initial data. Then, we have

  1. (1)

For a fixed $t,$ $y_{*}(x,t)$ and $y^{*}(x,t)$ are monotonically increasing in
$x$ and for $x_{1}<x_{2},$ $y^{*}(x_{1},t)\leq y_{*}(x_{2},t).$ Furthermore,
$y_{*}(x,t)$ is lower semicontinuous and $y^{*}(x,t)$ is upper semicontinuous.

  2. (2)

For fixed $(x,t)$, let the $\min\limits_{y\in[0,\infty)}F(y,x,t)$ be attained
at $y_{1}$. Then for any given point $(x^{\prime},t^{\prime})$ on the curve
joining $(y_{1},0)$ and $(x,t)$, we have
$F(y,x^{\prime},t^{\prime})>F(y_{1},x^{\prime},t^{\prime})$ for $y\neq y_{1}.$

  3. (3)

The function $F:\mathds{R}\times[0,\infty)\to\mathds{R},$ i.e.,$(x,t)\mapsto
F(x,t)$ is locally Lipschitz continuous.

  4. (4)

Let $t>0$ be fixed, and $x_{1}\neq x_{2}$ be arbitrary. Then the
characteristic triangles associated with $(x_{1},t)$ and $(x_{2},t)$ do not
intersect in $\mathds{R}\times[0,\infty).$ Consequently, if two characteristic
triangles intersect in $\mathds{R}\times[0,\infty)$, then one is contained in
the other.

  5. (5)

For any time $t_{0}>0$, we have

$\displaystyle\bigcup_{x\in\mathds{R}}\Delta(x,t_{0})=\left\\{(x,t)\big{|}x\in\mathds{R},0\leq
t\leq t_{0}\right\\}.$

Now we are ready to derive the formula (6.6).

######  Lemma 6.2.

Let $t_{1}>0$. Each point $(x_{1},t_{1})$ uniquely determines a Lipschitz
continuous curve $x=X(t)$, for $t\geq t_{1}$ with $x_{1}=X(t_{1})$ such that
the characteristic triangles associated with points on the curve form an
increasing family of sets. At every $t\geq t_{1}$, $u(x,t)$ is defined as the
slope of the curve $X^{\prime}(t).$

###### Proof.

Let $t<t^{\prime}<t^{\prime\prime}$ and
$x^{\prime}=X(t^{\prime}),x^{\prime\prime}=X(t^{\prime\prime}).$ Applying
$(1)$ of Lemma 6.1, we have

$\displaystyle y_{*}(x^{\prime\prime},t^{\prime\prime})\leq
y_{*}(x^{\prime},t^{\prime})\leq y_{*}(x,t)\leq y^{*}(x,t)\leq
y^{*}(x^{\prime},t^{\prime})\leq y^{*}(x^{\prime\prime},t^{\prime\prime})$

and
$\\{y_{*}(x^{\prime\prime},t^{\prime\prime}),y_{*}(x^{\prime},t^{\prime})\\}$
and
$\\{y^{*}(x^{\prime\prime},t^{\prime\prime}),y^{*}(x^{\prime},t^{\prime})\\}$
tend to $y_{*}(x,t)$ and $y^{*}(x,t)$, respectively as
$t^{\prime\prime},t^{\prime}\to t.$ Now, we consider the following two cases.
Case I. Let $y_{*}(x,t)=y^{*}(x,t).$ Take any two points $(x_{1},t^{\prime})$
and $(x_{2},t^{\prime})$ on the backward characteristics
$X_{l}(x^{\prime\prime},t^{\prime\prime},t)$ and
$X_{r}(x^{\prime\prime},t^{\prime\prime},t)$, respectively. Then we have

$\displaystyle\frac{x^{\prime\prime}-x_{2}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}\leq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}\leq\frac{x^{\prime\prime}-x_{1}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}.$

Using (6.7) and simplifying, we obtain

$\displaystyle\frac{x^{\prime\prime}-x_{1}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}=$
$\displaystyle\frac{x^{\prime\prime}}{t^{\prime\prime}-t^{\prime}}\left[1-\frac{(t^{\prime}+\upkappa)-\frac{\upkappa^{2}}{t^{\prime}+\upkappa}}{(t^{\prime\prime}+\upkappa)-\frac{\upkappa^{2}}{t^{\prime\prime}+\upkappa}}\right]$
$\displaystyle-\frac{y_{*}(x^{\prime\prime},t^{\prime\prime})}{2(t^{\prime\prime}-t^{\prime})}\left[\left(\frac{t^{\prime}+\upkappa}{\upkappa}+\frac{\upkappa}{t^{\prime}+\upkappa}\right)-\frac{\left(\frac{t^{\prime\prime}+\upkappa}{\upkappa}+\frac{\upkappa}{t^{\prime\prime}+\upkappa}\right)\left((t^{\prime}+\upkappa)-\frac{\upkappa^{2}}{t^{\prime}+\upkappa}\right)}{(t^{\prime\prime}+\upkappa)-\frac{\upkappa^{2}}{t^{\prime\prime}+\upkappa}}\right].$
(6.8)

Similarly,
$\frac{x^{\prime\prime}-x_{2}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}$ can
be written in the above form by replacing
$y_{*}(x^{\prime\prime},t^{\prime\prime})$ by
$y^{*}(x^{\prime\prime},t^{\prime\prime})$ in (6.1). Now passing to the limit
as $t^{\prime\prime},~{}t^{\prime}\to t$ in (6.1), we get

$\displaystyle X^{\prime}(t)\leq
x\left[\frac{\left(1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}\right)}{(t+\upkappa)-\frac{\upkappa^{2}}{t+\upkappa}}\right]-\frac{y_{*}(x,t)}{2}\left[\frac{\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)}{\left((t+\upkappa)-\frac{\upkappa^{2}}{t+\upkappa}\right)}\left(1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}\right)-\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right)\right].$

Rearranging the above expression we get (6.6) for the case
$y_{*}(x,t)=y^{*}(x,t).$
Case II. Let $y_{*}(x,t)\neq y^{*}(x,t).$ First we note that

$\displaystyle
F(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-F(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})\leq
F(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})-F(y_{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime}).$
(6.9)

Inserting the potential (6.4) in (6.9) and simplifying, we get

$\displaystyle\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{y_{*}(x^{\prime},t^{\prime})}^{y^{*}(x^{\prime\prime},t^{\prime\prime})}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)(t^{\prime\prime}-t^{\prime})+\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\left(\frac{\upkappa}{(t^{\prime\prime}+\upkappa)}-\frac{\upkappa}{t^{\prime}+\upkappa}\right)\right]v_{0}(\eta)\mathrm{d}{\eta}$
$\displaystyle\leq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}\int_{y_{*}(x^{\prime},t^{\prime})}^{y^{*}(x^{\prime\prime},t^{\prime\prime})}v_{0}(\eta)\mathrm{d}{\eta}.$
(6.10)

Passing to the limit as $t^{\prime\prime},t^{\prime}\to t$ in (6.1) we obtain

$\displaystyle
X^{\prime}(t)\geq\frac{1}{\int_{y_{*}}^{y^{*}}v_{0}(\eta)\mathrm{d}{\eta}}\int_{y_{*}}^{y^{*}}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\mathrm{d}{\eta}.$
(6.11)

Again, considering the inequality

$\displaystyle
F(y^{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime})-F(y_{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})\leq
F(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})-F(y_{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime}),$

we get

$\displaystyle
X^{\prime}(t)\leq\frac{1}{\int_{y_{*}}^{y^{*}}v_{0}(\eta)\mathrm{d}{\eta}}\int_{y_{*}}^{y^{*}}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\mathrm{d}{\eta}.$
(6.12)

Combining (6.11)-(6.12), we obtain (6.6). ∎

The next result shows that the curve $X(t)$ can actually be started from
${t=0}.$ The proof can be completed using the arguments of [21, 57] and also
see [31] for a more general case extended to the initial-boundary value
problem.

######  Theorem 6.1.

Let $X(\eta,t)$ be a curve defined in Lemma 6.2 and $S$ be a countable set of
points on the $x$-axis. Then for all $(\eta,0)\notin S$ there exists a unique
Lipschitz continuous curve $x=X(\eta,t),t\geq 0$ such that $X(\eta,0)=\eta$
and the characteristics triangles associated to the points form an increasing
family of sets. Also, for all $(\eta,0)\notin S$, we have

$\displaystyle\frac{\partial}{\partial
t}X(\eta,t)=u(X(\eta,t),t)\,\,\,\,\text{for a.e}\,\,\,t>0.$

###  6.2. Verification of weak formulation, entropy criterion, and initial
condition

The goal of this section is to prove Theorem 1.3. We verify that the pair
$(m,u)$ satisfies the weak formulation, Lax entropy criterion, and the initial
condition.

####  6.2.1. Verification of weak formulation Eq. 1.13-(1.14)

Verification of weak formulation (1.13)-(1.14) consists of several steps.
First, let us define the momentum and energy potentials as follows:

$\displaystyle
q(x,t)=\int_{0}^{y_{*}(x,t)}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\mathrm{d}{\eta},$
$\displaystyle
E(x,t)=\frac{1}{2}\int_{0}^{y_{*}(x,t)}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)u(X(\eta,t),t)\mathrm{d}{\eta},$

and the functionals

$\displaystyle
H(y,x,t)=\int_{0}^{y}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\left(X(\eta,t)-x\right)\mathrm{d}{\eta},$
$\displaystyle
I(y,x,t)=\int_{0}^{y}\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{2\upkappa
v_{0}(\eta)}{(t+\upkappa)^{3}}\left(X(\eta,t)-x\right)\mathrm{d}{\eta},$
$\displaystyle
J(y,x,t)=\int_{0}^{y}\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{2\upkappa
v_{0}(\eta)}{(t+\upkappa)^{3}}\mathrm{d}{\eta}.$

Step 1. The following relations hold:

$\displaystyle(i)\,\,\,\mathrm{d}{q}=u\mathrm{d}{m},\,\,\,\,(ii)\,\,\,\mathrm{d}{E}=\frac{1}{2}u^{2}\mathrm{d}{m}.$
(6.13)

in the sense of Radon-Nikodym derivatives in $x.$
Proof of Step 1. If $y_{*}(x,t)$ is a constant in a neighbourhood of $(x,t),$
then (6.13) holds trivially. Assume that $y_{*}(x,t)$ is not constant in a
neighbourhood of $(x,t)$ and $y_{*}(x,t)=y^{*}(x,t).$ For a fixed $t>0,$ let
$x_{1}<x<x_{2}.$ By using the definition, we have

$\displaystyle F(y_{*}(x_{1},t),x_{1},t)\leq F(y_{*}(x_{2},t),x_{1},t).$

Then, we have

$\displaystyle\int_{y_{*}(x_{1},t)}^{y_{*}(x_{2},t)}\frac{u_{0}(\eta)}{2}\left((t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}\right)v_{0}(\eta)\mathrm{d}{\eta}$
$\displaystyle\leq\int_{y_{*}(x_{1},t)}^{y_{*}(x_{2},t)}\left(x_{1}-\frac{\eta}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)\right)v_{0}(\eta)\mathrm{d}{\eta}$
$\displaystyle\leq\int_{y_{*}(x_{1},t)}^{y_{*}(x_{2},t)}\left(x_{1}-\frac{y_{*}(x_{1},t)}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)\right)v_{0}(\eta)\mathrm{d}{\eta}.$

This implies

$\displaystyle\int_{y_{*}(x_{1},t)}^{y^{*}(x_{2},t)}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\mathrm{d}{\eta}$
$\displaystyle\leq\frac{1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}}{(t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}}\left[x-\frac{1}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)y_{*}(x_{1},t)\right]+\frac{y_{*}(x_{2},t)}{2}\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right)\int_{y_{*}(x_{1},t)}^{y_{*}(x_{2},t)}v_{0}(\eta)\mathrm{d}{\eta}.$

Hence, we obtain

$\displaystyle\frac{q(x_{1},t)-q(x_{2},t)}{m(x_{1},t)-m(x_{2},t)}\leq\frac{1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}}{(t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}}\left[x-\frac{1}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)y_{*}(x_{1},t)\right]+\frac{y_{*}(x_{2},t)}{2}\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right).$
(6.14)

Now passing to the limit as $x_{1}\nearrow x$ and $x_{2}\searrow x$ in (6.14)
we get

$\displaystyle\lim_{x_{1},x_{2}\to
x}\frac{q(x_{1},t)-q(x_{2},t)}{m(x_{1},t)-m(x_{2},t)}\leq u(x,t).$

Similarly, considering the inequality

$\displaystyle F(y_{*}(x_{2},t),x_{2},t)\leq F(y_{*}(x_{1},t),x_{2},t),$

and following the same argument as above one can easily obtain the other way
inequality

$\displaystyle\lim_{x_{1},x_{2}\to
x}\frac{q(x_{1},t)-q(x_{2},t)}{m(x_{1},t)-m(x_{2},t)}\geq u(x,t),$

and this proves $(i).$
If $y_{*}(x,t)<y^{*}(x,t),$ then

$\displaystyle\lim_{x_{1},x_{2}\to
x}\frac{q(x_{1},t)-q(x_{2},t)}{m(x_{1},t)-m(x_{2},t)}$
$\displaystyle=\lim_{x_{1},x_{2}\to
x}\frac{\int_{y_{*}(x_{1},t)}^{y_{*}(x_{2},t)}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\mathrm{d}{\eta}}{\int_{y_{*}(x_{1},t)}^{y^{*}(x_{2},t)}v_{0}(\eta)\mathrm{d}{\eta}}=u(x,t).$

This completes the proof.

Proof of $(ii)$ follows a similar argument. First consider the case
$y_{*}(x,t)=y^{*}(x,t).$ For $t>0$ and $x_{1}<x<x_{2}$, we have

$\displaystyle
E(x_{2},t)-E(x_{1},t)=\frac{1}{2}\int_{y_{*}(x_{1},t)}^{y_{*}(x_{2},t)}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)-\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)u(X(\eta,t),t)\mathrm{d}{\eta}.$
(6.15)

On the other hand, for $y_{*}(x_{1},t)\leq\eta\leq y_{*}(x_{2},t)$, we have

$\displaystyle\frac{1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}}{(t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}}\left[x-\frac{1}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)y_{*}(x_{1},t)\right]+\frac{y_{*}(x_{1},t)}{2}\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right)$
$\displaystyle\leq
u(X(\eta,t),t)\leq\frac{1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}}{(t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}}\left[x-\frac{1}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)y_{*}(x_{2},t)\right]+\frac{y_{*}(x_{2},t)}{2}\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right).$
(6.16)

From (6.15)-(6.2.1), we get

$\displaystyle\frac{1}{2}\frac{1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}}{(t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}}\left[x-\frac{1}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)y_{*}(x_{1},t)\right]+\frac{y_{*}(x_{1},t)}{2}\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right)$
$\displaystyle\leq\frac{E(x_{2},t)-E(x_{1},t)}{q(x_{2},t)-q(x_{1},t)}\leq\frac{1}{2}\frac{1+\frac{\upkappa^{2}}{(t+\upkappa)^{2}}}{(t+\upkappa)-\frac{\upkappa^{2}}{(t+\upkappa)}}\left[x-\frac{1}{2}\left(\frac{t+\upkappa}{\upkappa}+\frac{\upkappa}{t+\upkappa}\right)y_{*}(x_{2},t)\right]+\frac{y_{*}(x_{2},t)}{2}\left(\frac{1}{\upkappa}-\frac{\upkappa}{(t+\upkappa)^{2}}\right).$
(6.17)

Passing to the limit $x_{1}\nearrow x$ and $x_{2}\searrow x$ in (6.2.1), we
obtain

$\displaystyle\lim_{x_{1},x_{2}\to
x}\frac{E(x_{2},t)-E(x_{1},t)}{q(x_{2},t)-q(x_{1},t)}=\frac{1}{2}u(x,t)$

and consequently, by $(i),$ we find
$\mathrm{d}{E}=\frac{1}{2}u^{2}\mathrm{d}{m}.$
Now if $y_{*}(x,t)<y^{*}(x,t),$ it is straightforward to see

$\displaystyle\lim_{x_{1},x_{2}\to
x}\frac{E(x_{2},t)-E(x_{1},t)}{q(x_{2},t)-q(x_{1},t)}=\frac{1}{2}u(x,t)$

where we used the fact that $y_{*}(x_{1},t)\to y_{*}(x,t)$ and
$y_{*}(x_{2},t)\to y^{*}(x,t)$ as $x_{1},x_{2}\to x$ and for
$\eta\in[y_{*}(x,t),y^{*}(x,t)],$ we have $u(X(\eta,t),t)=u(x,t).$ This
completes the proof of $(ii).$

Step 2. Define $F(x,t)=\min\limits_{y\in\mathds{R}}F(y,x,t).$ Then the
following hold

$\displaystyle(i)\,\,\,\frac{\partial}{\partial x}F(x,t)=-m(x,t),$ (6.18)
$\displaystyle(ii)\,\,\,\frac{\partial}{\partial t}F(x,t)=q(x,t).$ (6.19)

Proof of Step 2. To prove (6.18), first we fix $t>0$ and choose any two points
$x_{1},x_{2}\in\mathds{R}.$ We claim that

$\displaystyle\int_{x_{1}}^{x_{2}}m(x,t)\mathrm{d}{x}=F(x_{2},t)-F(x_{1},t).$
(6.20)

Take any $x,x^{\prime}\in[x_{1},x_{2}]$ with $x<x^{\prime}.$ It is enough to
prove

$\displaystyle(x^{\prime}-x)m(x,t)\leq
F(x,t)-F(x^{\prime},t)\leq(x^{\prime}-x)m(x^{\prime},t).$ (6.21)

The inequality (6.21) can be proved by considering

$\displaystyle F(x,t)-F(x^{\prime},t)$
$\displaystyle=F(y_{*}(x,t),x,t)-F(y_{*}(x^{\prime},t),x^{\prime},t)$
$\displaystyle=[F(y_{*}(x,t),x,t)-F(y_{*}(x,t),x^{\prime},t)]+[F(y_{*}(x,t),x^{\prime},t)-F(y_{*}(x^{\prime},t),x^{\prime},t)].$
(6.22)

It is worth noting that the second term in (6.22) is positive, leading us to
the following inequality,

$\displaystyle F(x,t)-F(x^{\prime},t)\geq
F(y_{*}(x,t),x,t)-F(y_{*}(x,t),x^{\prime},t).$

Similarly, we have

$\displaystyle F(x,t)-F(x^{\prime},t)\leq
F(y_{*}(x^{\prime},t),x,t)-F(y_{*}(x,t^{\prime}),x,t^{\prime}).$

Combining the above two inequalities, we conclude (6.21). Note that, since
$y_{*}(x,t)$ is increasing in $x$ and $v_{0}>0,$ we have $m(x,t)$ is also
increasing in $x.$ Therefore, $m(x,t)$ is Riemann integrable. Now taking
Riemann sum and using the inequality (6.21), we conclude (6.20) and hence
(6.18).

To prove (6.19), we first fix $x\in\mathds{R}$ and choose any
$t,t+h\in(0,\infty).$ Then we have

$\displaystyle F(x,t+h)-F(x,t)$
$\displaystyle=F(y_{*}(x,t+h),x,t+h)-F(y_{*}(x,t),x,t)$
$\displaystyle=[F(y_{*}(x,t+h),x,t+h)-F(y_{*}(x,t+h),x,t)]$
$\displaystyle+[F(y_{*}(x,t+h),x,t)-F(y_{*}(x,t),x,t)].$

Again, using minimization of $F,$ we obtain

$\displaystyle F(x,t+h)-F(x,t)\geq F(y_{*}(x,t+h),x,t+h)-F(y_{*}(x,t+h),x,t).$

Similarly, we have

$\displaystyle F(x,t+h)-F(x,t)\leq F(y_{*}(x,t),x,t+h)-F(y_{*}(x,t),x,t).$

Combining the above two inequalities, we get

$\displaystyle\int_{0}^{y_{*}(x,t+h)}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)+\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{1}{h}\left(\frac{\upkappa}{t+h+\upkappa}-\frac{\upkappa}{t+\upkappa}\right)\right]v_{0}(\eta)\mathrm{d}{\eta}$
$\displaystyle\leq\frac{F(x,t+h)-F(x,t)}{h}\leq\int_{0}^{y_{*}(x,t)}\left[\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)+\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{1}{h}\left(\frac{\upkappa}{t+h+\upkappa}-\frac{\upkappa}{t+\upkappa}\right)\right]v_{0}(\eta)\mathrm{d}{\eta}.$
(6.23)

As we know $y_{*}(x,t+h)\to y_{*}(x,t)$ as $h\to 0$. Then passing to the limit
as $h\to 0$ in the above inequality (6.2.1), we obtain (6.19). This completes
the proof.

Step 3. Now we show that $(m,u)$ satisfies the first equation of the weak
formulation (1.13). For a test function $\varphi$ with compact support in
$\mathds{R}\times]0,\infty[$ we infer using the step 1 and step 2:

$\displaystyle 0$
$\displaystyle=\iint[F_{x}\varphi_{t}(x,t)-F_{t}\varphi_{x}(x,t)]\mathrm{d}{x}\mathrm{d}{t}=\iint[m(x,t)\varphi_{t}(x,t)-q(x,t)\varphi_{x}(x,t)]\mathrm{d}{x}\mathrm{d}{t}$
$\displaystyle=\iint[m(x,t)\varphi_{t}(x,t)\mathrm{d}{x}\mathrm{d}{t}-\iint
u(x,t)\varphi(x,t)\mathrm{d}{m}\mathrm{d}{t}\,.$ (6.24)

This identity proves that $(m,u)$ satisfies the first equation of the system
(6.1).

Step 4. Define $H(x,t)=\min\limits_{y\in\mathds{R}}H(y,x,t)$ and
$I(x,t)=\min\limits_{y\in\mathds{R}}I(y,x,t).$ Then the following relations
hold:

$\displaystyle(i)\,\,\,\frac{\partial}{\partial x}H(x,t)=-q(x,t),$ (6.25)
$\displaystyle(ii)\,\,\,\frac{\partial}{\partial t}H(x,t)=2E(x,t)+I(x,t).$
(6.26)

Proof of Step 4. To prove (6.25), following the previous argument, we fix
$t>0$ and choose any two points $x_{1},x_{2}\in\mathds{R}$ and claim that

$\displaystyle\int_{x_{1}}^{x_{2}}q(x,t)\mathrm{d}{x}=H(x_{2},t)-H(x_{1},t).$

Take any $x,x^{\prime}\in[x_{1},x_{2}]$ with $x<x^{\prime}.$ Exactly as in the
proof of (6.18), we show

$\displaystyle(x^{\prime}-x)q(x,t)\leq
H(x,t)-H(x^{\prime},t)\leq(x^{\prime}-x)q(x^{\prime},t),$

and this completes the proof of (6.25).
To prove (6.26), first we fix $x$ and for $t,t+h\in(0,\infty),$ and following
the proof of (6.19), we have

$\displaystyle H(x,t+h)-H(x,t)\geq H(y_{*}(x,t+h),x,t+h)-H(y_{*}(x,t+h),x,t).$

Therefore, we have

$\displaystyle\frac{H(x,t+h)-H(x,t)}{h}$
$\displaystyle\geq\int_{0}^{y_{*}(x,t+h)}v_{0}(\eta)\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)\frac{\left(X(\eta,t+h)-X(\eta,t)\right)}{h}\mathrm{d}{\eta}$
$\displaystyle-\int_{0}^{y_{*}(x,t+h)}v_{0}(\eta)\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{1}{h}\left[\frac{\upkappa
X(\eta,t+h)}{(t+h+\upkappa)^{2}}-\frac{\upkappa
X(\eta,t)}{(t+\upkappa)^{2}}\right]\mathrm{d}{\eta}$
$\displaystyle+x\int_{0}^{y_{*}(x,t+h)}\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{1}{h}\left[\frac{\upkappa}{(t+h+\upkappa)^{2}}-\frac{\upkappa}{(t+\upkappa)^{2}}\right]v_{0}(\eta)\mathrm{d}{\eta}.$
(6.27)

Now passing to the limit as $h\to 0$ in (6.27) and using
$\frac{\mathrm{d}}{\mathrm{d}{t}}X(\eta,t)=u(X(\eta,t),t),$ we get

$\displaystyle\frac{\partial}{\partial t}H(x,t)$
$\displaystyle\geq\int_{0}^{y_{*}(x,t)}v_{0}(\eta)\left(\frac{\eta}{2\upkappa}+\frac{u_{0}(\eta)}{2}\right)u(X(\eta,t),t)\mathrm{d}{\eta}$
$\displaystyle-\int_{0}^{y_{*}(x,t)}v_{0}(\eta)\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\left[\frac{\upkappa
u(X(\eta,t),t)}{(t+\upkappa)^{2}}-\frac{2\upkappa
X(\eta,t)}{(t+\upkappa)^{3}}\right]\mathrm{d}{\eta}$
$\displaystyle-x\int_{0}^{y_{*}(x,t)}v_{0}(\eta)\left(\frac{\eta}{2}-\frac{u_{0}(\eta)\upkappa}{2}\right)\frac{2\upkappa}{(t+\upkappa)^{3}}\mathrm{d}{\eta}$
$\displaystyle=2E(x,t)+I(x,t).$

Similarly, considering the inequality

$\displaystyle H(x,t+h)-H(x,t)\leq H(y_{*}(x,t),x,t+h)-H(y_{*}(x,t),x,t),$

we obtain

$\displaystyle\frac{\partial}{\partial t}H(x,t)\leq 2E(x,t)+I(x,t),$

and this completes the proof.

Step 5. Define $J(x,t)=\min\limits_{y\in\mathds{R}}J(y,x,t).$ Then we have the
following relations:

$\displaystyle(i)\,\,\,\frac{\partial}{\partial x}I(x,t)=-J(x,t),$ (6.28)
$\displaystyle(ii)\,\,\,\mathrm{d}{J}=\frac{1}{t+\upkappa}\left(\frac{x}{t+\upkappa}-u\right)\mathrm{d}{m},$
(6.29)

in the sense of Radon-Nikodym derivatives in $x.$
Proof of Step 5. The proof of (6.28) is exactly same as the proof of (6.18)
and thus we omit it.

To prove (6.29), we consider the following two cases.
Case I. Let $y_{*}(x,t)=y^{*}(x,t).$ First of all, after a simplification, we
obtain

$\displaystyle\frac{1}{t+\upkappa}\left(\frac{x}{t+\upkappa}-u(x,t)\right)=\frac{2\upkappa}{(t+\upkappa)\left((t+\upkappa)^{2}-k^{2}\right)}\left[y_{*}(x,t)-\frac{\upkappa}{(t+\upkappa)}x\right].$

On the other hand, fixing $t>0$ and taking $x_{1}<x<x_{2},$ by the definition
of potential we have

$\displaystyle F(y_{*}(x_{2},t),x_{2},t)\leq F(y_{*}(x_{1},t),x_{2},t).$

This implies