footnote

11footnotetext: Département de mathématiques et statistique, Université de
Montréal, Montréal, Québec, Canada, H3C 3A7.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Laboratoire Jacques-Louis Lions & Centre National de la Recherche
Scientifique, Sorbonne Université, 4 Place Jussieu, 75252 Paris, France.
Email<EMAIL_ADDRESS>
2000 AMS Classification Primary. 35L65, 74XX. Secondary. 76N10, 76L05. Key
Words and Phrases. Nonlinear hyperbolic system; nonclassical Riemann solver;
undercompressive shock wave; kinetic relation; nucleation criterion;
splitting-merging pattern.

# The mathematical theory of splitting-merging patterns
in phase transition dynamics

Eva Kardhashi1, Marc Laforest1, and Philippe G. LeFloch3

(May 2021)

###### Abstract

For nonlinear hyperbolic systems of conservation laws in one space variable,
we establish the existence of nonclassical entropy solutions exhibiting
nonlinear interactions between shock waves with strong strength. The proposed
theory is relevant in the theory of phase transition dynamics, and the
solutions under consideration enjoy a splitting–merging pattern, consisting of
(compressive) classical and (undercompressive) nonclassical waves, interacting
together as well as with classical waves of smaller strength. Our analysis is
based on three novel ideas. First, a generalization of Hayes–LeFloch’s
nonclassical Riemann solver is introduced for systems and is based on
prescribing, on one hand, a kinetic relation for the propagation of
nonclassical undercompressive shocks and, on the other hand, a nucleation
criterion that selects between classical and nonclassical behavior. Second, we
extend LeFloch-Shearer’s theorem to systems and we prove that the presence of
a nucleation condition implies that only a finite number of splitting and
merging cycles can occur. Third, our arguments of nonlinear stability build
upon recent work by the last two authors who identified a natural total
variation functional for scalar conservation laws and, specifically, for
systems of conservation laws we introduce here novel functionals which measure
the total variation and wave interaction of nonclassical and classical waves.

###### Contents

  1. 1 Introduction
    1. 1.1 Objective of this paper
    2. 1.2 Background on nonclassical shocks and phase boundaries
    3. 1.3 Main results in this paper
  2. 2 Nonclassical Riemann solver with kinetics and nucleation
    1. 2.1 Background and notation
    2. 2.2 Construction based on Lax shock inequalities
    3. 2.3 Nonclassical wave sets based on a single entropy inequality
    4. 2.4 Kinetic relation and nucleation criterion
  3. 3 Splitting-merging patterns with kinetics and nucleation
    1. 3.1 Objectives and notation
    2. 3.2 Generalized wave strength and interaction patterns
    3. 3.3 Interaction estimates
    4. 3.4 Interaction potential
    5. 3.5 Global estimates
  4. 4 Splitting-merging solutions with nucleation
    1. 4.1 Preliminaries
    2. 4.2 Nucleation and the splitting-merging cycle
    3. 4.3 Total variation estimate for the splitting-merging cycle

##  1 Introduction

###  1.1 Objective of this paper

We consider the initial value problem associated with nonlinear hyperbolic
systems of conservation laws in one space variable, and we establish the
existence of nonclassical entropy solutions (in the sense introduced in [28]),
which, for instance, arise in complex fluid flows in the limit of vanishing
viscosity and capillarity. The solutions under consideration in the present
paper exhibit a “splitting–merging pattern” and consist of finitely many
(classical or nonclassical) big waves and classical small waves. More
precisely, our analysis relies on a new nonclassical Riemann solver which is
motivated by the theory of phase transition dynamics and is based on

  * •

a kinetic relation for the propagation of nonclassical undercompressive shocks
and,

  * •

optionally, a nucleation criterion that selects between classical and
nonclassical behavior.

All solutions satisfy a single entropy inequality associated with a given
entropy pair, and we note that the kinetic relation is equivalent to imposing
the rate of entropy dissipation across nonclassical shocks. The existence of
nonclassical entropy solutions for scalar equations was recently established
for the initial value problem by the last two authors [25] who, building on
the earlier work [3, 4, 5, 28, 30], succeeded in identifying a “natural” total
variation functional adapted to deal with nonclassical shock waves and their
interactions. By natural, we mean that the construction of the total variation
relies explicitly on the main property of the kinetic function, namely on the
contraction property of its iterates. The present paper provides a new and
significant development to this theory and we are able to

  * •

construct nonclassical entropy solutions to nonlinear strictly hyperbolic
systems, and

  * •

exhibit the splitting–merging phenomena for systems, which illustrates the
complexity of fluid flows involving phase transitions.

To achieve these results, we introduce here new functionals that suitably
measure the total variation and wave interaction of nonclassical and classical
waves. When the nucleation criterion is introduced, we show that these
functionals are strictly decreasing by an amount proportional to the strength
of the nucleation criteria.

Recall that the kinetic relation is an additional jump relation required only
for nonclassical undercompressive shocks. It generalizes to general systems of
conservation laws a notion originally introduced for a (hyperbolic-elliptic)
model of phase transition dynamics first studied by Abeyaratne and Knowles [1,
2], and Truskinovsky [37, 38], in relation to earlier work by Slemrod [34,
35]. The notion of a kinetic relation was introduced for general hyperbolic
systems by LeFloch and collaborators, and was then investigated mathematically
[20, 21, 22, 26, 27, 28, 29]. In addition, for scalar conservation laws
LeFloch and Shearer [30] observed that the classical solution may be
physically relevant in certain applications even in a range where a
nonclassical solution is available. This observation led them to propose a
nonclassical Riemann solver with kinetics and nucleation, which allows for
nonclassical behavior only in Riemann problems having sufficiently ‘large”
initial jump. This proposal was motivated by studies on the dynamics of thin
films, in which capillarity effects play a driving role on propagating
interfaces; cf. Bertozzi, Münch, and Shearer [8] and Bertozzi and Shearer [9].

###  1.2 Background on nonclassical shocks and phase boundaries

We thus consider nonlinear hyperbolic systems of conservation laws in one
space dimension, of the general form

${\partial}_{t}u+{\partial}_{x}f(u)=0,\quad
u=u(t,x)\in\mathcal{B}({\delta_{0}})$ (1.1)

and, as is customary, we focus attention on weak solutions understood in the
sense of distributions which are assumed to take values near a given constant
state in $\mathbb{R}^{N}$, normalized to be the origin. Throughout,
$\mathcal{B}({\delta_{0}})\subset\mathbb{R}^{N}$ denotes the ball centered at
the origin and with small radius $\delta_{0}>0$, while the flux
$f:\mathcal{B}({\delta_{0}})\subset\mathbb{R}^{N}$ is a smooth ($C^{3}$, say)
vector-valued map, In addition, weak solutions of interest should satisfy the
entropy inequality

${\partial}_{t}U(u)+{\partial}_{x}F(u)\leq 0,$ (1.2)

in which the entropy $U:\mathcal{B}({\delta_{0}})\to\mathbb{R}$ is assumed to
be strictly convex, and the associated entropy flux
$F:\mathcal{B}({\delta_{0}})\subset\mathbb{R}^{N}$ satisfies, by definition,
$\nabla F(u)^{T}=\nabla U(u)^{T}\,Df(u)$.

Let us first stress that our motivation for imposing the single entropy
inequality (1.2) comes from the fact that most (if not all) regularizations of
(1.1) arising in applications, of the form

${\partial}_{t}u^{\epsilon}+{\partial}_{x}f(u^{\epsilon})=R^{\epsilon}=R^{\epsilon}\big{(}u^{\epsilon},\epsilon\,u_{x}^{\epsilon},\epsilon^{2}\,u_{xx}^{\epsilon},\ldots\big{)}$

are “conservative” and ”entropy dissipative” in the limit $\epsilon\to 0$ and,
formally at least, lead to weak solutions satisfying (1.1)–(1.2). (See [28,
Chap. 1] for further details.)

Furthermore, we assume that (1.1) is a strictly hyperbolic system, for which
the Jacobian matrix $A(u):=Df(u)$ has real distinct eigenvalues
$\lambda_{1}(u)<\ldots<\lambda_{N}(u)$ and corresponding basis of left– and
right–eigenvectors $l_{j}(u),r_{j}(u)$. Following Hayes and LeFloch [21, 22],
a nonlinearity condition on the flux $f$ is also imposed which roughly
speaking requires that the flux $f$ has “at most one inflection point” along
every eigenvector direction — as this is a typical feature of many systems of
interest in continuum physics, especially phase transition models. More
precisely, we assume that each characteristic field of (1.1) is either (with
$u\in\mathcal{B}({\delta_{0}})$)

  * •

genuinely nonlinear (GNL), $j\in J_{1}$, that is, $\nabla\lambda_{j}(u)\cdot
r_{j}(u)>0$ after normalization, or

  * •

linearly degenerate (LD), $j\in J_{2}$, that is, $\nabla\lambda_{j}(u)\cdot
r_{j}(u)\equiv 0$, or

  * •

concave-convex (C-C), $i\in J_{3}$.

In the latter case by definition (see [21, 22]), the scalar-valued function
$m_{i}(u):=(\nabla\lambda_{i}\cdot r_{i})(u)$ does not keep a constant sign,
but
${\mathcal{M}}_{i}=\big{\\{}u\in\mathcal{B}({\delta_{0}})\,|\,m_{i}(u)=0\big{\\}}$
is a smooth manifold with dimension $N-1$, such that the vector field $r_{i}$
is transverse to ${\mathcal{M}}_{i}$ with

$\text{concave-convex (C--C):}\quad\nabla m_{i}(u)\cdot r_{i}(u)>0.$ (1.3)

Convex-concave fields can be defined similarly and, for simplicity in the
presentation, will not be considered in the present paper.

We are now in a position to consider the initial value problem associated with
(1.1), whose simplest version is provided by the so–called Riemann problem,
for which the initial data consist of two constant states
$u_{l},u_{r}\in\mathcal{B}({\delta_{1}})$ initially separated by a single jump
discontinuity:

$u(0,x)=u_{0}(x)=\begin{cases}u_{l},&x<0,\\\ u_{r},&x>0.\end{cases}$ (1.4)

Hayes and LeFloch [21, 22] considered weak solutions to the Riemann problem
(in the class of piecewise Lipschitz continuous functions, say) associated
with general systems with GNL, LD, or CC fields, and observed that solutions
satisfying the single entropy inequality (1.2), are not uniquely determined by
their initial data. In order to select a unique nonclassical solution to the
Riemann problem, it is necessary to impose an additional selection criterion,
called a kinetic relation, which selects, within a two–dimensional wave set, a
unique wave curve associated with any state (in $\mathcal{B}({\delta_{1}})$
with $\delta_{1}\leq\delta_{0}$, say). This selection criteria for the CC
fields, together with the usual solutions for the GNC and LD fields, yields
the nonclassical Riemann solver, as described in [21, 22].

###  1.3 Main results in this paper

In the present paper, we go beyond Hayes and LeFloch’s construction and we
introduce a new nonclassical Riemann solver for systems, referred to as the
nonclassical Riemann solver with kinetics and nucleation. We investigate its
properties, especially from the point of view of wave interactions. A main
result established below is the existence of a large class of splitting-
merging solutions containing one or two big (classical or nonclassical) waves,
propagating within a background of small waves. In short, we will establish
the following result: Consider the initial value problem associated with a
strictly hyperbolic system (1.1) with GNL, LD, or CC fields. Suppose that the
initial data is a perturbation of a large classical wave which splits into a
nonclassical wave and a big classical. Then, there exists a globally defined
in time solution with finitely many large waves. The two big waves can merge
and, later on, split due to interactions between themselves or with small
waves. In presence of the nucleation criterion, this splitting-merging
phenomena only takes place finitely many times, but can continue forever if
the nucleation criterion is removed.

Let us stress that the long-term objective is to establish an existence result
for initial data of small total variation with nonclassical shocks. As a step
towards this objective, this research has treated in detail small
perturbations to a single strong isolated nonclassical shocks using LeFloch’s
theory of kinetic functions in order to constrain the family of admissible
shocks. The main novelties are the new measure of shock strength, extending a
previous study of scalar problems [25], and a new weighted interaction
potential inspired by earlier works [31, 23]. This work explicitly identifies
the relationship between the strength of the nonclassical shocks and of the
perturbation, even as the strength of the nonclassical wave vanishes. Finally,
it is shown that a nucleation condition provides additional stability to the
splitting-merging process, thereby generalizing a result of LeFloch and
Shearer for scalar conservation laws [30]. The techniques described here
appear to be applicable to the analysis of solutions to problems with general
initial data.

An outline of this paper is as follows. In Section 2, we introduce our
nonclassical Riemann solver with kinetics and nucleation. Section 3 contains
the construction of the splitting-merging functional and presents our
generalized wave strength and total variation functionals. Section 4 is
devoted to showing that the nucleation condition implies that only finitely
many splittings and mergings may occur.

##  2 Nonclassical Riemann solver with kinetics and nucleation

###  2.1 Background and notation

Solutions to the Riemann problem are made of shock waves and rarefaction
waves, and we will first introduce some basic notation relative to such waves.
The results can be found in [22] and in the monograph of LeFloch [28].

A left-hand state $u_{-}$ being fixed, we consider the Hugoniot set consisting
of all right-hand states $u_{+}$ satisfying, for some $\lambda$, the condition
$-\lambda\,\bigl{(}u_{+}-u_{-}\bigr{)}+f(u_{+})-f(u_{-})=0.$ It is easily
checked that there exist $\delta_{1}<\delta_{0}$ and $\epsilon>0$ such that
for each $u_{-}\in\mathcal{B}({\delta_{1}})$ the following holds. The Hugoniot
set can be decomposed into $N$ curves
${\mathcal{H}}_{j}(u_{-})=\Big{\\{}v_{j}(s;u_{-})\,/\,s\in(-\epsilon,\epsilon)\Big{\\}}$
($1\leq j\leq N$) depending smoothly on $s$ and $u_{-}$, such that

$v_{j}(s;u_{-})=u_{-}+s\,r_{j}(u_{-})+{1\over
2}\,s^{2}\,\big{(}Dr_{j}(u_{-})\,r_{j}\big{)}(u_{-})+O(s^{3})$ (2.1)

and, for the shock speed $\lambda=\overline{\lambda}_{j}(s;u_{-})$,

$\displaystyle\overline{\lambda}_{j}(s;u_{-})=$
$\displaystyle\lambda_{j}(u_{-})+{1\over 2}\,s\,\bigl{(}\nabla\lambda_{j}\cdot
r_{j}\bigr{)}(u_{-})$ $\displaystyle+{1\over
6}\,s^{2}\,\Big{(}\bigl{(}\nabla(\nabla\lambda_{j}\cdot r_{j})\cdot
r_{j}\bigr{)}+{\nabla\lambda_{j}\cdot r_{j}\over
2}\,l_{i}(Dr_{j}\,r_{j})\Big{)}(u_{-})+O(s^{3}).$

Recall that we have assumed a normalization of the eigenvectors $r_{j}(u)$ for
genuinely nonlinear fields for which $\nabla\lambda_{j}\cdot r_{j}(u)>0$ for
all $u\in{\mathcal{B}}(\delta_{0})$.

Rarefaction waves are Lipschitz continuous, self-similar solutions of (1.1),
determined by integral curves of the vector fields $r_{j}$. It is easily
checked that, for $\delta_{1}<\delta_{0}$ and $\epsilon>0$, and for all
$u_{-}\in{\mathcal{B}}(\delta_{1})$ the following holds. For each
$j\in\bigl{\\{}1,\ldots,N\bigr{\\}}$ the integral curve
${\mathcal{O}}_{j}(u_{-})=\Big{\\{}w_{j}(s;u_{-})\,/\,s\in(-\epsilon,\epsilon)\Big{\\}}$
depends smoothly upon $u_{-}$ and $s$ and satisfies

$w_{j}(s;u_{-})=u_{-}+s\,r_{j}(u_{-})+{1\over
2}\,s^{2}\,(Dr_{j}\,r_{j})(u_{-})+O(s^{3})$ (2.2)

and

$\lambda_{j}(w_{j}(s;u_{-}))=\lambda_{j}(u_{-})+s\,\big{(}\nabla\lambda_{j}\cdot
r_{j}\big{)}(u_{-})+{1\over 2}\,s^{2}\bigl{(}\nabla(\nabla\lambda_{j}\cdot
r_{j})\cdot r_{j}\bigr{)}(u_{-})+O(s^{3}).$

To ensure the uniqueness of the weak solution, we impose the entropy
inequality (1.2) associated with a given strictly convex entropy pair $(U,F)$.
For a shock wave connecting $u_{-}$ to $u_{+}$, the entropy inequality imposes
that the rate of entropy dissipation be non-positive, i.e.

$E(u_{-},u_{+}):=-\lambda\,\bigl{(}U(u_{+})-U(u_{-})\bigr{)}+F(u_{+})-F(u_{-})\leq
0.$ (2.3)

Hence, we have a constraint on the sign of the entropy dissipation
$\overline{E}(s;u_{-}):=E(u_{-},v_{i}(s;u_{-}))$ along the $j$-Hugoniot curve
issuing from $u_{-}$. In fact, one has

$\overline{E}(s;u_{-})={s^{3}\over
12}\left(r_{j}^{T}\,D^{2}U\,r_{j}\right)(u_{-})\,\big{(}\nabla\lambda_{j}\cdot
r_{j}\big{)}(u_{-})+O(s^{4}).$ (2.4)

Recalling Lax shock inequalities

$\lambda_{j}(u_{-})\geq\lambda\geq\lambda_{j}(u_{+})$ (2.5)

for a shock wave connecting $u_{-}$ to $u_{+}$ at the speed $\lambda$, we
state the following standard result about genuinely nonlinear fields. Consider
the $j$-Hugoniot curve $s\mapsto u_{+}=v_{j}(s;u_{-})$ issuing from a state
$u_{-}$ and associated with a genuinely nonlinear family. Then, the shock
speed $s\mapsto\overline{\lambda}_{j}(s;u_{-})$ is an increasing function of
$s$, and Lax shock inequalities are equivalent to imposing the entropy
inequality (2.3) and select the part $s<0$ of the Hugoniot curve. Both
inequalities above are violated for $s>0$. The composite curve
${\mathcal{W}}_{j}(u_{-}):={\mathcal{S}}_{j}(u_{-})\cup{\mathcal{O}}_{j}(u_{-})$,
called the $j$-wave curve issuing from $u_{-}$, is defined as

$\displaystyle{\mathcal{W}}_{j}(u_{-})={\mathcal{S}}_{j}(u_{-})\cup{\mathcal{O}}_{j}(u_{-})=:\bigl{\\{}\psi_{j}(s;u_{-})\,/\,s\in(-\epsilon,\epsilon)\bigr{\\}},$
$\displaystyle\psi_{j}(s;u_{-})=\begin{cases}v_{j}(s;u_{-}),&s\in(-\epsilon,0],\\\
w_{j}(s;u_{-}),&s\in[0,\epsilon).\end{cases}$

The mapping
$\psi_{j}:(-\epsilon,\epsilon)\times\mathcal{B}({\delta_{1}})\to\mathbb{R}^{N}$
admits continuous derivatives up to second-order and bounded third-order
derivatives in $s$ and $u_{-}$, with

$\psi_{j}(s;u_{-})=u_{-}+s\,r_{j}(u_{-})+{1\over
2}\,s^{2}\,(Dr_{j}\,r_{j})(u_{-})+O(s^{3}).$

###  2.2 Construction based on Lax shock inequalities

We now turn our attention to the characteristic fields $i\in J_{3}$ that are
not globally genuinely nonlinear and such that genuine nonlinearity condition
fails at one point along each wave curve. In this case, the Lax shock
inequalities (2.5) still single out a unique Riemann solution, but the entropy
inequality (2.3) is no longer sufficiently discriminating in itself (as we
will discuss in the following subsection). It is convenient now to use a
global parameter defined below.

We now assume that the scalar-valued function
$m_{i}(u):=\nabla\lambda_{i}(u)\cdot r_{i}(u)$ does not keep a constant sign
and that
${\mathcal{M}}_{i}=\big{\\{}u\in{\mathcal{B}}(\delta_{0})\,|\,m_{i}(u)=0\big{\\}}$
is a smooth affine manifold with dimension $N-1$ such that the vector field
$r_{i}$ is transverse to the manifold ${\mathcal{M}}_{i}$ and that

$\text{Concave-convex (CC) field: }\nabla m_{i}(u)\cdot r_{i}(u)>0,\quad
u\in{\mathcal{B}}(\delta_{0}).$

The transversality assumption above implies that the wave speed $\lambda_{i}$
has one critical point (a minimum) along each wave curve, that is,
$\nabla\lambda_{i}(u)\cdot r_{i}(u)=0$ if and only if $m_{i}(u)=0$. To ensure
that ${\mathcal{M}}_{i}\cap\mathcal{B}({\delta_{1}})\neq\emptyset$, and hence
that the problem is not reduced to the genuinely nonlinear case, we assume
that the origin belongs to ${\mathcal{M}}_{i}$, possibly at the cost of an
additional translation of $u$.

To parametrize the Hugoniot curves
${\mathcal{H}}_{i}(u_{-})=\big{\\{}v_{i}(m;u_{-})\big{\\}}$ and integral
curves ${\mathcal{O}}_{i}(u_{-})=\big{\\{}w_{i}(m;u_{-})\big{\\}}$, we
introduce a global parameter $\mu_{i}(u)\in\mathbb{R}$ which depends smoothly
upon $u$ and is strictly monotone along the wave curves, specifically

$\nabla\mu_{i}(u)\cdot r_{i}(u)\neq 0,\quad u\in\mathcal{B}({\delta_{0}})$

and

$\mu_{i}(u)=0\quad\text{ if and only if }\quad m_{i}(u)=0.$

The obvious choice is to set the parameter $\mu_{i}$ now coincides with the
parameter $m_{i}$, that is,

$\mu_{i}\bigl{(}v_{i}(m;u_{-})\bigr{)}=m,\text{ for $m\leq
m_{i}(u_{-})$},\quad\mu_{i}\bigl{(}w_{i}(m;u_{-})\bigr{)}=m,\text{ for $m\geq
m_{i}(u_{-})$.}$ (2.6)

We observe that although the eigenvector $r_{i}$ associated to the CC family
$i$ can be parametrized smoothly throughout ${\mathcal{B}}(\delta_{0})$, the
sign of $m_{i}=\nabla\lambda_{i}\cdot r_{i}(u)$ changes and hence the Hugoniot
curve $v_{i}$ can no longer be parametrized everywhere so that negative $s$
corresponds to the entropy admissible portion of the curve.

From now on we consider a concave-convex field $i\in J_{3}$ and we restrict
attention to shocks that satisfy Lax shock inequalities. Let $u_{-}$ be a
state with $\mu_{i}(u_{-})>0$ and we consider the integral and Hugoniot
curves.

######  Lemma 2.1 (Characteristic speed along the integral curve).

The $i$-characteristic speed along the integral curve
$m\mapsto\lambda_{i}(w_{i}(m;u_{-}))$ is a strictly convex function of $m$
which achieves its minimum value at $m=0$. In particular, it is decreasing for
$m<0$ and increasing for $m>0$.

The part of the classical wave curve ${\mathcal{W}}_{i}^{c}(u_{-})$ associated
with rarefaction waves is the part $m\geq\mu_{i}(u_{-})$ of the integral curve
${\mathcal{O}}_{i}(u_{-})$.

######  Lemma 2.2 (Characteristic speed and shock speed along the Hugoniot
curve).

The $i$-characteristic speed along the Hugoniot curve
$m\mapsto\lambda_{i}(v_{i}(m;u_{-}))$ is a strictly convex function. On the
other hand, the $i$-shock speed along the Hugoniot curve

$m\mapsto\overline{\lambda}_{i}(m;u_{-}):=\overline{\lambda}_{i}\left(u_{-},v_{i}(m;u_{-})\right)$

is a strictly convex function, which is either globally increasing or else
achieves a minimum value at a point, denoted by $\mu_{i}^{\natural}(u_{-})$.
At this critical value, the characteristic speed and the shock speed coincide:

$\overline{\lambda}_{i}(m;u_{-})=\lambda_{i}(v_{i}(m;u_{-}))\quad\text{ at
}m=\mu_{i}^{\natural}(u_{-})$

and, for some smooth function $e=e(u_{-})<0$,

${\partial}_{m}v_{i}(\mu_{i}^{\natural}(u_{-});u_{-})=e(u_{-})\,r_{i}\bigl{(}v_{i}(\mu_{i}^{\natural}(u_{-});u_{-})\bigr{)}.$

For definiteness, from now on we assume that the shock speed does admit a
minimum value, that is to say that
$v_{i}(\mu_{i}^{\natural}(u_{-}),u_{-})\in\mathcal{B}({\delta_{1}})$. The
characteristic speed and the shock speed satisfy

$\displaystyle\overline{\lambda}_{i}(m;u_{-})-\lambda_{i}(v_{i}(m;u_{-}))>0,\quad
m\in\big{(}\mu_{i}^{\natural}(u_{-}),\mu_{i}(u_{-})\big{)},$
$\displaystyle\overline{\lambda}_{i}(m;u_{-})-\lambda_{i}(v_{i}(m;u_{-}))<0,\quad
m<\mu_{i}^{\natural}(u_{-})\text{ or }m>\mu_{i}(u_{-}).$

######  Proposition 2.3 (Classical wave curves).

There exists $\delta_{1}<\delta_{0}$ and $\epsilon>0$ with the following
property. For $i\in J_{3}$, $u_{-}\in\mathcal{B}({\delta_{1}})$ and
$\mu_{i}(u_{-})>0$, the $i$-wave curve of right-hand states connected to
$u_{-}$ by a combination of $i$-elementary waves satisfying Lax shock
inequalities is composed of the following three pieces:

${\mathcal{W}}_{i}^{nc}(u_{-})\,=\,\begin{cases}{\mathcal{O}}_{i}(u_{-}),&\mu_{i}(u)\geq\mu_{i}(u_{-}),\\\
{\mathcal{H}}_{i}(u_{-}),&\mu_{i}^{\natural}(u_{-})\leq\mu_{i}(u)\leq\mu_{i}(u_{-}),\\\
{\mathcal{O}}_{i}(u_{-}^{\natural}),&\mu_{i}(u)\leq\mu_{i}^{\natural}(u_{-}).\end{cases}$

where $u_{-}^{\natural}:=v_{i}(\mu_{i}^{\natural}(u_{-});u_{-})$. It is
continuously differentiable with bounded second-order derivatives in $m$ and
$u_{-}$ and satisfies

$\psi_{i}(m;u_{-})=u_{-}+{m-\mu_{i}(u_{-})\over\bigl{(}\nabla\mu_{i}\cdot
r_{i}\bigr{)}(u_{-})}\,r_{i}(u_{-})+O\bigl{(}m-\mu_{i}(u_{-})\bigr{)}^{2}.$
(2.7)

The second-order derivatives of wave curves for non-genuinely nonlinear fields
are not continuous, in general.

It is easy to check that the expansion (2.7) still holds for states $u_{-}$
satisfying $\mu_{i}(u_{-})<0$ for our extension of $r_{i}$ to all of
${\mathcal{B}}(\delta_{0})$.

###  2.3 Nonclassical wave sets based on a single entropy inequality

This section generalizes Hayes-LeFloch’s construction [21, 22] of a
nonclassical Riemann solver based on a kinetic function and presents our new
Riemann solver based on, both, kinetics and nucleation.

We now investigate the consequences of a single entropy inequality for the
solutions of the Riemann problem. A shock wave is called a nonclassical shock
if it satisfies the single entropy inequality (2.3) but not Lax shock
inequalities (2.5). It is called a classical shock if both (2.3) and (2.5)
hold. Before imposing any entropy condition at this stage, we have that a
$i$-shock wave connecting a left-hand state $u_{-}$ to a right-hand state
$u_{+}$ can be:

  1. 1.

a Lax shock, satisfying
$\lambda_{i}(u_{-})\geq\overline{\lambda}_{i}(u_{-},u_{+})\geq\lambda_{i}(u_{+})$,

  2. 2.

a slow undercompressive shock:
$\overline{\lambda}_{i}(u_{-},u_{+})\,\leq\,\min\bigl{(}\lambda_{i}(u_{-}),\lambda_{i}(u_{+})\bigr{)},$

  3. 3.

a fast undercompressive shock:
$\overline{\lambda}_{i}(u_{-},u_{+})\,\geq\,\max\bigl{(}\lambda_{i}(u_{-}),\lambda_{i}(u_{+})\bigr{)},$

  4. 4.

or a rarefaction shock:
$\lambda_{i}(u_{-})<\overline{\lambda}_{i}(u_{-},u_{+})<\lambda_{i}(u_{+}).$

Let $u_{-}$ be such that $\mu_{i}(u_{-})>0$ and denote by
$\mu_{i}^{-\natural}(u_{-})$ the point of ${\mathcal{H}}_{i}(u_{-})$ such that

$\overline{\lambda}_{i}(\mu_{i}^{-\natural}(u_{-});u_{-})=\lambda_{i}(u_{-}),\quad\mu_{i}^{-\natural}(u_{-})<\mu_{i}^{\natural}(u_{-}),$
(2.8)

whenever such a point exists. For simplicity in the presentation we assume
that both points $u_{-}^{\natural}:=\mu_{i}^{\natural}(u_{-})$ and
$u_{-}^{-\natural}:=\mu_{i}^{-\natural}(u_{-})$ belong to
$\mathcal{B}({\delta_{1}})$, since the present discussion would be much
simpler otherwise. Note also that the shock wave connecting $u_{-}$ to
$u_{-}^{\natural}$ is a right-contact, while the shock connecting $u_{-}$ to
$u_{-}^{-\natural}$ is a left-contact.

Note also that the function $\mu_{i}^{\natural}$ depends smoothly upon its
argument, and

${\nabla\mu_{i}^{\natural}\cdot r_{i}}\sim-{1\over 2}\,{\nabla\mu_{i}\cdot
r_{i}}\quad\text{ near the manifold }{\mathcal{M}}_{i},$ (2.9)

as established in [28]. In particular, (2.9) implies that
$\mu_{i}^{\natural}<0$. Let $u_{-}$ be given with $\mu_{i}(u_{-})\geq 0$ and
consider a point $u_{+}$ on the Hugoniot curve ${\mathcal{H}}_{i}(u_{-})$, say
$u_{+}=v_{i}(m;u_{-})$ with $m=\mu_{i}(u_{+})$. It can be checked that the
shock is

$\displaystyle-\text{ a rarefaction shock if }m>\mu_{i}(u_{-})\,\text{ or
}m<\mu_{i}^{-\natural}(u_{-}),$ $\displaystyle-\text{ a Lax shock if
}m\in\big{[}\mu_{i}^{\natural}(u_{-}),\mu_{i}(u_{-})\big{]},$
$\displaystyle-\text{ an undercompressive shock if
}m\in\big{[}\mu_{i}^{-\natural}(u_{-}),\mu_{i}^{\natural}(u_{-})\big{)}.$

######  Lemma 2.4 (Entropy dissipation along the Hugoniot curve).

  1. (i)

The entropy dissipation $\overline{E}(m;u_{-}):=E(u_{-},v_{i}(m;u_{-}))$
vanishes at $\mu_{i}(u_{-})$ and at a point

$\mu^{\flat}_{i0}(u_{-})\in\big{(}\mu_{i}^{-\natural}(u_{-}),\mu_{i}^{\natural}(u_{-})\big{)}.$

The entropy dissipation is decreasing for $m<\mu_{i}^{\natural}(u_{-})$,
increasing for $m>\mu_{i}^{\natural}(u_{-})$, and achieves a negative maximum
value at the critical point of the wave speed $\mu_{i}^{\natural}(u_{-})$.

  2. (ii)

A shock satisfying the entropy inequality (2.3) cannot be a rarefaction shock.
A nonclassical shock is undercompressive and satisfies
$m\in\big{(}\mu^{\flat}_{i0}(u_{-}),\mu_{i}^{\natural}(u_{-})\big{)}.$

  3. (iii)

Any shock satisfying Lax shock inequalities also satisfies the entropy
inequality (2.3).

We refer to $\mu^{\flat}_{i0}$ as the zero-entropy dissipation function
associated with the $i$-characteristic field. In view of Lemma 2.4 we can also
define the companion function $\mu^{\sharp}_{i0}$ by

$\mu^{\flat}_{i0}(u_{-})<\mu_{i}^{\natural}(u_{-})<\mu^{\sharp}_{i0}(u_{-}),\quad\overline{\lambda}_{i}\bigl{(}\mu^{\sharp}_{i0}(u_{-});u_{-}\bigr{)}=\overline{\lambda}_{i}\bigl{(}\mu^{\flat}_{i0}(u_{-});u_{-}\bigr{)}.$

It can be checked by the implicit function theorem that $\mu^{\flat}_{i0}$ and
$\mu^{\sharp}_{i0}$ are at least as smooth as $f,F$ and $U$.

Given $u_{-}$, we will also denote by $\Phi_{i0}^{\flat}(u_{-})$,
$\Phi_{i}^{\natural}(u_{-})$, and $\Phi_{i}^{\sharp}(u_{-})$ on the shock
curve ${\mathcal{H}}_{i}(u_{-})$ the states with coordinates
$\mu^{\flat}_{i0}(u_{-})$, $\mu_{i}^{\natural}(u_{-})$, and
$\mu_{i}^{\sharp}(u_{-})$, respectively. It is a fundamental identity of the
theory that the second iterate of the map $\Phi_{i0}^{\flat}$ is the identity
map $id$ [28], that is,

$\Phi_{i0}^{\flat}\circ\Phi_{i0}^{\flat}=id.$ (2.10)

Given a left-hand state $u_{-}$, the set of all states that can be reached
using only $i$-waves is called the nonclassical $i$-wave set issuing from
$u_{-}$ and denoted by ${\mathcal{X}}_{i}(u_{-})$.

######  Proposition 2.5 (Nonclassical wave set).

In addition to the classical one, the $i$-wave fan may contain a nonclassical
$i$-shock connecting $u_{-}$ to some intermediate state
$u_{+}\in{\mathcal{H}}_{i}(u_{-})$ with
$\mu_{i}(u_{+})\in\big{[}\mu^{\flat}_{i0}(u_{-}),\mu_{i}^{\natural}(u_{-})\big{)}$
followed by

  1. (a)

either a non-attached rarefaction connecting $u_{+}$ to
$u\in{\mathcal{O}}_{i}(u_{+})$ if $\mu_{i}(u)<\mu_{i}(u_{+})$,

  2. (b)

or by a classical shock connecting $u_{+}$ to $u\in{\mathcal{H}}_{i}(u_{+})$
if $\mu_{i}(u)>\mu_{i}(u_{+})$.

This defines a two-parameter family of right-hand states $u$ which can be
reached from $u_{-}$ by nonclassical solutions.

###  2.4 Kinetic relation and nucleation criterion

The nonclassical wave set ${\mathcal{X}}_{i}(u_{-})$ is a two-dimensional
manifold within which we will select the nonclassical wave curve
${\mathcal{W}}_{i}^{nc}(u_{-})$. One parameter should be prescribed for each
wave family in $J_{3}$, and we postulate that for all $u_{-}$ a single right-
hand state $u_{+}$ can be reached from $u_{-}$ across a nonclassical shock.

For each concave-convex family $i$, a kinetic function for the
$i$-characteristic field is a Lipschitz continuous mapping
$\mu_{j}^{\flat}:\mathcal{B}({\delta_{1}})\to\mathbb{R}$ satisfying

$\displaystyle\mu^{\flat}_{i0}(u)<\mu^{\flat}_{i}(u)\leq\mu_{i}^{\natural}(u),\quad\mu_{i}(u)>0,$
(2.11)
$\displaystyle\mu_{i}^{\natural}(u)\leq\mu^{\flat}_{i}(u)<\mu^{\flat}_{i0}(u),\quad\mu_{i}(u)<0.$

We impose the following kinetic relation: for every $i$-nonclassical shock the
right-hand state $u_{+}$ is determined from the left-hand state $u_{-}$ by

$u_{+}=u_{-}^{\flat}:=v_{i}(m_{-}^{\flat};u_{-}),\qquad
m_{-}^{\flat}=\mu^{\flat}_{i}(u_{-}).$ (2.12)

To the kinetic function we shall associate its companion function
$\mu_{i}^{\sharp}$ by

$\overline{\lambda}_{i}(u_{-},u_{-}^{\sharp})=\overline{\lambda}_{i}(u_{-},u_{-}^{\flat}),\qquad
u_{-}^{\sharp}:=v_{i}(\mu_{i}^{\sharp}(u_{-});u_{-}).$ (2.13)

It can be checked that the map $\mu_{i}^{\sharp}$ depends Lipschitz
continuously upon its argument and

$\displaystyle\mu_{i}^{\natural}(u)\leq\mu_{i}^{\sharp}(u)<\mu_{i}(u)\quad\text{
when }\mu_{i}(u)>0,$ (2.14)
$\displaystyle\mu_{i}(u)<\mu_{i}^{\sharp}(u)\leq\mu_{i}^{\natural}(u)\quad\text{
when }\mu_{i}(u)<0.$

In the present paper, we need one additional map which allows us to select
between the classical and the nonclassical construction of the wave fans. We
introduce a Lipschitz continuous, nucleation threshold function $\mu_{i}^{n}$
associated with the kinetic function and satisfying

$\displaystyle\mu_{i}^{\natural}(u_{-})\leq\mu_{i}^{n}(u_{-})\leq\mu_{i}^{\sharp}(u_{-})\quad\text{
when }\mu_{i}(u_{-})\geq 0,$ (2.15)
$\displaystyle\mu_{i}^{\sharp}(u_{-})\leq\mu_{i}^{n}(u_{+})\leq\mu_{i}^{\natural}(u_{-})\quad\text{
when }\mu_{i}(u_{-})\leq 0.$

The nucleation criterion is stated in the following way:

For $|\mu_{i}(u_{-})-\mu_{i}(u_{+})|<|\mu_{i}(u_{-})-\mu_{i}^{n}(u_{-})|$,
classical $i$-shocks are always preferred, (2.16) whenever they are available.

When $\mu_{i}^{n}(u_{-})=\mu_{i}^{\sharp}(u_{-})$, then the nucleation
criteria is superfluous.

The Riemann solver with kinetics and nucleation is defined as the solution to
the Riemann problem that satisfies the entropy inequality, the kinetic
relation, and the nucleation criterion. In other words, a large initial jump
always “nucleates” and produces a nonclassical shock wave. This is consistent
with observations made in phase transition dynamics, especially in material
science where nucleations in austenite-martensite transformations are observed
[2].

######  Theorem 2.6 (Nonclassical Riemann solver with kinetics and
nucleation).

Suppose that all fields of the strictly hyperbolic system (1.1) are genuinely
nonlinear, linearly degenerate, or concave-convex.

  1. 1.

Consider a concave-convex field $i\in J_{3}$ and fix a Lipschitz continuous,
kinetic function $\mu^{\flat}_{i}$ satisfying (2.11). Then, for each
$u_{-}\in\mathcal{B}({\delta_{1}})$ the kinetic relation (2.12) and the
nucleation criterion (2.15) select a unique nonclassical $i$-wave curve
${\mathcal{W}}_{i}^{nc}(u_{-})$ within the nonclassical wave set
${\mathcal{X}}_{i}(u_{-})$. When $\mu_{i}(u_{-})>0$ it is composed of the
following two connected components:

${\mathcal{W}}_{i}^{nc}(u_{-})\,=\,\begin{cases}{\mathcal{O}}_{i}(u_{-}),&\mu_{i}(u)\geq\mu_{i}(u_{-}),\\\
{\mathcal{H}}_{i}(u_{-}),&\mu_{i}^{n}(u_{-})\leq\mu_{i}(u)\leq\mu_{i}(u_{-}),\\\
{\mathcal{H}}_{i}(u_{-}^{\flat}),&\mu^{\flat}_{i}(u_{-})\leq\mu_{i}(u)<\mu_{i}^{\sharp}(u_{-}),\\\
{\mathcal{O}}_{i}(u_{-}^{\flat}),&\mu_{i}(u)\leq\mu^{\flat}_{i}(u_{-}),\end{cases}$

where $u_{-}^{\flat}:=v_{i}\big{(}\mu^{\flat}_{i}(u_{-});u_{-}\big{)}$. The
Riemann solution is a single rarefaction shock, or a single classical shock,
or a nonclassical shock followed by a classical shock, or finally a
nonclassical shock followed by a rarefaction, respectively. Each connected
component of ${\mathcal{W}}_{i}^{nc}(u_{-})$ is a continuous and monotone
curve in the parameter $m=\mu_{i}(u)$. When
$\mu_{i}^{n}(u_{-})=\mu_{i}^{\sharp}(u_{-})$, it has bounded second-order
derivatives for all $m\neq\mu_{i}^{\sharp}(u_{-})$ and is Lipschitz continuous
(at least) at $m=\mu_{i}^{\sharp}(u_{-})$.

  2. 2.

For all $u_{l}$ and $u_{r}$ in $\mathcal{B}({\delta_{1}})$ the Riemann problem
(1.1) and (1.2) admits a unique solution satisfying a single entropy
inequality, the kinetic relation, and the nucleation criterion, which is
determined by the intersection of the classical wave curves
${\mathcal{W}}_{j}^{c}$ for genuinely nonlinear fields and the nonclassical
wave curves ${\mathcal{W}}_{i}^{nc}$ for concave-convex fields $i$. The
solution is a self-similar solution made of $N+1$ constant states

$u_{l}=u^{0},\,u^{1},\ldots,\,u^{N}=u_{r}$

separated by $j$ wave fans. The intermediate constant states satisfy

$u^{j}\in{\mathcal{W}}_{j}(u^{j-1}),\quad u^{j}=\psi_{j}(m_{j};u^{j-1}),$

with $m_{j}:={\mu_{j}}(u^{j-1})$, and are connected with combination of
classical or nonclassical waves.

We recover the classical wave curve ${\mathcal{W}}_{i}^{c}(u_{-})$ with the
trivial choice $\mu^{\flat}_{i}\equiv\mu_{i}^{n}\equiv\mu_{i}^{\natural}$.
Observe that, contrary to the scalar case, the curve
${\mathcal{H}}_{i}(u_{-}^{\flat})$ is continued up to the value
$\mu_{i}^{\sharp}(u_{-})$, since the intersection with a wave curve from
another family (which enters in the construction of the Riemann solution) may
generate a point $u$ which has $\mu_{i}(u)>\mu_{i}^{n}(u_{-})$, while the
intersection of the classical curve ${\mathcal{H}}_{i}(u_{-})$ (with
$\mu_{i}^{n}(u_{-})\leq\mu_{i}(u)\leq\mu_{i}(u_{-})$) is empty.

It is natural to ask about the physical relevance of the proposed hyperbolic
theory with kinetics and nucleation. It is expected that the nucleation
criterion is required when instabilities (one-wave/two-wave patterns) are
observed numerically or in regularization involving competition between
viscosity, capillarity, or relaxation effects. It is conceivable that the
nucleation function (as is the case for the kinetic relation) could be
determined numerically within a range where both classical and nonclassical
behaviors co-exist. Applications could be sought in the fields of phase
transition dynamics, thin film dynamics, and Camassa-Holm–type models.

##  3 Splitting-merging patterns with kinetics and nucleation

###  3.1 Objectives and notation

In the case that the nucleation threshold is chosen to coincide with the
critical value determined by the kinetic mapping, the nonclassical Riemann
solution constructed in the previous section depends continuously in the
$L^{1}$ norm upon its initial data, but not so in a pointwise sense. Observe
that, along a nonclassical wave curve, the speeds of the (rarefaction or
shock) waves change continuously since

$\lim_{m\to{\mu_{j}}^{\sharp}+}\lambda_{j}\big{(}v_{j}(m,u_{-})\big{)}=\lim_{m\to{\mu_{j}}^{\sharp}-}\lambda_{j}\big{(}v_{j}(m,u_{-}^{\flat})\big{)},$

and

$\lim_{m\to{\mu_{j}}^{\flat}-}\lambda_{j}\big{(}w_{j}(m,u_{-}^{\flat})\big{)}=\lim_{m\to{\mu_{j}}^{\flat}+}\lambda_{j}\big{(}v_{j}(m,u_{-}^{\flat})\big{)}.$

However, at $\mu_{j}^{\sharp}(u_{-})$, one has to compare the shock speed of
the nonclassical shock with the shock speeds of the nonclassical and classical
shocks. The wave speeds (only) are continuous but the total variation of the
nonclassical Riemann solution is not a continuous function of its end points.
This lack of continuity makes it delicate to control the strengths of waves at
interactions. This motivates the study made in the present section which
covers general nucleation thresholds, including the “trivial” choice.

We consider three distinct states $u_{*}$, $u_{*}^{\sharp}$, and
$u_{*}^{\flat}$ associated with a given wave family (say $i\in J_{3}$), so
that $u_{*}$ can be connected to $u_{*}^{\sharp}$ by a classical shock
(denoted by $C_{*}^{\downarrow}$), $u_{*}$ can be connected to $u_{*}^{\flat}$
by a nonclassical shock (denoted by $N_{*}$), and $u_{*}^{\flat}$ can be
connected to $u_{*}^{\sharp}$ by a classical shock (denoted by
$C_{*}^{\uparrow}$); moreover, we assume that all three speeds coincide, that
is,

$\overline{\lambda}_{i}(u_{*},u_{*}^{\sharp})=\overline{\lambda}_{i}(u_{*},u_{*}^{\flat})=\overline{\lambda}_{i}(u_{*}^{\flat},u_{*}^{\sharp})=:\overline{\lambda}_{*}.$
(3.1)

Such a structure is directly determined by the kinetic relation that was
prescribed in Section 2.

######  Definition 3.1.

The triplet $\big{(}u_{*},u_{*}^{\sharp},u^{\flat}_{*}\big{)}$ satisfying the
condition above is called a classical–nonclassical three state pattern.

The nonclassical shock is stable by perturbation in the phase space, in the
sense that for every $u$ in a sufficiently small neighborhood of $u_{*}$ there
exists a state $u^{\flat}\in{\mathcal{H}}_{i}(u)$ which lies in a small
neighborhood of $u^{\flat}_{*}$ and still satisfies the kinetic relation. The
classical shock connecting $u_{*}^{\flat}$ to $u_{*}^{\sharp}$ is also stable
in the same manner. However, the classical shock connecting $u_{*}$ to
$u_{*}^{\sharp}$ is not a stable structure when its end points are perturbed.
Actually, depending upon the size of the perturbation the shock connecting
$u_{*}$ to $u_{*}^{\sharp}$ may either keep its structure, or else transform
into a perturbation $N^{\downarrow}$ of the nonclassical shock
$N_{*}^{\downarrow}$, plus a perturbation $C^{\uparrow}$ of the classical
shock $C_{*}^{\uparrow}$. When the nucleation condition (2.15) is present,
then the instability of the classical shock connecting $u_{*}$ to
$u_{*}^{\sharp}$ requires the perturbation to be strong enough to overcome the
difference $\mu_{i}^{\sharp}(u_{*})-\mu_{i}^{n}(u_{*})$. In general, we will
write $u_{*}^{n}$ to denote the state $\psi_{i}(\mu_{i}N(u_{*});u_{*})$, which
will coincide with $u_{*}^{\sharp}$ when nucleation is absent.

Following LeFloch and Shearer [30], we are interested here in the effect of
adding a perturbation to this double wave structure and establishing a
global–in–time stability result, when the perturbation has sufficiently small
total variation. Indeed, we are going to establish the existence of a large
class of solutions undergoing repeated splitting and merging, whose global
structure at each time consists of a single shock or a double-shock structure
plus small waves. As was the case for scalar conservation laws, we will see
that the splitting/merging feature can take place infinitely many times when
the nucleation function coincides with the kinetic function but only finitely
many times when a non-trivial nucleation criterion is selected. The analysis
for systems is much more involved, since now interactions between different
wave families may take place and even interactions between waves of the same
family may contribute to increase the wave strengths.

The analysis of hyperbolic systems of conservation laws, since the work of
Glimm [19], proceeds by constructing an approximation within a simple class of
solutions in which all waves, in particular rarefactions, are localized to a
point. Following the new standard Ansatz of front-tracking approximations [11,
32], it suffices to construct piecewise constant approximations $u_{h}$ in
space with a finite number of discontinuities which depend on a small
parameter $h$, and for which the initial data $u_{0}$ converge in the
$\operatorname{TV}$ norm in space and in $L^{1}(\mathbb{R})$ to the initial
data as $h$ converges to zero. The main argument is then to show that the
sequence of approximations satisfy the hypothesis of Helley’s theorem. For
these reasons, the analysis in the rest of the paper will suppose the
solution, denoted $u_{h}=u_{h}(t,x)$, is in fact a front-tracking
approximation, i.e. a piecewise constant approximation with discontinuities
connecting states along waves, all up to a small error in
$L^{\infty}(\operatorname{TV})\cap L^{1}(\mathbb{R}\times[0,T])$. Usually, the
dependence $h$ will be implicit and the front-tracking approximation will
simply be denoted by $u$.

The front-tracking approximation $u$ we construct is associated with piecewise
constant initial data. Given the state $u_{*}$ consider initial data
$u(x,0)=u_{0}(x)$ having the following specific structure:

$u_{0}(x)={\overline{u}}_{0}(x)+v_{0}(x),$ (3.2)

where

${\overline{u}}_{0}(x):=\begin{cases}u_{*},&x<0,\\\
u_{*}^{n},&x>0.\end{cases}$ (3.3)

We assume that the small perturbation $v_{0}$ is piecewise constant, formed of
a collection of discontinuities $\gamma\in\mathcal{D}_{0}$ located at
$x_{\gamma}$, and of small total variation:

$\epsilon_{0}>TV(v_{0}):=\sum_{\gamma\in\mathcal{D}_{0}}\big{|}v_{0}(x_{\gamma}+)-v_{0}(x_{\gamma}-)\big{|}.$
(3.4)

The total variation above uses the euclidean distance between the states in
phase-space to mesure the strength of the discontinuity. In order to
distinguish the strength of the initial perturbation from the strength of the
initial strong wave in the $i$-th family, we assume that $v_{0}(0)=0$. The
strength of the large unperturbed shock $|u^{n}_{*}-u_{*}|$ provides a first
length scale, the size of the perturbation $\epsilon_{0}$ provides a second,
but smaller, length scale, and the nucleation criteria provides the third and
final length scale

$\eta:=\mu_{i}^{\sharp}(u_{*})-\mu_{i}^{n}(u_{*}),$ (3.5)

which we also assume to be the smallest of all three scales. Note that
${\overline{u}}_{0}$ is a single step function which admits two distinct
Riemann solutions ${\overline{u}}={\overline{u}}(t,x)$ made of admissible
waves.

For instance, if the perturbation $v_{0}$ is simply chosen to be a single step
function having a jump discontinuity located at $x=0$ and connecting $0$ (say)
to $v_{0}^{+}$, then one just need to solve a Riemann problem with left-hand
state $u_{*}$ and right-hand state $u_{*}^{n}+v_{0}^{+}$. In view of the
properties of the nonclassical Riemann solver (Section 2), we can find a cone
${\mathcal{A}}$ centered at the point $u_{*}^{n}$ such that:

  1. $\bullet$

If $\mu_{i}(u_{*}^{n}+v_{0}^{+})>\mu_{i}(u_{*}^{n})$ and
$u_{*}^{n}+v_{0}^{+}\in{\mathcal{A}}$ then the Riemann solution contains a
single big shock $C^{\downarrow}$ plus small waves in all families.

  2. $\bullet$

If $\mu_{i}(u_{*}^{n}+v_{0}^{+})<\mu_{i}(u_{*}^{n})$ and
$u_{*}^{n}+v_{0}^{+}\in{\mathcal{A}}$ the Riemann solution contains a big
nonclassical shock $N^{\downarrow}$ plus a faster big classical shock
$C^{\uparrow}$ plus small waves in other families $j\neq i$.

  3. $\bullet$

When $u_{*}^{n}+v_{0}^{+}\notin{\mathcal{A}}$ either one of the above
behaviors may be observed.

###  3.2 Generalized wave strength and interaction patterns

Waves will be measured by a generalized strength which extends the earlier
definitions for scalar equations [28, 30, 25]. For genuinely nonlinear and
linearly degenerate families, say indexed $j$, the strength of a simple wave
connecting $u_{-}$ on the left to $u_{+}$ on the right will be given by the
classical definition

$\sigma(u_{-},u_{+})=\mu_{j}(u_{+})-\mu_{j}(u_{-}).$

For every concave-convex family indexed $i$, we propose here to use the
zero–dissipation map $\Phi^{\flat}_{i0}$ and identify any state in the
$\mu_{i}<0$ region to a state in the $\mu_{i}>0$ region. Precisely, we
introduce the map

$\widetilde{\Phi^{\flat}_{i0}}(u)=\begin{cases}u&\text{if }\mu_{i}(u)>0,\\\
\Phi^{\flat}_{i0}(u)&\text{if }\mu_{i}(u)<0.\end{cases}$

######  Definition 3.2.

The signed strength of a wave joining states $u_{-}$ and $u_{+}$ in a concave-
convex family indexed $i$ is measured by

$\sigma(u_{-},u_{+})=\mu_{i}\bigl{(}\widetilde{\Phi^{\flat}_{i0}}(u_{+})\bigr{)}-\mu_{i}\bigl{(}\widetilde{\Phi^{\flat}_{i0}}(u_{-})\bigr{)}.$
(3.6)

In other words, we have

$\sigma(u_{-},u_{+})=\begin{cases}\mu_{i}(u_{+})-\mu_{i}(u_{-})&\text{if
}\mu_{i}(u_{-}),\mu_{i}(u_{+})>0,\\\
\mu_{i}\bigl{(}\Phi^{\flat}_{i0}(u_{+})\bigr{)}-\mu_{i}(u_{-})&\text{if
}\mu_{i}(u_{+})\leq 0<\mu_{i}(u_{-}),\\\
\mu_{i}(u_{+})-\mu_{i}\bigl{(}\Phi^{\flat}_{i0}(u_{-})\bigr{)}&\text{if
}\mu_{i}(u_{-})\leq 0<\mu_{i}(u_{+}),\\\
\mu_{i}\bigl{(}\Phi^{\flat}_{i0}(u_{+})\bigr{)}-\mu_{i}\bigl{(}\Phi^{\flat}_{i0}(u_{-})\bigr{)}&\text{if
}\mu_{i}(u_{-}),\mu_{i}(u_{+})\leq 0.\end{cases}$

This definition will be used for both strong and weak waves of the $i$-th
family, that is the concave-convex family. In particular, for weak waves of
the $i$-th family only the first and last cases above apply. Finally, we
observe that the weak rarefactions and weak shocks have strengths that are
respectively positive and negative.

We can now summarize our assumptions on the kinetic map $\Phi^{\flat}_{i}$ for
each concave-convex characteristic field, as follows:

  * (H1)

$\Phi^{\flat}_{i}(u)\in{\mathcal{W}}_{i}^{nc}(u)$ for all
$u\in\mathcal{B}({\delta_{1}})$. Throughout, we always assume that
$\delta_{1}\leq\delta_{0}$ is small enough so that, for all
$u\in\mathcal{B}(\delta_{1})$, the two states $\Phi^{\sharp}(u)$ and
$\Phi^{\flat}(u)$ are well-defined (and, of course, belong to
$\mathcal{B}({\delta_{0}})$).

  * (H2)

$\Phi^{\flat}_{i}:\mathcal{B}({\delta_{1}})\to\mathbb{R}^{n}$ is Lipschitz
continuous and one–to–one on its image.

  * (H3)

The restriction of $\Phi^{\flat}_{i}$ to the hypersurface ${\mathcal{M}}_{i}$
is the identity map, and the function
$s\mapsto\mu_{i}(\Phi^{\flat}_{i}(\psi_{i}(u;s)))$ is monotone descreasing for
all $u\in\mathcal{B}({\delta_{1}})$, provided $s$ satisfies
$\psi_{i}(u;s)\in\mathcal{B}({\delta_{1}})$.

  * (H4)

There exists a constant $C^{\flat\flat}\in(0,1)$ such that
$\big{|}\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u)\big{)}\big{|}\leq
C^{\flat\flat}\,|\mu_{i}(u)|$ for all $u\in\mathcal{B}({\delta_{1}})$.

The property (H4) was expressed in a form that will be convenient for later
analysis, but it should be pointed out that it is equivalent, because of
property (2.10), to the strict and uniform contractivity of
$\Phi^{\flat}_{i}\circ\Phi^{\flat}_{i}$.

######  Lemma 3.3.

Under the conditions (H1)–(H4), the generalized shock strength satisfies the
following properties:

  * •

When the nucleation map $\Phi^{n}_{i}$ coincides with the map
$\Phi^{\sharp}_{i}$ determined by the kinetic map, the generalized wave
strength is continuous (up to perturbation due to small waves) when the two
large waves combine together or when a classical shock splits.

  * •

The generalized wave strength is equivalent to the standard wave strength.

###### Proof.

Consider the states $\big{(}u_{*},u_{*}^{\sharp},u^{\flat}_{*}\big{)}$ forming
a classical-non-classical three state wave pattern. For an arbitrarily small
perturbation, the classical shock connecting $u_{*}$ to $u_{*}^{\sharp}$ can
be split into a non-classical shock between $u_{*}$ and $u_{*}^{\flat}$
followed by a classical shock connecting $u_{*}^{\flat}$ to $u_{*}^{\sharp}$.
Using the definition of wave strength, we verify

$\displaystyle\sigma(u_{*},u_{*}^{\sharp})=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\sharp}_{i}(u_{*})\big{)}-\mu_{i}(u_{*})$
$\displaystyle=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\sharp}_{i}(u_{*})\big{)}-\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{*})\big{)}+\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{*})\big{)}-\mu_{i}(u_{*})$
$\displaystyle=$
$\displaystyle\,\sigma(u_{*},u_{*}^{\flat})+\sigma(u_{*}^{\flat},u_{*}^{\sharp}),$

which establishes the continuity property.

To prove the equivalence, we must show that there exists strictly positive
constants $c$ and $C$ such that for every $j$-wave connecting two states
$u_{-},u_{+}$, then

$c\big{|}\mu_{j}(u_{+})-\mu_{j}(u_{-})\big{|}\leq\big{|}\sigma(u_{-},u_{+})\big{|}\leq
C\big{|}\mu_{j}(u_{+})-\mu_{j}(u_{-})\big{|}.$

If $j\neq i$, that is if the wave does not belong to the same concave-convex
family of the strong shock, then the definition of wave strength is the same
as usual and there is nothing to show. Consider therefore the case with $j=i$
with $u_{+}\in\mathcal{W}^{nc}_{i}(u_{-})$. If both $\mu_{i}(u_{-})$ and
$\mu_{i}(u_{+})$ are positive, then
$|\sigma(u_{-},u_{+})|=|\mu_{i}(u_{+})-\mu_{i}(u_{-})|$ is the usual
definition. If both $\mu_{i}(u_{-})$ and $\mu_{i}(u_{+})$ are negative, then
because $\Phi^{\flat}_{i}$ is Lipschitz and one-to-one there exists $c$ and
$C$ such that

$c\big{|}\mu_{i}(u_{+})-\mu_{i}(u_{-})\big{|}\leq|\sigma(u_{-},u_{+})|=\Big{|}\mu_{i}\big{(}\Phi^{\flat}_{i}(u_{+})\big{)}-\mu_{i}\big{(}\Phi^{\flat}_{i}(u_{-})\big{)}\Big{|}\leq
C\big{|}\mu_{i}(u_{+})-\mu_{i}(u_{-})\big{|}.$

Consider now the last case where $\mu_{i}(u_{-})>0>\mu_{i}(u_{+})$. First of
all, the upper bound obviously holds
$|\sigma(u_{-},u_{+})|<|\mu_{i}(u_{+})-\mu_{i}(u_{-})|$. To derive the lower
bound, we observe that for a single wave,
$\mu_{i}(u_{+})>\mu_{i}(\Phi^{\flat}_{i}(u_{-}))$. Given that
$\mu_{i}(\Phi^{\flat}_{i}(\psi_{i}(u_{-};m)))$ is monotone decreasing, clearly

$\displaystyle\big{|}\sigma(u_{-},u_{+})\big{|}=$
$\displaystyle\,\Big{|}\mu_{i}\big{(}\Phi^{\flat}_{i0}(u_{+})\big{)}-\mu_{i}(u_{-})\Big{|}$
$\displaystyle\geq$
$\displaystyle\,\Big{|}\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{-})\big{)}-\mu_{i}(u_{-})\Big{|}\geq(1-C^{\flat\flat})\big{|}\mu_{i}(u_{-})\big{|}.$

The Lipschitz continuity of $\Phi^{\flat}_{i}$ implies that there exists a
constant $B^{\flat}$ such that

$\big{|}\mu_{i}(\Phi^{\flat}_{i}(u_{-}))\big{|}\leq
B^{\flat}\big{|}\mu_{i}(u_{-})\big{|}.$

Continuing from our earlier lower bound, we find

$\displaystyle\big{|}\sigma(u_{-},u_{+})\big{|}\geq$
$\displaystyle\,\frac{1-C^{\flat\flat}}{1+B^{\flat}}(1+B^{\flat})\big{|}\mu_{i}(u_{-})\big{|}$
$\displaystyle\geq$
$\displaystyle\,\frac{1-C^{\flat\flat}}{1+B^{\flat}}\Big{(}\big{|}\mu_{i}(u_{-})\big{|}+\big{|}\mu_{i}(\Phi^{\flat}_{i}(u_{-}))\big{|}\Big{)}\geq\frac{1-C^{\flat\flat}}{1+B^{\flat}}\big{|}\mu_{i}(u_{+})-\mu_{i}(u_{-})\big{|}.$

This completes the proof of the equivalence. ∎

We are now in a position to state our main result.

######  Definition 3.4.

A splitting–merging pattern is a nonclassical entropy solution to (1.1) which
contains at most two big waves located at $y=y(t)\leq z=z(t)$ separated by
small classical waves, so that at each time the solution contains either (i)
only a single large classical shock, or (ii) a large nonclassical shock
followed by a large classical shock.

######  Theorem 3.5 (Nonlinear stability of splitting–merging patterns).

Given any strictly hyperbolic system (1.1) with genuinely nonlinear, linearly
degenerate, or concave-convex characteristic families, consider a kinetic
mapping $\Phi^{\flat}_{i}$ associated with a concave-convex family (indexed by
$i$), uniformly satisfying the conditions (H1)–(H4). Then, any two wave
pattern $\big{(}u_{*},\Phi^{\flat}_{i}(u_{*}),\Phi^{\sharp}_{i}(u_{*})\big{)}$
is globally-in-time nonlinearly stable, that is, for every perturbation with
sufficiently small total variation there exists a global–in–time solution
$u=u(t,x)$ to (1.1) which is a splitting–merging pattern satisfying the
initial condition (3.2).

######  Theorem 3.6 (Nonlinear stability of splitting–merging patterns with
arbitrarily small strength).

Given any strictly hyperbolic system (1.1) with genuinely nonlinear, linearly
degenerate, or concave-convex characteristic families, there exist positive
constants $\kappa_{*}$ and $\delta_{*}<\delta_{0}$ depending only upon the
flux function and the constant $C^{\flat\flat}$ arising in (H4), so that the
following property holds: for every family of kinetic mappings
$\Phi^{\flat}_{i}$ associated with concave-convex families (indexed by $i$),
provided they satisfy the conditions (H1)–(H4), any three wave pattern
$\big{(}u_{*},\Phi^{\flat}_{i}(u_{*}),\Phi^{\sharp}_{i}(u_{*})\big{)}\in\mathcal{B}(\delta_{*})^{3}$
is globally-in-time nonlinearly stable, that is, for every perturbation with
total variation
$\epsilon_{0}<\kappa_{*}|\sigma(u_{*},\Phi^{\sharp}_{i}(u_{*}))|$ there exists
a global–in–time solution $u=u(t,x)$ to (1.1) which is a splitting–merging
pattern satisfying the initial condition (3.2).

Observe that the above theorem applies to a broad class of kinetic functions
and allows the strength of the classical/nonclassical pattern to have
arbitrarily small strength, while the perturbation is allowed to be a small
(but fixed) ratio of those waves.

###  3.3 Interaction estimates

The purpose of this section is to introduce a functional $W$ that measures the
total variation of the weak waves and provides estimates for its variation
during the interactions which occur in splitting-merging solutions. This will
require an extension of Glimm’s interaction estimates [19, 36, 15] to the
interactions involving non-classical waves.

We begin with our construction of a functional equivalent to the total
variation of the weak waves and depending on parameters whose values will be
determined in Section 3.5. Henceforth, it will be convenient to use the
symbols ${N^{\downarrow}_{\pm}},{C^{\uparrow}},{C^{\downarrow}}$ to denote
respectively, the non-classical wave, its classical wave partner connecting
increasing states, and the classical wave present when the non-classical wave
is absent. We define $y^{h}(t)$ to be the position of either crossing wave,
${C^{\downarrow}_{\pm}}$ or ${N^{\downarrow}_{\pm}}$, in the front-tracking
approximations and, when ${C^{\uparrow}}$ is present, we define $z^{h}(t)$ to
be the position of ${C^{\uparrow}}$. When ${C^{\uparrow}}$ doesn’t appear, we
assume $z^{h}=y^{h}$.

The shock positions allow us to define three functionals, each associated to
specific families of waves,

$\displaystyle V_{L,k}(t)=\sum_{\begin{subarray}{c}\text{$k$-waves}\\\
x<y_{h}(t)\end{subarray}}|\sigma_{k,x}|,\qquad
V_{M,k}(t)=\sum_{\begin{subarray}{c}\text{$k$-waves}\\\
y^{h}(t)<x<z^{h}(t)\end{subarray}}|\sigma_{k,x}|,\qquad
V_{R,k}(t)=\,\,\sum_{\begin{subarray}{c}\text{$k$-waves}\\\
z^{h}(t)<x\end{subarray}}|\sigma_{k,x}|.$

In practice, a linear combination of these measures of variation will be used.
Assuming the strong waves belong to the $i$-th family of waves, then we
introduce the positive constants $k_{L},k_{M},k_{R}$,
$k^{\lessgtr}_{L},k^{\lessgtr}_{M}$ and $k^{\lessgtr}_{R}$, and the
functionals

$\displaystyle V_{L}(t)=$
$\displaystyle\,\,k_{L}V_{L,i}(t)+k_{L}^{<}\sum_{j<i}V_{L,j}(t)+k_{L}^{>}\sum_{j>i}V_{L,j}(t),$
$\displaystyle V_{M}(t)=$
$\displaystyle\,\,k_{M}V_{M,i}(t)+k_{M}^{<}\sum_{j<i}V_{M,j}(t)+k_{M}^{>}\sum_{j>i}V_{M,j}(t),$
$\displaystyle V_{R}(t)=$
$\displaystyle\,\,k_{R}V_{R,i}(t)+k_{R}^{<}\sum_{j<i}V_{M,j}(t)+k_{R}^{>}\sum_{j>i}V_{R,j}(t),$
$\displaystyle W(t)=$ $\displaystyle\,\,V_{L}(t)+V_{R}(t)+V_{M}(t).$

For convenience, we will also be writing $|\,\alpha|:=|\sigma(\alpha)|$ and

$\operatorname{sign}(\alpha)=\begin{cases}-1,&\text{ $\alpha$ is a shock,}\\\
+1,&\text{ otherwise,}\end{cases}$

We start by listing all possible wave interactions which may occur within a
splitting-merging solution. All possible interactions in the systems case are,
up to the generation of weak waves in families $i\neq j$, a subset of those
which may be found in the scalar case and described in LeFloch’s monograph
[28]. Following the notation introduced there, let $C$ and $N$ denote
respectively classical and nonclassical waves. Outgoing waves will be
distinguished by a prime superscript, such as $C^{\prime}$, and if waves
connect states $u_{-}>u_{+}$ ($u_{-}<u_{+}$) then we will write
${C^{\downarrow}}$ (${C^{\uparrow}}$) depending on the type of wave.
Sometimes, we may also include a subscript $\pm$ ($\mp$) to indicate more
precisely that a wave $C_{\pm}$ ($C_{\mp}$) connects $u_{-}>0>u_{+}$
($u_{-}<0<u_{+}$). Finally, lowercase greek letters, like $\alpha_{i}$ and
$\beta_{j}$, will denote weak waves in respectively the $i$-th and $j$-th
family. Using the notation above, interactions involving two $i$-waves are the
following.

  1. Case 1.

Splitting of a big wave: $C^{\downarrow}\alpha_{i}\longrightarrow
NC^{\uparrow}$ or $\alpha_{i}C^{\downarrow}\longrightarrow NC^{\uparrow}$.

  2. Case 2.

Merging of two big waves: $NC^{\uparrow}\longrightarrow C^{\downarrow}$.

  3. Case 3.

Perturbation of a big classical wave: $C\alpha_{i}\longrightarrow C$ or
$\alpha_{i}C\longrightarrow C$ with $C={C^{\downarrow}_{\pm}},{C^{\uparrow}}$.

  4. Case 4.

Perturbation of a big nonclassical wave: $\alpha_{i}N\longrightarrow
N\alpha_{i}^{\prime}$.

Secondly, interactions may involve a big $i$-wave and a small wave of another
family $i\neq j$:

  1. Case 5.

Splitting of a big wave: $C^{\downarrow}\beta_{j}\longrightarrow
NC^{\uparrow}$ or $\beta_{j}C^{\downarrow}\longrightarrow NC^{\uparrow}$.

  2. Case 6.

Perturbation of a big nonclassical wave:
$N\beta_{j}\longrightarrow\beta_{j}^{\prime}N$.

  3. Case 7.

Perturbation of a big classical wave:
$C\beta_{j}\longrightarrow\beta_{j}^{\prime}C$ or $\beta_{j}C\longrightarrow
C\beta_{j}^{\prime}$ with $C={C^{\downarrow}_{\pm}},{C^{\uparrow}}$.

That these are the only possible interactions within a splitting-merging
solution is clear for very short times but will only hold at all times when we
show that the strengths of the weak waves remain small.

We now introduce an extension of Glimm’s interaction estimates to the
interactions given above not previously covered by interactions between
classical waves. Informally, these estimates show that up to a quadratic
error, the strengths of the outgoing waves are the sum of the strengths of the
incoming waves. Classically, if
$\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{N})^{T}\in\mathbb{R}^{N}$ is
the vector of the signed strengths of the $N$ waves connecting $u_{\ell}$ to
$u_{m}$ according to the Riemann problem, then it means that there are $N-1$
intermediate states connecting $u_{\ell}$ to $u_{m}$ defined by

$\displaystyle\widetilde{u}_{1}=$
$\displaystyle\,\psi_{1}(\alpha_{1};u_{\ell}),$
$\displaystyle\widetilde{u}_{2}=$
$\displaystyle\,\psi_{1}(\alpha_{1};\widetilde{u}_{1}),$
$\displaystyle\vdots\,$ $\displaystyle u_{m}=$
$\displaystyle\,\psi_{1}(\alpha_{1};\widetilde{u}_{N-1}).$

Hence, if $\boldsymbol{\alpha}\in\mathbb{R}^{N}$ describes the solution to the
Riemann problem connecting $u_{\ell}$ to $u_{m}$ and
$\boldsymbol{\beta}\in\mathbb{R}^{n}$ does the same between $u_{m}$ and
$u_{r}$, then the solution $\boldsymbol{\gamma}$ to the Riemann problem
connecting $u_{\ell}$ to $u_{r}$ satisfies

$\boldsymbol{\gamma}=\boldsymbol{\alpha}+\boldsymbol{\beta}+\big{(}\text{
quadratic }\big{)}.$

Unfortunately, it is not trivial to extend the previous estimate to simple
waves from a concave-convex family $i$ connecting $u_{-}$ and $u_{+}$ because
the map $\psi_{i}$ does not satisfy

$u_{+}=\psi_{i}\big{(}\sigma(u_{-},u_{+});u_{-}\big{)},$ (3.7)

for $\sigma$ defined by (3.6). Hence, care will be needed when discussing the
parametrization of the wave curves. Before proceeding, recall that two simple
waves $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$, with
$\boldsymbol{\alpha}$ to the left of $\boldsymbol{\beta}$ and belonging
repesctively to families $k$ and $\ell$, are said to be approaching if
$k>\ell$ or if $k=\ell$ but at least one of the waves is a shock.

######  Theorem 3.7 (Glimm’s interaction estimates for a nonclassical Riemann
solver).

Consider any strictly hyperbolic system (1.1) with genuinely nonlinear,
linearly degenerate, or concave-convex characteristic families, and assume
there exists a kinetic mapping $\Phi^{\flat}_{i}$ associated with a concave-
convex family (indexed by $i$), uniformly satisfying the conditions (H1)–(H4).
Then there exists a sufficiently small $\delta_{1}$ such that for any states
$u_{\ell},u_{m},u_{r}\in\mathcal{B}({\delta_{1}})$, connecting simple waves
$\boldsymbol{\alpha},\boldsymbol{\beta}\in\mathbb{R}^{N}$ and interacting
according to Case 1-Case 7, or connecting only only weak waves, we have the
estimates

$\boldsymbol{\gamma}=\boldsymbol{\alpha}+\boldsymbol{\beta}+\mathcal{O}\bigg{(}\sum_{\begin{subarray}{c}\alpha_{k},\beta_{\ell}\\\
\text{approaching}\end{subarray}}|\alpha_{k}|\cdot|\beta_{\ell}|\bigg{)},$

where $\boldsymbol{\gamma}$ is the vector of signed strengths for the waves in
the solution to the Riemann problem for the consecutive states $u_{\ell}$ and
$u_{r}$.

###### Proof.

The literature contains a few proofs of this result for interactions between
classical waves [19, 36, 15]; here we will adapt the proof given in Smoller
[36]. The proof proceeds in two steps. The first one is to demonstrate the
simpler quadratic estimate

$\boldsymbol{\gamma}=\boldsymbol{\alpha}+\boldsymbol{\beta}+\mathcal{O}\big{(}\big{[}|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|\big{]}^{2}\big{)}.$
(3.8)

In a second step, one uses the additivity of simple rarefaction waves to
demonstrate inductively that the more accurate estimate holds. A careful
analysis of the second part shows that the argument also applies to
nonclassical waves if the weaker estimate (3.8) still holds. Our objective
will therefore be to demontrate the weaker estimate for the new interactions
present in a splitting-merging pattern. There are in fact five new
interactions that need to be discussed, from among the interactions listed
previously, but the cases 1 and 4 are similar to the cases 5 and 6
respectively. Furthermore, Case 3 and Case 7 are entirely classical and easy
to treat, despite the failure of (3.7). For the sake of brevity, we will
therefore only treat Case 1, 2 and 4.

Case 1. (${C^{\downarrow}_{\pm}}\alpha_{i}\rightarrow
N^{\prime}{{C^{\uparrow}}}^{\prime}$ or
$\alpha_{i}{C^{\downarrow}_{\pm}}\rightarrow
N^{\prime}{{C^{\uparrow}}}^{\prime}$). We consider the first interaction where
${C^{\downarrow}_{\pm}}$ is connecting states $u_{\ell}$ and $u_{m}$ while
$\alpha_{i}$ connects $u_{m}$ and $u_{r}$. For convenience, we will write
$\delta_{C}=\mu_{i}(u_{m})-\mu_{i}(u_{\ell})$ and
$\delta_{\alpha}=\mu_{i}(u_{r})-\mu_{i}(u_{m})$. Using the expansion (2.7), we
find

$\displaystyle u_{m}=$
$\displaystyle\,u_{\ell}+\frac{\delta_{C}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}r_{i}(u_{\ell})+\mathcal{O}\big{(}\delta_{C}^{2}\big{)},$
(3.9a) $\displaystyle u_{r}=$
$\displaystyle\,u_{m}+\frac{\delta_{\alpha}}{(\nabla\mu_{i}\cdot
r_{i})(u_{m})}r_{i}(u_{m})+\mathcal{O}\big{(}\delta_{\alpha}^{2}\big{)}.$
(3.9b)

Substituting the first expansion for $u_{m}$ into the second, and accounting
for the dependence on $u_{m}$ in the second expansion, we find

$\displaystyle u_{r}=$
$\displaystyle\,\Big{(}u_{\ell}+\frac{\delta_{C}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}r_{i}(u_{\ell})\Big{)}+\frac{\delta_{\alpha}}{(\nabla\mu_{i}\cdot
r_{i})(u_{m})}r_{i}(u_{m})+\mathcal{O}\big{(}\delta_{C}^{2}+\delta_{\alpha}^{2}\big{)}$
$\displaystyle=$
$\displaystyle\,u_{\ell}+\frac{\delta_{C}+\delta_{\alpha}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}r_{i}(u_{\ell})+\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)}.$
(3.10)

This expansion used the following two simple identities

$\displaystyle\frac{1}{(\nabla\mu_{i}\cdot r_{i})(u_{m})}=$
$\displaystyle\frac{1}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}+\mathcal{O}(|\delta_{C}|),\qquad\quad
r_{i}(u_{m})=r_{i}(u_{\ell})+\mathcal{O}(|\delta_{C}|).$

The solution to the Riemann problem is described by the vector
$\boldsymbol{\gamma}$ of signed strengths of the waves, leading to the
expansion

$u_{r}=u_{\ell}+\sum_{k}\gamma_{k}r_{k}(u_{\ell})+\mathcal{O}(|\boldsymbol{\gamma}|^{2}).$
(3.11)

Because a Taylor’s expansion with respect to small parameters is unique, we
have that

$\displaystyle\gamma_{k}=$
$\displaystyle\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)}\quad\text{for
$k\neq
i$;}\qquad\quad\gamma_{i}=\frac{\delta_{C}+\delta_{\alpha}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}+\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)}.$

If $u_{\ell}^{\prime}$ and $u_{r}^{\prime}$ are the states connected by the
$i$-th wave in the Riemann problem, then

$\displaystyle|u_{\ell}-u_{\ell}^{\prime}|=$
$\displaystyle\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)},\qquad\quad|u_{r}-u_{r}^{\prime}|=\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)}.$

The Riemann solver of Proposition 2.6 tells us the outgoing waves of the
$i$-th family are a nonclassical shock $N^{\prime}$ followed by a classical
shock ${C^{\uparrow}}^{\prime}$. To check additivity, we compute

$\displaystyle\sigma({{N^{\downarrow}_{\pm}}}^{\prime})+\sigma({{C^{\uparrow}}}^{\prime})=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{\ell}^{\prime})\big{)}-\mu_{i}(u_{\ell}^{\prime})+\mu_{i}\big{(}\Phi^{\flat}_{i0}(u_{r}^{\prime})\big{)}-\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{\ell}^{\prime})\big{)}$
$\displaystyle=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}(u_{r})\big{)}-\mu_{i}(u_{\ell})+\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)}$
$\displaystyle=$
$\displaystyle\,\mu_{i}(\Phi^{\flat}_{i0}(u_{m}))-\mu_{i}(u_{\ell})+\mu_{i}\big{(}\Phi^{\flat}_{i0}(u_{r})\big{)}-\mu_{i}(\Phi^{\flat}_{i0}(u_{m}))+\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)}$
$\displaystyle=$
$\displaystyle\,\sigma({C^{\downarrow}_{\pm}})+\sigma(\alpha_{i})+\mathcal{O}\big{(}\big{[}|\delta_{C}|+|\delta_{\alpha}|\big{]}^{2}\big{)}.$

Case 2.
(${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{{C^{\downarrow}_{\pm}}}^{\prime}$).
We consider a nonclassical wave ${N^{\downarrow}_{\pm}}$, separated by states
$u_{\ell}$ and $u_{m}$, which interacts with a classical wave connecting
$u_{m}$ and $u_{r}$. For the scalars
$\delta_{N}=\mu_{i}(u_{m})-\mu_{i}(u_{\ell})$ and
$\delta_{C}=\mu_{i}(u_{r})-\mu_{i}(u_{m})$, the expansion (2.7) provides

$\displaystyle u_{m}=$ $\displaystyle
u_{\ell}+\frac{\delta_{N}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}r_{i}(u_{\ell})+\mathcal{O}\big{(}\delta_{N}^{2}\big{)},\qquad\quad
u_{r}=u_{m}+\frac{\delta_{C}}{(\nabla\mu_{i}\cdot
r_{i})(u_{m})}r_{i}(u_{m})+\mathcal{O}\big{(}\delta_{C}^{2}\big{)}.$

Substituting the first expansion into the second and following the procedure
established in the previous interaction, we find

$u_{r}=u_{\ell}+\frac{\delta_{N}+\delta_{C}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}r_{i}(u_{\ell})+\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)}.$

By comparing the solution $\boldsymbol{\gamma}$ to the Riemann problem between
$u_{\ell}$ and $u_{r}$, given by the expansion (3.11), to the previous
expansion we deduce that

$\displaystyle\gamma_{k}=$
$\displaystyle\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)},\qquad\text{for
$k\neq
i$;}\quad\qquad\gamma_{i}=\frac{\delta_{C}+\delta_{\alpha}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}+\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)}.$

If the outgoing wave ${C^{\downarrow}_{\pm}}^{\prime}$ connects the states
$u_{\ell}^{\prime}$ and $u_{r}^{\prime}$, then

$\displaystyle|u_{\ell}-u_{\ell}^{\prime}|=$
$\displaystyle\,\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)},$
$\displaystyle|u_{r}-u_{r}^{\prime}|=$
$\displaystyle\,\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)}.$

Using these observations, we verify the additivity of the wave strengths

$\displaystyle\sigma({{C^{\downarrow}_{\pm}}}^{\prime})=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}(u_{r}^{\prime})\big{)}-\mu_{i}(u_{\ell}^{\prime})$
$\displaystyle=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}(u_{r})\big{)}-\mu_{i}(u_{\ell})+\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)}$
$\displaystyle=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}(u_{r})\big{)}-\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\sharp}_{i}(u_{\ell})\big{)}+\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\sharp}_{i}(u_{\ell})\big{)}-\mu_{i}(u_{\ell})+\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)}$
$\displaystyle=$
$\displaystyle\,\sigma({N^{\downarrow}_{\pm}})+\sigma({C^{\downarrow}})+\mathcal{O}\big{(}\big{[}|\delta_{N}|+|\delta_{C}|\big{]}^{2}\big{)}.$

Case 4.
($\alpha_{i}{N^{\downarrow}_{\pm}}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}\alpha_{i}^{\prime}$).
Consider a pair of simple waves of the $i$-th family with $\alpha_{i}$
approaching a nonclassical shock ${N^{\downarrow}_{\pm}}$ from the left. We
suppose that the neighboring states of the waves are, from left to right,
$u_{\ell},u_{m}$ and $u_{r}$ and we introduce the scalars
$\delta_{\alpha}=\mu_{i}(u_{m})-\mu_{i}(u_{\ell})$,
$\delta_{N}=\mu_{i}(u_{r})-\mu_{i}(u_{m})$. With this notation, the expansion
(2.7) gives us

$\displaystyle u_{m}=$ $\displaystyle
u_{\ell}+\frac{\delta_{N}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}r_{i}(u_{\ell})+\mathcal{O}\big{(}\delta_{\alpha}^{2}\big{)},\qquad\quad
u_{r}=u_{m}+\frac{\delta_{C}}{(\nabla\mu_{i}\cdot
r_{i})(u_{m})}r_{i}(u_{m})+\mathcal{O}\big{(}\delta_{N}^{2}\big{)},$

which can be combined to obtain

$u_{r}=u_{\ell}+\frac{\delta_{\alpha}+\delta_{N}}{(\nabla\mu_{i}\cdot
r_{i})(u_{\ell})}r_{i}(u_{\ell})+\mathcal{O}\big{(}\big{[}|\delta_{\alpha}|+|\delta_{N}|\big{]}^{2}\big{)}.$

By comparing this expansion with the one given by the unique solution
$\boldsymbol{\gamma}$ to the Riemann problem, written previously as (3.11), we
conclude that

$\displaystyle|u_{\ell}-u_{\ell}^{\prime}|=$
$\displaystyle\mathcal{O}\big{(}\big{[}|\delta_{\alpha}|+|\delta_{N}|\big{]}^{2}\big{)},\qquad\quad|u_{r}-u_{r}^{\prime}|=\mathcal{O}\big{(}\big{[}|\delta_{\alpha}|+|\delta_{N}|\big{]}^{2}\big{)}.$

From these estimates, we can verify additivity in the following way.

$\displaystyle\sigma({{N^{\downarrow}_{\pm}}}^{\prime})+\sigma(\alpha_{i}^{\prime})=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{\ell}^{\prime})\big{)}-\mu_{i}(u_{\ell}^{\prime})+\mu_{i}(\Phi^{\flat}_{i0}(u_{r}^{\prime}))-\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{\ell}^{\prime})\big{)}$
$\displaystyle=$
$\displaystyle\,\mu_{i}(\Phi^{\flat}_{i0}(u_{r}^{\prime}))-\mu_{i}(u_{\ell}^{\prime})$
$\displaystyle=$
$\displaystyle\,\mu_{i}\big{(}\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{m})\big{)}-\mu_{i}(u_{\ell})+\mathcal{O}\big{(}\big{[}|\delta_{\alpha}|+|\delta_{N}|\big{]}^{2}\big{)}$
$\displaystyle=$
$\displaystyle\,\mu_{i}(u_{m})-\mu_{i}(u_{\ell})+\mu_{i}(\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{m}))-\mu_{i}(u_{m})+\mathcal{O}\big{(}\big{[}|\delta_{\alpha}|+|\delta_{N}|\big{]}^{2}\big{)}$
$\displaystyle=$
$\displaystyle\,\sigma(\alpha_{i})+\sigma({{N^{\downarrow}_{\pm}}})+\mathcal{O}\big{(}\big{[}|\delta_{\alpha}|+|\delta_{N}|\big{]}^{2}\big{)}.$

This completes the proof of Case 4. ∎

We will now describe a series of lemmas which quantify the changes in the
functional $W$ during the interactions presented in the last section. These
results will sometimes use the nomenclature presented in LeFloch [28], where
the $16$ possible nonclassical interactions were given names. For example, the
expression CR-4 was used to identify the $4$-th type of interaction which may
occur (under certain conditions) between a shock $C$ on the left and a
rarefaction $R$ on the right.

######  Lemma 3.8 (Case 1. $\alpha_{i}{C^{\downarrow}_{\pm}}\rightarrow
N^{\prime}{C^{\uparrow}}^{\prime}$ or
${C^{\downarrow}_{\pm}}\alpha_{i}\rightarrow
N^{\prime}{{C^{\uparrow}}}^{\prime}$).

Suppose a weak wave $\alpha_{i}$ interacts with ${C^{\downarrow}_{\pm}}$ to
generate a nonclassical and classical shock pair
${{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$. Three types of
interactions may occur. When the wave $\alpha_{i}$ is slower than
${C^{\downarrow}_{\pm}}$, then the interaction must be CR-$4$ and $\alpha_{i}$
must be a rarefaction. If the wave $\alpha_{i}$ is faster than
${C^{\downarrow}_{\pm}}$, then the interaction can be either RC-$3$, with a
rarefaction $\alpha_{i}$, or CC-3 with a shock $\alpha_{i}$. When $\alpha_{i}$
is slower than ${C^{\downarrow}_{\pm}}$, then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}\,|\alpha_{i}|+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{R}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$
(3.12)

When $\alpha_{i}$ is faster than ${C^{\downarrow}_{\pm}}$, then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}\,|\alpha_{i}|+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{L}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$
(3.13)

###### Proof.

Consider the case where the weak wave $\alpha_{i}$ approaches
${C^{\downarrow}_{\pm}}$ from the right and generates a CR-$4$ interaction.
The interaction may generate weak waves in the $j$-th family, $j\neq i$, of
strength $\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|)$, but
otherwise it is easy to see that

$\displaystyle[V_{L}]=$
$\displaystyle\,\,k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}\,|\alpha_{i}|+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|).$

We now study simultaneously the RC-3 and CC-3 interactions with a weak wave
$\alpha_{i}$ approaching ${C^{\downarrow}_{\pm}}$ from the left.

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}\,|\alpha_{i}|+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|).$

∎

######  Lemma 3.9 (Case 2.
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{C^{\downarrow}_{\pm}}$).

During the strong interaction
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{C^{\downarrow}_{\pm}}$, we
have

$\displaystyle[V_{L}]=$
$\displaystyle\,\,k_{L}^{<}\cdot|{N^{\downarrow}_{\pm}}||{C^{\uparrow}}|\mathcal{O}\big{(}|v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})|\big{)},$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,k_{R}^{>}\cdot|{N^{\downarrow}_{\pm}}||{C^{\uparrow}}|\mathcal{O}\big{(}|v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})|\big{)},$
$\displaystyle[W]=$
$\displaystyle\,\,(k_{L}^{<}+k_{R}^{>})\cdot|{N^{\downarrow}_{\pm}}||{C^{\uparrow}}|\mathcal{O}\big{(}|v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})|\big{)}.$
(3.14)

###### Proof.

Consider the strong
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{C^{\downarrow}_{\pm}}$
interaction. The strength of the weak outgoing waves is obviously proportional
to $|{N^{\downarrow}_{\pm}}|\cdot|{C^{\uparrow}}|$ but also to

$\displaystyle\big{|}v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})\big{|}.$

These weak waves only contribute to $V_{L}$ and $V_{R}$ with $V_{M}=0$ before
and after the ${N^{\downarrow}_{\pm}}{C^{\uparrow}}$ interaction. ∎

######  Lemma 3.10 (Case 3. $\alpha_{i}C\rightarrow{C}^{\prime}$ or
$C\alpha_{i}\rightarrow{C}^{\prime}$ ).

Consider a weak wave $\alpha_{i}$ of the $i$-th family. The strong shock $C$
can be either a decreasing shock ${C^{\downarrow}_{\pm}}$ or the increasing
shock ${C^{\uparrow}}$ that appears when a nonclassical shock is present. If
the weak wave crosses the classical strong shock ${C^{\downarrow}_{\pm}}$ from
the left, then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}\,|\alpha_{i}|+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{L}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$
(3.15)

If the weak wave crosses the shock ${C^{\downarrow}_{\pm}}$ from the right,
then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}\,|\alpha_{i}|+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{R}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$
(3.16)

If the weak wave crosses the classical strong shock ${C^{\uparrow}}$ from the
left, then

$\displaystyle[V_{L}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{M}]=$
$\displaystyle\,\,-k_{M}\,|\alpha_{i}|+k_{M}^{<}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{R}]=$
$\displaystyle\,\,+k_{R}^{>}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\alpha_{i}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{M}+\big{(}k_{M}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\uparrow}}|)\Big{)}.$
(3.17)

If the weak wave crosses the shock ${C^{\uparrow}}$ from the right, then

$\displaystyle[V_{L}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{M}]=$
$\displaystyle\,\,+k_{M}^{<}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}\,|\alpha_{i}|+k_{R}^{>}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\alpha_{i}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{R}+\big{(}k_{M}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\uparrow}}|)\Big{)}.$
(3.18)

###### Proof.

The proof is relatively straightforward and we treat only the first two cases.
When the wave $\alpha_{i}$ approaches from the left, then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}|\alpha_{i}|+k_{L}^{<}\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,k_{R}^{>}\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|).$

∎

######  Lemma 3.11 (Case 4.
$\alpha_{i}{N^{\downarrow}_{\pm}}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}\alpha_{i}^{\prime}$).

Suppose a weak wave $\alpha_{i}$ of the $i$-th family approaches
${N^{\downarrow}_{\pm}}$ from the left and that the nonclassical shock
${{N^{\downarrow}_{\pm}}}^{\prime}$ remains after the interaction. Then the
only two possible interactions are RN and CN-$3$, and in both cases, we have

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}\,|\alpha_{i}|+k_{L}^{<}\,\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{M}]=$
$\displaystyle\,\,+k_{M}\,|\alpha_{i}^{\prime}|+\big{(}k_{M}+k_{M}^{>}\big{)}\,\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),$
$\displaystyle[V_{R}]=$ $\displaystyle\,\,0,$

With the constant $C^{\flat\flat}<1$ provided by property (H4) of
$\Phi^{\flat}_{i}$, we have

$\displaystyle[W]\leq$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{L}+C^{\flat\flat}k_{M}+\big{(}k_{M}+k_{L}^{<}+k_{M}^{>}\big{)}\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\Big{)}.$
(3.19)

###### Proof.

We will consider the changes in the total variation with respect to the
Definition 3.2 of wave strength. When the interaction occurs it generates
secondary waves in the families $j\neq i$ of strength
$\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|)$, and there is an
outgoing wave $\alpha_{i}^{\prime}$ to the right of
${{N^{\downarrow}_{\pm}}}^{\prime}$ and of the same type as $\alpha_{i}$. This
implies that

$\displaystyle|u_{\ell}-u_{\ell}^{\prime}|=\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),\quad|u_{r}-u_{r}^{\prime}|=\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|),\quad|\Phi^{\flat}_{i}(u_{m})-u_{m}^{\prime}|=\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|).$

Recall that only RN and CN-$3$ interactions are admissible and so $\alpha_{i}$
could be a rarefaction or a shock. We always have $[V_{R}]=0$ but the
calculation of the two other variations is less obvious.

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}|\alpha_{i}|+k_{L}^{<}\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|)$
$\displaystyle[V_{M}]=$
$\displaystyle\,\,+k_{M}|\alpha_{i}^{\prime}|+k_{M}^{>}\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|)$
$\displaystyle=$
$\displaystyle\,\,+k_{M}\big{|}\mu_{i}(\Phi^{\flat}_{i0}(u_{r}^{\prime}))-\mu_{i}(\Phi^{\flat}_{i0}(u_{m}^{\prime}))\big{|}+k_{M}^{>}\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|)$
$\displaystyle=$
$\displaystyle\,\,+k_{M}\big{|}\mu_{i}(\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{m}))-\mu_{i}(\Phi^{\flat}_{i0}\circ\Phi^{\flat}_{i}(u_{\ell}))\big{|}+(k_{M}+k_{M}^{>}\big{)}\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|)$
$\displaystyle=$
$\displaystyle\,\,+k_{M}C^{\flat\flat}\big{|}\mu_{i}(u_{m})-\mu_{i}(u_{\ell})\big{|}+(k_{M}+k_{M}^{>}\big{)}\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|)$

where we have used $C_{i}$ of condition (H4). ∎

######  Lemma 3.12 (Case 5.
$\beta_{j}{C^{\downarrow}_{\pm}}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$
or
${C^{\downarrow}_{\pm}}\beta_{j}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$
).

Consider a weak wave $\beta_{j}$, with $j\neq i$, which after interacting with
${C^{\downarrow}_{\pm}}$ leads to ${N^{\downarrow}_{\pm}}{C^{\uparrow}}$. When
the weak wave is faster than ${C^{\downarrow}_{\pm}}$, that is $j>i$, then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}^{>}\,|\beta_{j}|+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,+k_{R}^{>}\,|\beta_{j}|+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\Big{(}-k_{L}^{>}+k_{R}^{>}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$
(3.20)

On the other hand, when $j<i$ we have

$\displaystyle[V_{L}]=$
$\displaystyle\,\,+k_{L}^{<}\,|\beta_{j}|+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}^{<}\,|\beta_{j}|+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\Big{(}-k_{R}^{<}+k_{L}^{<}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$
(3.21)

###### Proof.

We begin with the case where $\beta_{j}$ is faster than
${C^{\downarrow}_{\pm}}$. The weak wave $\beta_{j}$ belongs to the family $j$,
hence up to a quadratic error, the strength of the outgoing wave $j$-wave is
the same strength as that of the incoming wave $\beta_{j}$. We begin by
computing the change when the wave $\beta_{j}$ belongs to a family $j>i$.

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}^{>}|\beta_{j}|+k_{L}^{<}\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|)$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,+k_{R}^{>}|\beta_{j}|+k_{R}^{>}\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|)$

In the second case, that is when $j<i$, then the wave $\beta_{j}$ approaches
from the right. We now remark that

$\displaystyle[V_{L}]=$
$\displaystyle\,\,+k_{L}^{<}|\beta_{j}|+k_{L}^{<}\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}^{<}|\beta_{j}|+k_{R}^{>}\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|).$

∎

######  Lemma 3.13 (Case 6.
$\beta_{j}{N^{\downarrow}_{\pm}}\rightarrow{N^{\downarrow}_{\pm}}^{\prime}\beta_{j}^{\prime}$
or
${N^{\downarrow}_{\pm}}\beta_{j}\rightarrow\beta_{j}^{\prime}{N^{\downarrow}_{\pm}}^{\prime}$).

Suppose a weak wave $\beta_{j}$ from the $j$-th family, $j>i$, crosses
${N^{\downarrow}_{\pm}}$ from the left and generates a small wave
$\beta_{i}^{\prime}$ in the $i$-th family, then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}^{>}\,|\beta_{j}|+k_{L}^{<}\,\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{M}]=$
$\displaystyle\,\,+k_{M}^{>}\,|\beta_{j}|+(k_{M}+k_{M}^{>})\,\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{R}]=$ $\displaystyle\,\,0,$ $\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\big{(}-k_{L}^{>}+k_{M}^{>}+(k_{M}+k_{L}^{<}+k_{M}^{>})\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\big{)}.$
(3.22)

Suppose a weak wave $\beta_{j}$ from the $j$-th family, $j<i$, crosses
${N^{\downarrow}_{\pm}}$ from the right. Then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,+k_{L}^{<}\,|\beta_{j}|+k_{L}^{<}\,\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{M}]=$
$\displaystyle\,\,-k_{M}^{<}\,|\beta_{j}|+(k_{M}+k_{M}^{>})\,\mathcal{O}(|{N^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{R}]=$ $\displaystyle\,\,0,$ $\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\big{(}-k_{M}^{<}+k_{L}^{<}+(k_{M}+k_{L}^{<}+k_{M}^{>})\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\big{)}.$
(3.23)

######  Lemma 3.14 (Case 7. $\beta_{j}C\rightarrow
C^{\prime}\beta_{j}^{\prime}$ or
$C\beta_{j}\rightarrow\beta_{j}^{\prime}C^{\prime}$).

Consider a weak wave $\beta_{j}$ belonging to the family $j>i$. The strong
shock $C$ can be either decreasing ${C^{\downarrow}_{\pm}}$ or increasing
${C^{\uparrow}}$. Suppose that $\beta_{j}$ crosses ${C^{\downarrow}_{\pm}}$
from the left. Then we have the estimates

$\displaystyle[V_{L}]=$
$\displaystyle\,\,-k_{L}^{>}\,|\beta_{j}|+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,+k_{R}^{>}\,|\beta_{j}|+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\big{(}-k_{L}^{>}+k_{R}^{>}+(k_{L}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\big{)}.$
(3.24)

Suppose a weak wave $\beta_{j}$ from the $j$-th family, $j<i$, crosses
${C^{\downarrow}_{\pm}}$ from the right. Then

$\displaystyle[V_{L}]=$
$\displaystyle\,\,+k_{L}^{<}\,|\beta_{j}|+k_{L}^{<}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{M}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}^{<}\,|\beta_{j}|+k_{R}^{>}\,\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\big{(}-k_{R}^{<}+k_{L}^{<}+(k_{L}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\big{)}.$
(3.25)

Suppose that $\beta_{j}$, with $j>i$, crosses ${C^{\uparrow}}$ from the left,
then we have the estimates

$\displaystyle[V_{L}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{M}]=$
$\displaystyle\,\,-k_{M}^{>}\,|\beta_{j}|+k_{M}^{<}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{R}]=$
$\displaystyle\,\,+k_{R}^{>}\,|\beta_{j}|+k_{R}^{>}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\beta_{j}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\big{(}-k_{M}^{>}+k_{R}^{>}+(k_{M}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\uparrow}}|)\big{)}.$
(3.26)

Consider a weak wave $\beta_{j}$ belonging to the family $j<i$ and crossing
${C^{\uparrow}}$ from the right. Then the interactions are such that

$\displaystyle[V_{L}]=$ $\displaystyle\,\,0,$ $\displaystyle[V_{M}]=$
$\displaystyle\,\,+k_{M}^{<}\,|\beta_{j}|+k_{M}^{<}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\beta_{j}|),$
$\displaystyle[V_{R}]=$
$\displaystyle\,\,-k_{R}^{<}\,|\beta_{j}|+k_{R}^{>}\,\mathcal{O}(|{C^{\uparrow}}|\cdot|\beta_{j}|),$
$\displaystyle[W]=$
$\displaystyle\,\,|\beta_{j}|\big{(}-k_{R}^{<}+k_{M}^{<}+(k_{M}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\uparrow}}|)\big{)}.$
(3.27)

###  3.4 Interaction potential

We propose to study the following quadratic interaction potential. It will
involve the usual product of the strength of the waves weighted by the the
positive part of the difference in wave speeds. Weighted functionals were
first introduced and studied by Liu [31] and later by Iguchi and LeFloch [23],
as well as Bianchini and Bressan [10]. As is usually the case, the interaction
potential is defined only for front-tracking approximations, with waves
localized at point $x$, in which case, we may speak of its strength
$\sigma_{x}$ and of its family $i_{x}$,

$\displaystyle Q\big{(}u(t)\big{)}:=\sum_{\begin{subarray}{c}\text{waves at
$x<y$ }\\\ \text{approaching }\\\ \text{family $i_{x}\neq i_{y}\neq i$
}\end{subarray}}\big{|}\sigma_{x}\cdot\sigma_{y}\big{|}+\sum_{\begin{subarray}{c}\text{weak
approaching waves $x<y$}\\\ \text{$i_{x}=i$ and/or}\\\
\text{$i_{y}=i$}\end{subarray}}|v_{x}-v_{y}|^{+}\cdot\big{|}\sigma_{x}\cdot\sigma_{y}\big{|}.$
(3.28)

The second term includes potential interactions involving both weak and strong
i-waves.

This section will require a measure of the strength of the perturbation as a
function of time, but this is not immediately an obvious quantity to define
since the strong waves travel in time. We define the strength of the
perturbation at some time $t$ to be

$\epsilon(t):=\sum_{\begin{subarray}{c}\text{weak waves $\beta$}\\\ \text{in
$u(\cdot,t)$}\end{subarray}}|\beta|.$ (3.29)

Given the position of the waves $x^{h}(t),y^{h}(t)$ at some time $t$, we
define the unperturbed waves as

$\bar{u}(x,t)=\begin{cases}u_{*},&\text{ $x<x^{h}(t)$,}\\\
\Phi^{\flat}_{i0}(u_{*}),&\text{ $x^{h}(t)\leq x<y^{h}(t)$,}\\\
u_{*}^{n},&\text{ $y^{h}(t)\leq x$.}\end{cases}$

It is tempting to state that the strength (3.29) is equivalent to

$\epsilon(t)=\operatorname{TV}\big{(}u(\cdot,t)-\bar{u}(\cdot,t)\big{)},$

but if $u(\cdot,t)-\bar{u}(\cdot,t)$ takes non-zero values at $x^{h}(t)$ or
$y^{h}(t)$, then the total variation of the difference may include the
strength of the small perturbation of the strong waves away from the
classical-nonclassical three state pattern. Throughout the analysis below, we
will study only the strength of the weak waves. If it is proven that
$\epsilon(t)$ remains small for the weak waves, then the strong waves will
remain small perturbations of the strong waves $\bar{u}$.

First, we will study here interactions involving waves from the $i$-th family.
We distinguish between several cases of interactions. For the functional $Q$,
we extend the decomposition (3.28) to also treat the change in $Q$. In fact,
in certain cases $[Q]$ will not be negative but $[W]$ will contribute negative
first order terms to control $[Q]$, and so it is important to accurately
measure the change in $Q$. We will introduce a decomposition of the terms
appearing in the variation of $Q$ during interactions. The analysis will study
each term separately before accounting for the total change.

If an interaction occurs among weak waves of families different than $i$, the
concave-convex family, then the change in $Q$ is quadratic in the strength of
the waves interacting and the usual argument of Glimm applies and shows that
for some small total variation $\epsilon$ and a large constant $K$, then
$W+KQ$ is decreasing. Henceforth, we will focus exclusively on the second sum
in the interaction potential $Q$.

If an interaction occurs which involves at least one wave from the $i$-th
family, then the change in $Q$ involves (1) quadratic terms in the strength of
the interacting waves weighted by their difference in wave speeds and (2), the
sum of the changes in terms $|v_{x}-v_{y}|^{+}\sigma_{x}\sigma_{y}$ where one
of them was involved in the interaction but the other wave was not. More
concretely, consider two incoming waves $C_{\ell},C_{r}$, respectively the
left and right incoming waves, which generate
$C_{\ell}^{\prime},C_{r}^{\prime}$, respectively the left and right outgoing
waves. Then, abusing our notation, we write

$\displaystyle\big{[}Q(u(\cdot,t))\big{]}=[Q_{1}]+[Q_{2}],$

where

$\displaystyle[Q_{1}]=\,\,\big{|}v(C_{\ell}^{\prime})-v(C_{r}^{\prime})\big{|}^{+}|C_{\ell}^{\prime}||C_{r}^{\prime}|-\big{|}v(C_{\ell})-v(C_{r})\big{|}^{+}|C_{\ell}||C_{r}|,$

and

$\displaystyle[Q_{2}]=$ $\displaystyle\,\,\sum_{\begin{subarray}{c}\text{weak
$\beta_{j}$ at $y$ }\\\ j\geq i\text{ and
}y<x\end{subarray}}\Big{\\{}\big{|}v(\beta_{j})-v(C_{\ell}^{\prime})\big{|}^{+}|C_{\ell}^{\prime}||\beta_{j}|+\big{|}v(\beta_{j})-v(C_{r}^{\prime})\big{|}^{+}|C_{r}^{\prime}||\beta_{j}|$
$\displaystyle\,\phantom{\sum_{\begin{subarray}{c}\text{weak $\beta_{j}$ at
$y$ }\\\ j\geq i\text{ and
}y<x\end{subarray}}}-\big{|}v(\beta_{j})-v(C_{\ell})\big{|}^{+}|C_{\ell}||\beta_{j}|-\big{|}v(\beta_{j})-v(C_{r})\big{|}^{+}|C_{r}||\beta_{j}|\Big{\\}}$
$\displaystyle\,\,+\sum_{\begin{subarray}{c}\text{weak $\beta_{j}$ at $y$ }\\\
j\leq i\text{ and
}x<y\end{subarray}}\Big{\\{}\big{|}v(C_{\ell}^{\prime})-v(\beta_{j})\big{|}^{+}|C_{\ell}^{\prime}||\beta_{j}|+\big{|}v(C_{r}^{\prime})-v(\beta_{j})\big{|}^{+}|C_{r}^{\prime}||\beta_{j}|$
$\displaystyle\,\phantom{\sum_{\begin{subarray}{c}\text{weak $\beta_{j}$ at
$y$ }\\\ j\geq i\text{ and
}y<x\end{subarray}}}-\big{|}v(C_{\ell})-v(\beta_{j})\big{|}^{+}|C_{\ell}||\beta_{j}|-\big{|}v(C_{r})-v(\beta_{j})\big{|}^{+}|C_{r}||\beta_{j}|\Big{\\}}.$

The analysis of $[Q]$ will demonstrate that $[Q_{1}]$ is always negative and
that if the perturbation of the solution $u$ at some time $t$ away from the
unperturbed state $\bar{u}$ satisfies an a priori bound with respect to the
strength of the strong waves, then the potential increase in $[Q_{2}]$ can be
compensated by $[Q_{1}]$ and result in the sum $[Q]$ being negative by a known
negative quantity.

Among the two variations above, there are generalities that can be said about
$[Q_{2}]$. In most cases, the variation in $Q_{2}$ will be caused by the
interaction of two incoming waves and two outgoing waves, and for the sake of
argument, we will consider only weak waves $\beta$ to the left of the
interaction. Then for every such weak wave $\beta$, there is a contribution to
$[Q_{2}]$ of the form

$\displaystyle[Q_{2}]_{\beta}:=$
$\displaystyle\,|\beta|\Big{(}\big{|}v(\beta)-v(C_{\ell}^{\prime})\big{|}^{+}|C_{\ell}^{\prime}|+\big{|}v(\beta)-v(C_{r}^{\prime})\big{|}^{+}|C_{r}^{\prime}|$
$\displaystyle\,\phantom{|\beta|\Big{(}}-\big{|}v(\beta)-v(C_{\ell})\big{|}^{+}|C_{\ell}|-\big{|}v(\beta)-v(C_{r})\big{|}^{+}|C_{r}|\Big{)}.$

As a function of $v(\beta)$, this expression is continuous and piecewise
linear with four points of discontinuity in the derivative. Clearly, if
$v(\beta)<m_{v}:=\min v(C)$ then all the weights vanish and
$[Q_{2}]_{\beta}=0$. We claim that over the unbounded domain $[M_{v},\infty[$,
with $M_{v}=\max v(C)$, the variation $[Q_{2}]_{\beta}$ is decreasing and
therefore the maximum value is taken at $v(\beta)=M_{v}$. This follows from
the fact that during interactions between waves of the family $i$,

$|C_{\ell}|+|C_{r}|-|C_{\ell}^{\prime}|-|C_{r}^{\prime}|\leq 0,$

with equality only when all waves are shocks. Each weight can be written as

$\big{|}v(\beta)-v(C)\big{|}^{+}=|v(\beta)-M_{v}|+|M_{v}-v(C)|$

and therefore we have

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle\,|\beta||v(\beta)-M_{v}|\Big{(}|C_{\ell}^{\prime}|+|C_{r}^{\prime}|-|C_{\ell}|-|C_{r}|\Big{)}$
$\displaystyle\,+|\beta|\Big{(}\Delta v_{1}|C_{\ell}^{\prime}|+\Delta
v_{2}|C_{r}^{\prime}|-\Delta v_{3}|C_{\ell}|-\Delta v_{4}|C_{r}|\Big{)}.$

Each of the new weights $\Delta v_{i}$ is constant with respect to $v(\beta)$
but the first term is negative. Hence the maximum of $[Q_{2}]_{\beta}$ must
occur at one of the three largest speeds where the derivative is
discontinuous. When the weak wave $\beta$ is approaching from the right,
similar arguments show that the maximum value will occur when $v(\beta)$
coincides with one of the the three lowest speeds in the interaction. Below,
when studying $[Q_{2}]_{\beta}$ it will be easy to infer the speed $v(\beta)$
for which the value will be largest.

######  Lemma 3.15 (Case 1. $\alpha_{i}{C^{\downarrow}_{\pm}}\rightarrow
N{C^{\uparrow}}$ or ${C^{\downarrow}_{\pm}}\alpha_{i}\rightarrow
N{C^{\uparrow}}$).

Consider at some time $t$ a weak wave $\alpha_{i}$ interacting with
${C^{\downarrow}_{\pm}}$ to generate a nonclassical and a classical shock pair
${{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$. If the
perturbation $\epsilon(t)$ satisfies an a priori bound

$\epsilon(t)\leq\mathcal{O}(|{C^{\downarrow}_{\pm}}|),$ (3.30)

then

$[Q]\leq-\big{|}\mathcal{O}(|\alpha_{i}|\cdot|{C^{\downarrow}_{\pm}}|^{2})\big{|}.$

###### Proof.

There are three types of interactions that occur, RC-$3$, CC-3, and CR-$4$.
All wave speeds, except $v(\alpha_{i})$, are close. In all these cases,

$\displaystyle[Q_{1}]=-\big{|}v(\alpha_{i})-v({C^{\downarrow}_{\pm}})\big{|}\,|\alpha_{i}|\cdot|{C^{\downarrow}_{\pm}}|=-\big{|}\mathcal{O}(|\alpha_{i}|\cdot|{C^{\downarrow}_{\pm}}|^{2})\big{|}.$
(3.31)

Consider first the case where $\alpha_{i}$ is a rarefaction and the shock
strengths satisfy

$|{N^{\downarrow}_{\pm}}^{\prime}|+|{C^{\downarrow}}^{\prime}|=|{C^{\downarrow}}|-|\alpha_{i}|+\mathcal{O}(|{C^{\downarrow}}|\cdot|\alpha_{i}|),$

which implies

$|{N^{\downarrow}_{\pm}}^{\prime}|+|{C^{\downarrow}}^{\prime}|<|{C^{\downarrow}}|+\mathcal{O}(|{C^{\downarrow}}|\cdot|\alpha_{i}|).$

An inspection of the normalized wave speeds associated to the states $u_{l}$
and $u_{r}$ shows that

$v({N^{\downarrow}_{\pm}}^{\prime})<\min\big{[}v({C^{\downarrow}_{\pm}}),v({C^{\uparrow}}^{\prime})\big{]}<v(\alpha_{i}),$

and that
$v({N^{\downarrow}_{\pm}}^{\prime})-v({C^{\downarrow}_{\pm}})=\mathcal{O}(|\alpha_{i}|)$.
We will now analyze the interaction term $[Q_{2}]$ in (3.28). Suppose there is
a weak wave $\beta$ to the left of the interaction, then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\beta)-v({C^{\uparrow}}^{\prime})\big{|}^{+}|{C^{\uparrow}}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v(\beta)-v(\alpha_{i})\big{|}^{+}|\alpha_{i}|-\big{|}v(\beta)-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{)}.$

If $v(\beta)<v({N^{\downarrow}_{\pm}}^{\prime})$, then $[Q_{2}]_{\beta}<0$.
The largest value of this function of $v(\beta)$ occurs when
$v({N^{\downarrow}_{\pm}}^{\prime})<v({C^{\uparrow}}^{\prime})<v(\beta)=v({C^{\downarrow}_{\pm}})<v(\alpha_{i})$,
but in that case

$\displaystyle[Q_{2}]_{\beta}\leq$
$\displaystyle|\beta|\Big{(}\big{|}v({C^{\downarrow}_{\pm}})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v({C^{\downarrow}_{\pm}})-v({C^{\uparrow}}^{\prime})\big{|}|{C^{\uparrow}}^{\prime}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\alpha_{i})\big{|}|\alpha_{i}|\Big{)}$
$\displaystyle\leq$
$\displaystyle|\beta|\Big{(}\mathcal{O}(|\alpha_{i}|)|{N^{\downarrow}_{\pm}}^{\prime}|+\mathcal{O}(|\alpha_{i}|)|{C^{\uparrow}}^{\prime}|\Big{)}\leq\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|).$
(3.32)

Still in the case RC-$3$, suppose now there is a weak wave $\beta$ to the
right of the interaction, then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v(\beta)\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v({C^{\uparrow}}^{\prime})-v(\beta)\big{|}^{+}|{C^{\uparrow}}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v(\alpha_{i})-v(\beta)\big{|}^{+}|\alpha_{i}|-\big{|}v|({C^{\downarrow}_{\pm}})-v(\beta)\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{)}.$

The variation is largest when
$v({N^{\downarrow}_{\pm}}^{\prime})<v({C^{\downarrow}_{\pm}})=v(\beta)<v({C^{\uparrow}}^{\prime})<v(\alpha_{i})$,
in which case

$\displaystyle[Q_{2}]_{\beta}\leq$
$\displaystyle|\beta|\Big{(}\big{|}v({C^{\uparrow}}^{\prime})-v({C^{\downarrow}_{\pm}})\big{|}|{C^{\uparrow}}^{\prime}|-\big{|}v(\alpha_{i})-v({C^{\downarrow}_{\pm}})\big{|}|\alpha_{i}|\Big{)}$
$\displaystyle\leq$
$\displaystyle|\beta|\Big{(}\mathcal{O}(|\alpha_{i}|)|{C^{\downarrow}_{\pm}}|+\mathcal{O}(|{C^{\downarrow}_{\pm}}|)|\alpha_{i}|\Big{)}=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|).$
(3.33)

Combining (3.31) with either (3.32) or (3.33), we find

$\displaystyle[Q]=[Q_{1}]+\sum_{\beta}[Q_{2}]_{\beta}\leq-\Big{|}\mathcal{O}(|\alpha_{i}|\cdot|{C^{\downarrow}_{\pm}}|^{2})\Big{|}+\sum_{\beta}\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|),$

which is negative if a bound of the form
$\epsilon(t)<\mathcal{O}(|{C^{\downarrow}_{\pm}}|)$ is satisfied by the
perturbation. In the case CC-$3$, the weak wave $\alpha_{i}$ can come either
from the left or the right of ${C^{\downarrow}_{\pm}}$. For the sake of
brevity, we will consider only left incoming weak waves $\alpha_{i}$. The
speeds satisfy

$v({C^{\downarrow}_{\pm}})<v({N^{\downarrow}_{\pm}}^{\prime})<v({C^{\uparrow}}^{\prime})<v(\alpha_{i}),$
(3.34)

and for the strengths we have

$|{{N^{\downarrow}_{\pm}}}^{\prime}|+|{{C^{\uparrow}_{-}}}^{\prime}|=|{C^{\downarrow}_{\pm}}|-\operatorname{sign}(\alpha_{i})|\alpha_{i}|+\mathcal{O}(|{C^{\downarrow}_{\pm}}|\cdot|\alpha_{i}|).$

Suppose that there is a weak wave $\beta$ to the left of the interaction, then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}(v(\beta)-v({C^{\uparrow}}^{\prime})\big{|}^{+}|{C^{\uparrow}}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v(\beta)-v(\alpha_{i})\big{|}^{+}|\alpha_{i}|-\big{|}v(\beta)-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{)}.$

The worst case occurs when $v(\beta)=v(\alpha_{i})$ and the change is

$\displaystyle[Q_{2}]_{\beta}\leq$
$\displaystyle|\beta|\Big{(}\big{|}v(\alpha_{i})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\alpha_{i})-v({C^{\uparrow}}^{\prime})\big{|}^{+}|{C^{\uparrow}}^{\prime}|-\big{|}v(\alpha_{i})-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{)}.$
(3.35)

Suppose now that there is a weak wave $\beta$ to the right of the interaction,
then the largest value is found when $v(\beta)=v({C^{\downarrow}_{\pm}})$.

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v({C^{\uparrow}}^{\prime})-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\uparrow}}^{\prime}|-\big{|}v(\alpha_{i})-v({C^{\downarrow}_{\pm}})\big{|}^{+}|\alpha_{i}|\Big{)}$
$\displaystyle\leq$
$\displaystyle|\beta|\Big{(}\mathcal{O}(|\alpha_{i}|)|{N^{\downarrow}_{\pm}}^{\prime}|+\mathcal{O}(|\alpha_{i}|)|{C^{\uparrow}}^{\prime}|-\mathcal{O}(|{C^{\downarrow}_{\pm}}|)|\alpha_{i}|\Big{)}=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|).$

As in the case RC-$3$, imposing an a priori constraint
$\sum_{\beta}|\beta|\leq\mathcal{O}(|{C^{\downarrow}_{\pm}}|)$ implies that
$[Q]$ is negative. In the case CR-$4$, the speeds are ordered as

$v(\alpha_{i})<v({N^{\downarrow}_{\pm}}^{\prime})<\min\Big{[}v({C^{\downarrow}_{\pm}}),v({C^{\uparrow}}^{\prime})\Big{]}.$
(3.36)

Suppose there is a weak wave $\beta$ to the left of the interaction, then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\beta)-v({C^{\uparrow}}^{\prime})\big{|}^{+}|{C^{\uparrow}}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v(\beta)-v(\alpha_{i})\big{|}^{+}|\alpha_{i}|-\big{|}v(\beta)-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{)}.$

The worst case corresponds to
$v(\beta)=v({C^{\downarrow}_{\pm}})>v({C^{\uparrow}}^{\prime})$. Then, we
estimate

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({C^{\downarrow}_{\pm}})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v({C^{\downarrow}_{\pm}})-v({C^{\uparrow}}^{\prime})\big{|}^{+}|{C^{\uparrow}}^{\prime}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\alpha_{i})\big{|}|\alpha_{i}|\Big{)}$
$\displaystyle\leq$
$\displaystyle|\beta|\Big{(}\mathcal{O}(|\alpha_{i}|)|{N^{\downarrow}_{\pm}}^{\prime}|+\mathcal{O}(|\alpha_{i}|)|{C^{\uparrow}}^{\prime}|-\mathcal{O}(|{C^{\downarrow}_{\pm}}|)|\alpha_{i}|\Big{)}=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|).$

Taking into account (3.4) and an a priori bond
$|\beta|\leq\mathcal{O}(|{C^{\downarrow}_{\pm}}|)$, we deduce

$[Q_{2}]_{\beta}\leq|\beta|\big{(}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)|\alpha_{i}|\big{)}\leq\mathcal{O}(|\alpha_{i}||{C^{\downarrow}_{\pm}}|^{2}).$
(3.37)

Suppose now there is a weak wave $\beta$ to the right of the interaction, then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v(\beta)\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v({C^{\uparrow}}^{\prime})-v(\beta)\big{|}^{+}|{C^{\uparrow}}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v(\alpha_{i})-v(\beta)\big{|}^{+}|\alpha_{i}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\beta)\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{)}.$

The two extreme cases correspond to $v(\beta)=v({C^{\downarrow}_{\pm}})$ and
$v(\beta)=v(\alpha_{i})$, but we consider here only
$v(\alpha_{i})<v({N^{\downarrow}_{\pm}}^{\prime})<v(\beta)=v({C^{\downarrow}_{\pm}})<v({C^{\uparrow}}^{\prime})$.
If this happens,

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v({C^{\downarrow}_{\pm}})\big{|}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v({C^{\uparrow}}^{\prime})-v({C^{\downarrow}_{\pm}})\big{|}|{C^{\uparrow}}^{\prime}|-\big{|}v(\alpha_{i})-v({C^{\downarrow}_{\pm}})\big{|}|\alpha_{i}|\Big{)}$
$\displaystyle\leq|\beta|\Big{(}\mathcal{O}(|\alpha_{i}|)|{N^{\downarrow}_{\pm}}^{\prime}|+\mathcal{O}(|\alpha_{i}|)|{C^{\uparrow}}^{\prime}|-\mathcal{O}(|{C^{\downarrow}_{\pm}}|)|\alpha_{i}|\Big{)}=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|).$

Again, combining (3.31) with an a priori bond
$\epsilon(t)\leq\mathcal{O}(|{C^{\downarrow}_{\pm}}|)$, we deduce that
$[Q]\leq 0$. ∎

######  Lemma 3.16 (Case 2.
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{{C^{\downarrow}_{\pm}}}^{\prime}$).

If during the strong interaction
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{{C^{\downarrow}_{\pm}}}^{\prime}$
at some time $t$, a bound of the form

$\epsilon(t)\leq\mathcal{O}(|{C^{\uparrow}}|),$ (3.38)

is satisfied by the perturbation, then

$[Q]\leq-\big{|}\mathcal{O}(\big{|}v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})\big{|}\,|{N^{\downarrow}_{\pm}}|\cdot|{C^{\uparrow}}|)\big{|}.$

###### Proof.

Consider the strong
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{{C^{\downarrow}_{\pm}}}^{\prime}$
interaction. We have

$\displaystyle|{C^{\downarrow}_{\pm}}^{\prime}|=|{N^{\downarrow}_{\pm}}|+|{C^{\uparrow}}|+\mathcal{O}(|{N^{\downarrow}_{\pm}}||{C^{\uparrow}}|),$
$\displaystyle
v({C^{\uparrow}})<v({N^{\downarrow}_{\pm}})<v({C^{\downarrow}_{\pm}}^{\prime}).$

All wave speeds are close and in fact, for example
$|v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})|=\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|-|{C^{\uparrow}}|-|{C^{\downarrow}_{\pm}}^{\prime}|\big{)}$,
is of the size of the perturbation $\epsilon(t)$. Clearly,

$\displaystyle[Q_{1}]=$
$\displaystyle-\,\big{|}v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})\big{|}\,|{N^{\downarrow}_{\pm}}|\cdot|{C^{\uparrow}}|.$
(3.39)

Consider a weak wave $\beta$ located to the left of the interaction. Then for
all speeds $v(\beta)$,

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({C^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v(\beta)-v({N^{\downarrow}_{\pm}})\big{|}^{+}|{N^{\downarrow}_{\pm}}|-\big{|}v(\beta)-v({C^{\uparrow}})\big{|}^{+}|{C^{\uparrow}}|\Big{)}\leq
0.$

Consider a weak wave $\beta$ located to the right of the interaction. The
variation is

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({C^{\downarrow}_{\pm}}^{\prime})-v(\beta)\big{|}^{+}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v({N^{\downarrow}_{\pm}})-v(\beta)\big{|}^{+}|{N^{\downarrow}_{\pm}}|-\big{|}v({C^{\uparrow}})-v(\beta)\big{|}^{+}|{C^{\uparrow}}|\Big{)},$

The worst case is $v(\beta)=v({N^{\downarrow}_{\pm}})$, then we have the
obvious overestimate

$\displaystyle[Q_{2}]_{\beta}\leq$
$\displaystyle|\beta|\big{|}v({C^{\downarrow}_{\pm}}^{\prime})-v({N^{\downarrow}_{\pm}})\big{|}|{C^{\downarrow}_{\pm}}^{\prime}|=\mathcal{O}(|\beta|\big{|}v({C^{\downarrow}_{\pm}}^{\prime})-v({N^{\downarrow}_{\pm}})\big{|}|{N^{\downarrow}_{\pm}}|).$

And so, if $\epsilon(t)\leq\mathcal{O}(|{C^{\uparrow}}|)$ then the result is
demonstrated. ∎

######  Lemma 3.17 (Case 3.
$\alpha_{i}{C^{\downarrow}_{\pm}}\rightarrow{{C^{\downarrow}_{\pm}}}^{\prime}$
or
${C^{\downarrow}_{\pm}}\alpha_{i}\rightarrow{{C^{\downarrow}_{\pm}}}^{\prime}$).

The interaction of a weak wave $\alpha_{i}$ of the $i$-th family with a strong
wave of the same family ${C^{\downarrow}_{\pm}}$ splits into four distinct
cases, depending on whether $\alpha_{i}$ is a shock or a rarefaction, and
whether the wave approaches from the left or the right. In all of these cases,
if a bound of the form

$\epsilon(t)\leq\mathcal{O}(|{C^{\downarrow}}|)$ (3.40)

is satisfied then

$[Q]\leq-\big{|}\mathcal{O}(|\alpha_{i}|\cdot|{C^{\downarrow}_{\pm}}|^{2})\big{|}.$

###### Proof.

We remark that the ordering of the velocities are all distinct and include all
cases with $v(\alpha_{i})$ being either the slowest or fastest. In all of
these cases only one wave is outgoing, so

$\displaystyle[Q_{1}]=$
$\displaystyle-\,\big{|}v(\alpha_{i})-v({C^{\downarrow}_{\pm}})\big{|}^{+}\,|\alpha_{i}|\cdot|{C^{\downarrow}_{\pm}}|=-\mathcal{O}(|\alpha_{i}|\cdot|{C^{\downarrow}_{\pm}}|^{2}).$
(3.41)

Consider the first interaction
$\alpha_{i}{C^{\downarrow}}$-${{C^{\downarrow}}}^{\prime}$ with $\alpha_{i}$ a
shock. We have the identities

$\displaystyle|{C^{\downarrow}}^{\prime}|=|\alpha_{i}|+|{C^{\downarrow}}|+\mathcal{O}(|\alpha_{i}||{C^{\downarrow}}|),$
$\displaystyle
v({C^{\downarrow}})<v({C^{\downarrow}}^{\prime})<v(\alpha_{i}).$

Suppose a weak wave $\beta$ is located to the left of the interaction, then
the largest value of

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({C^{\downarrow}}^{\prime})\big{|}^{+}|{C^{\downarrow}}^{\prime}|-\big{|}v(\beta)-v({C^{\downarrow}})\big{|}^{+}|{C^{\downarrow}}|-\big{|}v(\beta)-v(\alpha_{i})\big{|}^{+}|\alpha_{i}|\Big{)},$

occurs when $v(\beta)=v(\alpha_{i})$, with which we find

$\displaystyle[Q_{2}]_{\beta}\leq$
$\displaystyle|\beta|\Big{(}\big{|}v(\alpha_{i})-v({C^{\downarrow}}^{\prime})\big{|}|{C^{\downarrow}}^{\prime}|-\big{|}v(\alpha_{i})-v({C^{\downarrow}})\big{|}|{C^{\downarrow}}|\Big{)}$
$\displaystyle=$
$\displaystyle|\beta|\Big{(}\big{|}v(\alpha_{i})-v({C^{\downarrow}}^{\prime})\big{|}\big{(}|{C^{\downarrow}}^{\prime}|-|{C^{\downarrow}}|\big{)}-\big{|}v({C^{\downarrow}}^{\prime})-v({C^{\downarrow}})\big{|}|{C^{\downarrow}}|\Big{)}=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}}|).$

Suppose $\beta$ is to the right of the interaction.

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({C^{\downarrow}}^{\prime})-v(\beta)\big{|}^{+}|{C^{\downarrow}}^{\prime}|-\big{|}v({C^{\downarrow}})-v(\beta)\big{|}^{+}|{C^{\downarrow}}|-\big{|}v(\alpha_{i})-v(\beta)\big{|}^{+}|\alpha_{i}|\Big{)}$

The worst case is $v(\beta)=v({C^{\downarrow}})$, we can compute

$\displaystyle[Q_{2}]_{\beta}\leq|\beta|\Big{(}\big{|}v({C^{\downarrow}}^{\prime})-v({C^{\downarrow}})\big{|}|{C^{\downarrow}}^{\prime}|-\big{|}v(\alpha_{i})-v({C^{\downarrow}})\big{|}|\alpha_{i}|\Big{)}=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}}|).$

Again $[Q]$ can be shown to be decreasing if an a priori bound
$\sum_{\beta}|\beta|\leq\mathcal{O}(|{C^{\downarrow}}|)$ is satisfied.

Consider the second case
$\alpha_{i}{C^{\downarrow}}$-${{C^{\downarrow}}}^{\prime}$ with $\alpha_{i}$ a
rarefaction. We observed that the states of interest satisfy

$\displaystyle|{{C^{\downarrow}}}|-|\alpha_{i}|=|{C^{\downarrow}}^{\prime}|+\mathcal{O}(|\alpha_{i}||{C^{\downarrow}}|).$
$\displaystyle
v({C^{\downarrow}}^{\prime})<v({C^{\downarrow}})<v(\alpha_{i}).$

Suppose $\beta$ is a weak wave to the left of the interaction, then

$\displaystyle[Q_{2}]_{\beta}\leq$
$\displaystyle|\beta|\big{|}v({C^{\downarrow}})-v({C^{\downarrow}}^{\prime})\big{|}^{+}|{C^{\downarrow}}^{\prime}|=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}}|).$

This suffices to prove the result in that case.

The third interaction is
${C^{\downarrow}_{\pm}}\alpha_{i}$-${{C^{\downarrow}_{\pm}}}^{\prime}$ where
$\alpha_{i}$ is a shock. The following two identities hold

$\displaystyle|{C^{\downarrow}}^{\prime}|=|{C^{\downarrow}_{\pm}}|+|\alpha_{i}|+\mathcal{O}(|\alpha_{i}||{C^{\downarrow}}|),$
$\displaystyle
v(\alpha_{i})<v({C^{\downarrow}_{\pm}})<v({C^{\downarrow}_{\pm}}^{\prime}).$

Suppose $\beta$ is to the left of the interaction, then the variation in
$Q_{2}$ is

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({C^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v(\beta)-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\downarrow}_{\pm}}|-\big{|}v(\beta)-v(\alpha_{i})\big{|}^{+}|\alpha_{i}|\Big{)}.$

For $v(\beta)$,
$|v(\beta)-v({C^{\downarrow}_{\pm}}^{\prime})|<|v(\beta)-v({C^{\downarrow}_{\pm}})|$
and $|{C^{\downarrow}_{\pm}}^{\prime}|<|{C^{\downarrow}_{\pm}}|$ hence this
quantity is always negative. Suppose now that $\beta$ is to the right of the
interaction, then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({C^{\downarrow}_{\pm}}^{\prime})-v(\beta)\big{|}^{+}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\beta)\big{|}^{+}|{C^{\downarrow}_{\pm}}|-\big{|}v(\alpha_{i})-v(\beta)\big{|}^{+}|\alpha_{i}|\Big{)},$

and the worst case is $v(\beta)=v({C^{\downarrow}_{\pm}})$, for which we find

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\big{|}v({C^{\downarrow}_{\pm}}^{\prime})-v({C^{\downarrow}_{\pm}})\big{|}|{C^{\downarrow}_{\pm}}^{\prime}|=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|).$

In the third case, this shows that an a priori bound
$\sum_{\beta}|\beta|\leq\mathcal{O}(|{C^{\downarrow}_{\pm}}|)$ is sufficient
to show that $[Q]$ is decreasing.

The fourth and last interaction is
${C^{\downarrow}_{\pm}}{R^{\downarrow}_{-}}$-${{C^{\downarrow}_{\pm}}}^{\prime}$
where $\alpha_{i}$ is a rarefaction. The states of interest imply
$|{C^{\downarrow}_{\pm}}^{\prime}|<|{C^{\downarrow}_{\pm}}|$ and that the
speeds are ordered

$v(\alpha_{i})<v({C^{\downarrow}_{\pm}}^{\prime})<v({C^{\downarrow}_{\pm}}).$

Recall also that only one wave is outgoing, so we trivially have
$[Q_{1}]=-|v({C^{\downarrow}_{\pm}})-v(\alpha_{i})||\alpha_{i}|{C^{\downarrow}_{\pm}}|$.
Assume now that a weak wave $\beta$ is located to the left of the interaction,
then the least favorable case corresponds to the situation where
$v(\beta)=v({C^{\downarrow}_{\pm}})$, and

$\displaystyle[Q_{2}]_{\beta}\leq$
$\displaystyle|\beta|\Big{(}\big{|}v({C^{\downarrow}_{\pm}})-v({C^{\downarrow}_{\pm}}^{\prime})\big{|}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\alpha_{i})\big{|}|\alpha_{i}|\Big{)}=\mathcal{O}(|\beta||\alpha_{i}||{C^{\downarrow}_{\pm}}|).$

If $\beta$ is to the right, then the worst case is $v(\beta)=v(\alpha_{i})$
and we have

$[Q_{2}]_{\beta}\leq|\beta|\Big{(}\big{|}v({C^{\downarrow}_{\pm}}^{\prime})-v(\alpha_{i})\big{|}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\alpha_{i})\big{|}|{C^{\downarrow}_{\pm}}|\Big{)}\leq
0.$

In conclusion, four a priori bounds
$\epsilon(t)\leq\mathcal{O}(|{C^{\downarrow}_{\pm}}|)$ imply that $Q$ is
decreasing by an amount
$\mathcal{O}(|\alpha_{i}||{C^{\downarrow}_{\pm}}|^{2})$. ∎

######  Lemma 3.18 (Case 4.
$\alpha_{i}{N^{\downarrow}_{\pm}}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}\alpha_{i}^{\prime}$).

Consider at some time $t$ an interaction involving a weak wave $\alpha_{i}$
approaching ${N^{\downarrow}_{\pm}}$ from the left with outgoing waves
${N^{\downarrow}_{\pm}}^{\prime}$ and $\alpha_{i}^{\prime}$. If the
perturbation satisfies an a priori bound of the form

$\epsilon(t)\leq\mathcal{O}(|{N^{\downarrow}_{\pm}}|),$ (3.42)

then one has

$[Q]\leq-\big{|}\mathcal{O}(|\alpha_{i}||{N^{\downarrow}_{\pm}}|^{2})\big{|}.$

###### Proof.

Whether $\alpha_{i}$ is a shock or a rarefaction, we find

$\displaystyle[Q_{1}]=-\,\big{|}v(\alpha_{i})-v({N^{\downarrow}_{\pm}})\big{|}^{+}\,|\alpha_{i}|\cdot|{N^{\downarrow}_{\pm}}|=-\mathcal{O}(|\alpha_{i}||{N^{\downarrow}_{\pm}}|^{2}).$
(3.43)

When $\alpha_{i}$ is a rarefaction, then we have

$|{N^{\downarrow}_{\pm}}^{\prime}|<|{N^{\downarrow}_{\pm}}|,\qquad|\alpha_{i}^{\prime}|<|\alpha_{i}|,$

and the inequalities on the speeds

$\displaystyle
v({N^{\downarrow}_{\pm}}^{\prime})<v({N^{\downarrow}_{\pm}})<\min\big{\\{}v(\alpha_{i}),v(\alpha_{i}^{\prime})\big{\\}}.$

Using the observations above, we verify that if a weak wave $\beta$ is located
to the left of the interaction then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\beta)-v(\alpha_{i}^{\prime})\big{|}^{+}|\alpha_{i}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v(\beta)-v({N^{\downarrow}_{\pm}})\big{|}^{+}|{N^{\downarrow}_{\pm}}|-\big{|}v(\beta)-v(\alpha_{i})\big{|}^{+}|\alpha_{i}|\Big{)}.$

The largest value of $[Q_{2}]$ occurs when
$v(\beta)=v({N^{\downarrow}_{\pm}})$, in which case

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\big{|}v({N^{\downarrow}_{\pm}})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}|{N^{\downarrow}_{\pm}}^{\prime}|=\mathcal{O}(|\beta||\alpha_{i}||{N^{\downarrow}_{\pm}}|).$

When a weak wave $\beta$ is located to the right of the interaction, then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v(\beta)\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\alpha_{i}^{\prime})-v(\beta)\big{|}^{+}|\alpha_{i}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v({N^{\downarrow}_{\pm}})-v(\beta)\big{|}^{+}|{N^{\downarrow}_{\pm}}|-\big{|}v(\alpha_{i})-v(\beta)\big{|}^{+}|\alpha_{i}|\Big{)}.$

We find the largest value by taking $v(\beta)=v(\alpha_{i})$

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\big{|}v(\alpha_{i}^{\prime})-v(\alpha_{i})\big{|}|\alpha_{i}^{\prime}|=\mathcal{O}(|\beta||\alpha_{i}||{N^{\downarrow}_{\pm}}|).$

Once again, $[Q]$ is decreasing if all weak waves satisfy a condition
$\sum|\beta|\leq\mathcal{O}(|{N^{\downarrow}_{\pm}}|)$.

We now consider the interaction
$\alpha_{i}{N^{\downarrow}_{\pm}}\to{N^{\downarrow}_{\pm}}^{\prime}\alpha_{i}^{\prime}$
where $\alpha_{i}$ is a shock. We saw in the proof of Lemma 3.11 that

$|{N^{\downarrow}_{\pm}}|<|{N^{\downarrow}_{\pm}}^{\prime}|,\qquad|\alpha_{i}^{\prime}|<|\alpha_{i}|.$

For this interaction, we have

$v({N^{\downarrow}_{\pm}})<v({N^{\downarrow}_{\pm}}^{\prime})<\min\big{\\{}v(\alpha_{i}),v(\alpha_{i}^{\prime})\big{\\}}.$

For the functional $[Q_{2}]$, when there is a weak wave $\beta$ located to the
left of the interaction, we have

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\beta)-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\beta)-v(\alpha_{i}^{\prime})\big{|}^{+}|\alpha_{i}^{\prime}|$
$\displaystyle\phantom{|\beta|\Big{(}}-\big{|}v(\beta)-v({N^{\downarrow}_{\pm}})\big{|}^{+}|{N^{\downarrow}_{\pm}}|-\big{|}v(\beta)-v(\alpha_{i})\big{|}^{+}|\alpha_{i}|\Big{)}.$

The worst case occurs when $v(\alpha_{i}^{\prime})<v(\alpha_{i})=v(\beta)$,
and

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v(\alpha_{i})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\alpha_{i})-v(\alpha_{i}^{\prime})\big{|}|\alpha_{i}^{\prime}|-\big{|}v(\alpha_{i})-v({N^{\downarrow}_{\pm}})\big{|}|{N^{\downarrow}_{\pm}}|\Big{)}$
$\displaystyle=$
$\displaystyle|\beta|\Big{(}\big{|}v(\alpha_{i})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}\big{(}|{N^{\downarrow}_{\pm}}^{\prime}|-|{N^{\downarrow}_{\pm}}|\big{)}-\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v({N^{\downarrow}_{\pm}})\big{|}|{N^{\downarrow}_{\pm}}|+\big{|}v(\alpha_{i})-v(\alpha_{i}^{\prime})\big{|}|\alpha_{i}^{\prime}|\Big{)}$
$\displaystyle=$
$\displaystyle\mathcal{O}(|\beta||\alpha_{i}||{N^{\downarrow}_{\pm}}|).$

In a similar way, it is easy to verify that if there is a weak wave $\beta$
located to the right of the interaction, the least favorable case is
$v(\beta)=v({N^{\downarrow}_{\pm}}^{\prime})<v(\alpha_{i})<v(\alpha_{i}^{\prime})$

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle|\beta|\Big{(}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v(\beta)\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\alpha_{i}^{\prime})-v(\beta)\big{|}^{+}|\alpha_{i}^{\prime}|-\big{|}v({N^{\downarrow}_{\pm}})-v(\beta)\big{|}^{+}|{N^{\downarrow}_{\pm}}|-\big{|}v(\alpha_{i})-v(\beta)\big{|}^{+}|\alpha_{i}|\Big{)}$
$\displaystyle=$
$\displaystyle|\beta|\Big{(}\big{|}v(\alpha_{i}^{\prime})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}|\alpha_{i}^{\prime}|-\big{|}v({N^{\downarrow}_{\pm}})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}|{N^{\downarrow}_{\pm}}|-\big{|}v(\alpha_{i})-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}|\alpha_{i}|\Big{)}$
$\displaystyle=$
$\displaystyle\mathcal{O}(|\beta||\alpha_{i}||{N^{\downarrow}_{\pm}}|).$

∎

######  Lemma 3.19 (Case 5.
$\beta_{j}{C^{\downarrow}_{\pm}}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$
or
${C^{\downarrow}_{\pm}}\beta_{j}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$
).

Consider at some time $t$ an interaction between a weak wave $\beta_{j}$,
$j\neq i$, with a strong classical shock ${C^{\downarrow}_{\pm}}$ that leads
to a splitting ${N^{\downarrow}_{\pm}}^{\prime}{C^{\uparrow}}^{\prime}$. If
the perturbation satisfies an a priori bound

$\epsilon(t)\leq\mathcal{O}(1),$ (3.44)

then

$[Q]=-\big{|}\mathcal{O}\big{(}|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|\big{)}\big{|}.$

###### Proof.

The interaction between the weak wave $\beta_{j}$, $j\neq i$, with
${C^{\downarrow}_{\pm}}$ leads to ${N^{\downarrow}_{\pm}}{C^{\uparrow}}$. For
this interaction,
$|{C^{\downarrow}_{\pm}}|=|{N^{\downarrow}_{\pm}}^{\prime}|+|{C^{\uparrow}}^{\prime}|+\mathcal{O}(|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|)$
and the wave speeds are all close,

$v({C^{\downarrow}_{\pm}})=v({N^{\downarrow}_{\pm}}^{\prime})+\mathcal{O}(|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|)=v({C^{\uparrow}}^{\prime})+\mathcal{O}(|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|).$

First of all, we have

$[Q_{1}]=-\big{|}v(\beta_{j})-v({C^{\downarrow}_{\pm}})\big{|}\cdot|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|=-\Big{|}\mathcal{O}\big{(}|\beta_{j}||{C^{\downarrow}_{\pm}}|\big{)}\Big{|}.$

Let $\gamma$ be a weak wave at $x<y$ interacting with a strong wave. Then

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle\,|\gamma|\cdot\Big{[}\big{|}v(\gamma)-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\gamma)-v({C^{\uparrow}}^{\prime})\big{|}^{+}|{C^{\uparrow}}^{\prime}|-\big{|}v(\gamma)-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{]}$
$\displaystyle=$
$\displaystyle\,|\gamma|\cdot\big{|}v(\gamma)-v({C^{\downarrow}_{\pm}})\big{|}^{+}\Big{[}|{N^{\downarrow}_{\pm}}^{\prime}|+|{C^{\uparrow}}^{\prime}|-|{C^{\downarrow}_{\pm}}|\big{]}+\mathcal{O}(|\gamma||\beta_{j}||{C^{\downarrow}_{\pm}}|)$
$\displaystyle=$
$\displaystyle\,\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|).$

Similarly, if $\gamma$ is a weak wave to the right of the interaction, we have

$\displaystyle[Q_{2}]_{\beta}=$
$\displaystyle\,|\gamma|\cdot\Big{[}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v(\gamma)\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v({C^{\uparrow}}^{\prime})-v(\gamma)\big{|}^{+}|{C^{\uparrow}}^{\prime}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\gamma)\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{]}$
$\displaystyle=$
$\displaystyle\,\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|).$

By imposing an a priori condition on $\sum|\gamma|\leq\mathcal{O}(1)$ we can
conclude that
$[Q]\leq-\big{|}\mathcal{O}(|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|)\big{|}.$
∎

######  Lemma 3.20 (Case 6.
$\beta_{j}{N^{\downarrow}_{\pm}}\rightarrow{N^{\downarrow}_{\pm}}^{\prime}\beta_{j}^{\prime}$
or
${N^{\downarrow}_{\pm}}\beta_{j}\rightarrow\beta_{j}^{\prime}{N^{\downarrow}_{\pm}}^{\prime}$).

Consider at some time $t$ an interaction between a weak wave $\beta_{j}$,
$j\neq i$ with a strong classical shock ${N^{\downarrow}_{\pm}}$ that leaves
of the nonclassical shock unperturbed. If the perturbation satisfies an a
priori bound

$\epsilon(t)\leq\mathcal{O}(1),$ (3.45)

then one has

$[Q]=-\Big{|}\mathcal{O}\big{(}|\beta_{j}|\cdot|{N^{\downarrow}_{\pm}}|\big{)}\Big{|}.$

###### Proof.

The weak wave $\beta_{j}$ from the $j$-th family crosses
${N^{\downarrow}_{\pm}}$ from the left $(j>i)$ or from the right $(j<i)$,
possibly generating weak secondary waves of strength
$\mathcal{O}(|\beta_{j}||{N^{\downarrow}_{\pm}}|)$. We have the identity

$\displaystyle|{N^{\downarrow}_{\pm}}|=|{N^{\downarrow}_{\pm}}^{\prime}|+\mathcal{O}(|\beta_{j}|\cdot|{N^{\downarrow}_{\pm}}|).$

During such an interaction

$[Q_{1}]=-\big{|}v(\beta_{j})-v({N^{\downarrow}_{\pm}})\big{|}\cdot|{N^{\downarrow}_{\pm}}|\cdot|\beta_{j}|=-\big{|}\mathcal{O}\big{(}|\beta_{j}||{N^{\downarrow}_{\pm}}|\big{)}\big{|}.$

For a weak wave $\gamma$ interacting from the left

$\displaystyle[Q_{2}]=$
$\displaystyle\,|\gamma|\cdot\Big{[}\big{|}v(\gamma)-v({N^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\gamma)-v(\beta_{j}^{\prime})\big{|}^{+}|\beta_{j}^{\prime}|-\big{|}v(\gamma)-v({N^{\downarrow}_{\pm}})\big{|}^{+}|{N^{\downarrow}_{\pm}}|\Big{]}$
$\displaystyle=$