# Collapse of inelastic hard spheres in dimension $d\geq 2$

Théophile Dolmaire  Institute for Applied Mathematics, University of Bonn,
Endenicher Allee 60, D-53115 Bonn, Germany  Juan J. L. Velázquez  Institute
for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115
Bonn, Germany

###### Abstract

We investigate the collapse of three inelastic particles in dimension $d\geq
2$. We obtain general results of convergence and asymptotics concerning the
variables of the dynamical system describing a collapsing system of particles.
We prove a complete classification of the singularities when a collapse of
three particles takes place, obtaining only two possible orders of collisions
between the particles. In the first case we recover that the particles arrange
in a nearly-linear chain, already studied by Zhou and Kadanoff, and in the
second case we obtain that the particles arrange in a triangle, and we show
that, after sufficiently many collisions, the particles collide according to a
unique order of collisions, which is periodic. Finally, we construct an
initial configuration leading to a nearly-linear collapse, stable under
perturbations, and such that the angle between the particles at the time of
collapse can be chosen a priori, with an arbitrary precision.

Keywords. Inelastic Collapse; Inelastic Hard Spheres; Particle Systems.

###### Contents

  1. 1 Introduction
  2. 2 Obtaining the complete dynamical system
    1. 2.1 The model
      1. 2.1.1 The inelastic hard sphere flow
      2. 2.1.2 Dissipation of the kinetic energy
      3. 2.1.3 The problem of the well-posedness of the inelastic hard sphere flow
    2. 2.2 From the dynamics of the particles to a discrete dynamical system
      1. 2.2.1 Reducing the number of variables
      2. 2.2.2 Writing the evolution laws of the dynamical system
  3. 3 General properties of the inelastic collapse of three particles
    1. 3.1 Definition of the inelastic collapse
    2. 3.2 Converging quantities and vanishing quantities
      1. 3.2.1 Convergence of the distances
      2. 3.2.2 Convergence of the velocities
    3. 3.3 Asymptotic behaviour of the variables
      1. 3.3.1 Notations
      2. 3.3.2 Comparison of the variables I: the general case
      3. 3.3.3 Comparison of the variables II: the generic case, with non vanishing tangential velocities
      4. 3.3.4 Comparison of the variables III: the consequences of the regime $\tau\ll\eta$
      5. 3.3.5 Comparison of the variables IV: sufficient condition for the regime $\tau\ll\eta$
  4. 4 Sequence of collisions involving only two pairs of particles: the nearly-linear collapse
    1. 4.1 Converging quantities in the case of the nearly-linear collapse
    2. 4.2 The results of Zhou and Kadanoff concerning the nearly-linear collapse
    3. 4.3 Explicit construction of a stable set of initial configurations leading to the collapse
  5. 5 Sequence of collisions involving the three pairs of particles: the triangular collapse
    1. 5.1 Geometry of the triangular collapse
    2. 5.2 Order of the collisions of the triangular collapse
    3. 5.3 Formal study of the triangular collapse with the matrix of collisions
  6. 6 Conclusion and perspective
  7. A Characteristic polynomial of the matrix $\overline{\mathfrak{A}}_{\infty}$

##  1 Introduction

We consider in this article a system of three inelastic hard spheres, with
fixed restitution coefficient, in dimension $d\geq 2$. We assume that the
particles are spherical, of diameter $1$, and that a pair collides if and only
if the distance between the respective centers of the particles of the pair is
equal to $1$. The restitution coefficient, denoted by $r$, is a positive real
number such that $0<r<1$, which quantifies the dissipation of kinetic energy
during collisions between pairs of particles. More precisely, when a collision
between two particles of respective velocities $v_{1}$ and $v_{2}$ occurs, the
velocities are immediately modified into $v_{1}^{\prime}$ and $v_{2}^{\prime}$
such that:

$\displaystyle\left\\{\begin{array}[]{cccc}v_{1}^{\prime}+v_{2}^{\prime}&=&v_{1}+v_{2},&\text{(conservation
of the momentum)}\\\
\left[(v_{1}^{\prime}-v_{2}^{\prime})\cdot\omega_{1,2}\right]&=&-r\left[(v_{1}-v_{2})\cdot\omega_{1,2}\right],&\text{(dissipation
of a fixed fraction of the normal component)}\end{array}\right.$

where $\omega_{1,2}$ is the unitary vector pointing from the center of the
first particle towards the center of the second. We see that along the
collisions the momentum is conserved, as well as the tangential components of
the relative velocities, but a _fixed_ fraction, equal to $1-r$, of the normal
component of such relative velocities, is dissipated. Therefore, a positive
amount of kinetic energy is dissipated during each collision.

Our motivation to study such a system comes from kinetic theory, and more
precisely, granular materials, where inelastic hard spheres are used to model
such materials at the microscopic level. As usual in kinetic theory, one seeks
information on matter at a macroscopical scale by understanding the behaviour
of a large number of elementary particles. The elastic case, corresponding to
$r=1$, is relatively well understood. The system of particles, at the
microscopic level, is known as the system of _elastic hard spheres_ (see for
instance [26] where various questions concerning such a system are discussed).
For a system of $N$ elastic hard spheres, considering the first marginal of
the $N$-particles density leads to the Boltzmann equation when $N$ goes to
infinity. The solutions of the Boltzmann equation, that describes dilute
gases, are then in close relation with the behaviour of the elementary
components of such gases. This relation is rigorously described by Lanford’s
theorem ([18]).
In the case of inelastic hard spheres, the BBGKY hierarchy and the correlation
functions associated to the particle system are studied in [23] and [5], and
the associated kinetic equation is well-known: one recovers the inelastic
Boltzmann equation, which is the analog of the classical Boltzmann equation
for granular materials. A review on the recent results concerning such an
equation, stretching from the Cauchy theory to the long time behaviour, and
including also numerical investigations, can be found in [6]. The applications
of the inelastic Boltzmann equations are numerous and varied, for granular
materials appears everywhere in our environment: snow, dust, wheat in silos,
sand can all be described with such an equation. In general, every system
composed with a large number of elementary components that interact with
friction (or more generally, with dissipation of any sort of energy) is
susceptible to be described with the help of the theory of granular materials.
Granular materials exhibit behaviours that are typical, in some situations,
from liquids (they may flow), from solids (they might form a stable pile), or
from gases (as interstellar dust). In addition, some features of granular
materials are peculiar (for instance, the pressure in a silo filled with wheat
is an increasing function of the depth, until a certain limit, from which it
remains constant if one goes deeper). The interested reader may consult the
very informative review [17] about the extreme variety of the behaviour of
granular media.
However, the rigorous derivation of the kinetic equations to describe such
granular materials is still an open question: an analogous version of
Lanford’s theorem for inelastic hard spheres is missing. In the case of
elastic hard spheres, the main difficulty is to prove the convergence of the
marginals of the $N$-particles density towards the solutions of the Boltzmann
hierarchy. Such a density is described relying on the fact that the dynamics
of the elastic hard spheres is globally well-posed (almost everywhere), this
is Alexander’s theorem ([1], see also Proposition 4.1.1 page 28 in [12] for a
modern presentation). In the case of inelastic particles, even this very first
step of Lanford’s program is not under control. This open question is the
original motivation that led us to conduct the present work.

The original Alexander’s theorem relies on a crucial property of the flow of
elastic hard spheres: it preserves the measure in the phase space. Although
there exist models of inelastic particles that preserve the measure in the
phase space (that is, even if some kinetic energy is dissipated during the
collisions, see for example [11]), such a property is not true in the case of
the model we consider here. Therefore, in addition to the problems already
present in the elastic case (such as the possibility of triple collisions,
that is, when three or more particles collide together at the same time), one
might expect additional difficulties.
One of the first consequences of the dissipation of the kinetic energy during
the collisions is the spontaneous apparition of inhomogeneities in clouds of
inelastic particles, as it was numerically observed, for instance, in [13].
This behaviour is typical from inelastic particles, for a system of elastic
particles would have the opposite trend: the entropy is increasing, so that a
system of elastic particles tends to fill the space in a homogeneous way (at
least on time scales smaller than the enormous Poincaré’s return time).
Concerning the rich variety of the surprising behaviours of large systems of
inelastic particles, the reader may refer to [24]. When the restitution
coefficient is taken even smaller than the range producing spontaneous
inhomogeneities, the system of particles tends to concentrate in smaller and
smaller regions, and an infinite number of collisions take place in finite
time. In some sense, this is a regime of inhomogeneity pushed to its climax.
Such a phenomenon is called _inelastic collapse_. In particular, when an
inelastic collapse takes place, the numerical simulations of the system break
down, because the computers cannot determine the position of the particles
after the time of the collapse, which is a sort of horizon. We will see that
the inelastic collapse is the main obstruction to obtain an Alexander’s
theorem for inelastic particles, and we will study this phenomenon in great
detail in the present article.

The collapse of inelastic particles was first observed in [4], considering a
pair of one-dimensional inelastic particles in a domain with a vibrating
boundary. It remained to determine if such a phenomenon was peculiar to the
dimension one: in [21] was observed the first inelastic collapse of particles
in dimension $2$. From this point, it was clear that the collapse was a
constitutive feature of the general systems of inelastic particles.

Concerning the one-dimensional collapse, many independent directions were
investigated, although an important number of open questions remain, even for
this very simple case. In what follows, we describe only one-dimensional
systems evolving in a domain without boundary.
As a matter of fact, the collapse cannot take place for a system of only two
particles, but it was observed in [19] for three particles. With [9], where
the authors represented the system as a two-dimensional billiard in a corner
with appropriate reflection laws, the case of three one-dimensional particles
can be considered as fully understood.
Concerning the case of four particles, the two-particles system considered in
[4] can be interpreted as a symmetric system of four particles, for which the
collapse is also completely characterized. For general systems of four
particles though, a lot remains to be understood. If we label the collisions
between 1 and 2, and between 2 and 3, by respectively $a$ and $b$, for three
particles the order of the collisions is completely determined: it has to be
an infinite repetition of the pattern $ab$. With four particles or more, such
a property is not true anymore, and virtually all the orders are possibly
achieved by a well-chosen initial configuration. Inspired by preliminary
numerical simulations, the opposite approach is proposed in [8], where the
authors look for self-similar solutions, achieving patterns of the form
$(ab)^{n}(cb)^{n}$ (using the natural extension of the notations we introduced
for three particles). They observe first that the pattern $abcb$ can be
achieved, but that it is not stable, in the sense that a perturbation of the
initial data would eventually break this sequence of collisions. Then, they
observed that $(ab)^{2}(cb)^{2}$ is also feasible, and it is in addition
stable. For a thorough discussion of this result, the reader may consult [16].
Similarly, it seems that all the patterns of the form $(ab)^{n}(cb)^{n}$, with
$n\geq 2$ are feasible, and stable. But it is remarkable that, for each of
these patterns, corresponds a single interval $I_{n}\subset\,]0,1[$ of
restitution coefficients: for $(ab)^{n}(cb)^{n}$ to be stable, $r$ has to lie
in $I_{n}$, these intervals $I_{n}$ being disjoint, and accumulating as $n$
goes to infinity towards the upper critical value of $r$ for which the
collapse of three particles is feasible.
From these results, new questions emerge: what about the restitution
coefficients outside the intervals $I_{n}$? Is there stable periodic collision
patterns, different from $(ab)^{n}(cb)^{n}$? What about stable, but non-
periodic patterns? What is the maximal restitution coefficient for which a
collapse of four particles is possible? And in addition stable? Outside the
intervals $I_{n}$, it seems that when a collapse takes place, the order of
collisions is chaotic. These intervals $I_{n}$ seem to be the analogs of the
windows of stability for the logistic map. However, none of these last
statements are proved, nor disproved.
For larger numbers of particles, the picture is even less clear. As for the
case of four particles, the question of the largest restitution coefficient
$r$ for which an inelastic collapse is possible is still an open question. In
[3], explicit lower and upper bounds on such a critical restitution
coefficient are obtained. In the same article, the authors produce an explicit
collapse, with an arbitrary number of particles. Nevertheless, such a collapse
is likely not stable under perturbations, for it is symmetric. Therefore, the
question of finding stable collapses for systems with five or more particles
remains open. Let us also mention [7], in which collapses similar to the one
of [3] are constructed, with an elegant geometrical approach.
Finally, in [15], instead of considering inelastic particles on a line, the
system on a circle is considered, exhibiting a quite subtle dynamics.

The case of higher dimensions is less studied. There is essentially a single
reference dealing with our model: in [27], Zhou and Kadanoff provide the
first, and to the best of our knowledge, only study of a system of three
inelastic particles in dimension $d\geq 2$. In particular, two necessary
conditions for a collapse to be stable are determined. At the time of the
collapse, two particles being in contact with the third one (which is then in
a central position) and forming an angle $\overline{\theta}$, such a final
angle $\overline{\theta}$ has to be obtuse, and one needs:

$\displaystyle(-\cos\overline{\theta})>\frac{2r^{1/3}(1+r^{1/3})}{1+r}\cdotp$

In particular, if $r$ is too close to $1$, no collapse can take place. The
critical restitution coefficient preventing the apparition of the collapse
corresponds naturally to the optimal upper bound $r_{c}=7-4\sqrt{3}$, found in
[9]. On the other hand, if $r$ is small enough, collapses with final angles
arbitrarily close to $\pi/2$ are in theory feasible: the one-dimensional
result can be deformed in a notable way. The article of Zhou and Kadanoff will
be the starting point of our investigations.
Let us mention two articles concerning variations of our inelastic system:
[25] proves that three rotating inelastic particles can collapse even for
values of $r$ close to $1$, whereas [14] proves the impossibility of a
collapse of three particles on the line or on the circle, when the restitution
coefficient depends on the relative velocities, and tends to $1$ when these
velocities vanish.
To conclude this review of the literature, back to the classical model (with
fixed restitution coefficient), let us quote [21]. In this reference,
presenting numerical simulations for two-dimensional systems with a large
number of particles, it was observed that, a large number of collision take
place in a finite time, and the particles involved in the last collisions tend
to arrange into one-dimensional-like chains. This motivates very strongly the
study of the one-dimensional collapse. Let us emphasize that the numerical
approach of [21] does not provide information on the systems beyond the time
of collapse. In [24] are presented further investigations on that question, in
particular about the length of these linear structures, that they call
“collision chains” (see Section “Collision Chains” page 185).

As we mentioned already, our original motivation was to obtain an Alexander’s
theorem for inelastic particles. But it turns out that obtaining an
Alexander’s theorem, even for a system with a very small number of particles,
is a very challenging problem. Considering indeed the properties of the
collapse, that are intrinsic difficulties of the dynamics of inelastic
particles, it becomes clear that an Alexander’s theorem has to take into
account the possibility of an inelastic collapse, especially because such a
phenomenon may be stable. Therefore, the inelastic collapse cannot be
associated to a negligible set of initial data.
The configuration of a collapsing system of particles cannot be computed
explicitly in the neighbourhood of a collapse, due to the infinite number of
collisions. However, it does not mean one cannot overcome the time of the
collapse, since as it was already noticed by Zhou and Kadanoff ([27]), it is
possible, in theory, to compute the limiting velocities of the particles at
the time of collapse, and in principle, to continue the dynamics. Indeed, on
the one hand, the normal components of the relative velocities vanish. On the
other hand, if the tangential components of all the particles in contact are
non zero, we expect the particles to separate, and to be able to continue the
dynamics of the system in a unique way. If now the limiting relative velocity
of a pair of particles is zero, we expect that the dynamics cannot be
continued, for two particles would remain attached, and any further collisions
involving one of these two particles would yield an ill-posed problem.

In the present article, in order to understand this continuation principle, we
attached ourselves to establish a mathematical description of systems of
collapsing inelastic particles, and we hope that our results will be useful to
develop further the theory of such systems.

The main results are the following. We obtain general results of convergence
for collapsing systems of three inelastic particles, proving that the pairs of
particles that collide infinitely many times are in contact at the time of
collapse, and that the normal components of the relative velocities of such
pairs vanish. Such results are obtained without assuming any order on the
collisions, and hold therefore in full generality, under the only assumption
that an inelastic collapse takes place. We also obtain asymptotics on the
different variables describing the inelastic particles when the collapse takes
place, in particular we prove that all the vanishing variables can be
estimated only with the vanishing normal components of the relative
velocities, and the times between two consecutive collisions. We hope that
such an approach will prove helpful to understand better the collapsing
particle systems.
We establish a complete classification of the singularities when a collapse of
three inelastic particles takes place, proving that, after a sufficiently
large number of collisions, only two orders of collisions are possible,
namely:

  * •

either one recovers (up to relabel the particles) the infinite repetition of
0\- 1, 0\- 2 (corresponding to the setting of Zhou and Kadanoff in [27]), in
that case the particles arrange in a nearly-linear chain, and we call such a
situation a _nearly-linear collapse_ ,

  * •

or the three pairs are involved in infinitely many collisions, and then (up to
relabel the particles) the order becomes eventually the infinite repetition of
0\- 1, 0\- 2, 1\- 2. In that case, the three particles are in contact at the
time of collapse, forming a triangular structure. To the best of our
knowledge, this is the first description of such a collision order, that we
called a _triangular collapse_.

Finally, we construct explicitly a nearly-linear collapse, which is stable
under perturbation, where in addition the limiting angle between the particles
can be chosen a priori in $]\pi/2,\pi]$, with an arbitrary precision. To the
best of our knowledge, this is the first construction of an explicit, stable
collapse in dimension $d\geq 2$.

The plan of the article is as follows. In the second section of the paper we
introduce in full detail the dynamical system we will study, and the relevant
variables we will need.
The third section of the paper is devoted to establish general convergence and
asymptotic results concerning the different variables of a system of three
collapsing particles. Then, we describe a particular regime concerning the
asymptotic behaviour of the different variables of the system when a collapse
takes place, that was the setting that Zhou and Kadanoff investigated in [27].
We show that in such a regime, that we naturally called the _Zhou-Kadanoff_
-regime (or, in short, the _ZK_ -regime), the study of the dynamical system
simplifies. We expect such a regime to be the only stable one in the phase
space.
In the fourth section, we study in full detail the nearly-linear collapse,
presenting strengthened convergence results, we recall the results of Zhou and
Kadanoff concerning this collapse, and we finally construct explicitly a set
of initial data, with a positive measure, leading to a nearly-linear collapse,
in such a way that the limiting angle between the external particles belongs
to any arbitrary interval (as small as one wants), around any angle
$\overline{\theta}\in\,]\pi/2,\pi]$.
In the fifth section, we investigate the triangular collapse. We prove that,
eventually, only the infinite repetition of the collisions 0\- 1, 0\- 2, 1\- 2
can take place, and we study completely the limiting matrix that prescribes
the evolution of the velocities of the particles when such a collapse occurs.
Such a matrix encodes in particular important information concerning the
possible final configuration of the system when the time of the collapse is
reached.

In order to obtain an Alexander’s theorem for a system of three particles, it
seems in particular necessary to characterize the ZK-regime for linear
collapses. We obtained formal results in this directions, compiled in the
companion paper [10], where we present the study of simplified systems, that
govern, we believe, the behaviour of the full dynamical system, and that
encode in particular the characterization of the Zhou-Kadanoff regime. Such a
study is composed with the derivation of formal limiting systems, rigorous
investigations on such systems, concerning for instance a complete description
of all the fixed points (including also the unstable fixed points) or the long
time behaviour of the orbits of the simplified systems, as well as conjectures
suggested by numerical results. We conjecture in particular the existence of a
separatrix that characterizes the ZK-regime for the formal limiting systems,
and we hope to solve such a conjecture in the future, which should provide an
Alexander-type result for three inelastic particles.

##  2 Obtaining the complete dynamical system

###  2.1 The model

We consider a system of three inelastic particles in the Euclidean space
$\mathbb{R}^{d}$ ($d\geq 2$). Such particles will be denoted by i
($i\in\llbracket 1,3\rrbracket$). Let us denote $x_{i}\in\mathbb{R}^{d}$ and
$v_{i}\in\mathbb{R}^{d}$ the respective positions and velocities of the three
particles ($i\in\llbracket 1,3\rrbracket$). We assume that the particles are
identical spheres (that is, of the same diameter, equal to $1$) that cannot
overlap, as a consequence, our system will evolve in the phase space
$\mathcal{D}_{3}$ defined as

$\mathcal{D}_{3}=\left\\{Z=\left(x_{1},x_{2},x_{3},v_{1},v_{2},v_{3}\right)\in(\mathbb{R}^{d}_{x})^{3}\times(\mathbb{R}^{d}_{v})^{3}=\mathbb{R}^{6d}\
/\ |x_{i}-x_{j}|\geq 1\text{ for all }i<j\right\\}.$ (2.1)

####  2.1.1 The inelastic hard sphere flow

Let us denote as $t$ the time variable. We will assume that the particles
evolve according to the _inelastic hard sphere flow_. Let us describe such a
dynamics. Starting, at $t=0$, from an initial configuration

$Z_{0}=\left(x_{1}(0),x_{2}(0),x_{3}(0),v_{1}(0),v_{2}(0),v_{3}(0)\right)$

in the phase space $\mathcal{D}_{3}$, the evolution laws of the system are
defined as follows. When no pair of particles are in contact, that is, when
$|x_{i}-x_{j}|>1$ for all $1\leq i<j\leq 3$, and as long as such a condition
holds true, we will assume that the particles move in straight line, with
constant velocities (Newton’s law without any external force), that is

$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}v_{i}(t)$
$\displaystyle=0,$ $\displaystyle x_{i}(t)$
$\displaystyle=x_{i}(0)+tv_{i}(0).$ (2.2)

At the first moment two particles collide, that is, when there exists a time
$t_{1}\geq 0$ such that $|x_{i}(t_{1})-x_{j}(t_{1})|=1$ for a certain pair of
indices $i<j$, the velocities of the two particles $i$ and $j$ are immediately
changed from $v_{i}=v_{i}(0)$ and $v_{j}=v_{j}(0)$ into

$\left\\{\begin{array}[]{rl}v_{i}^{\prime}&=v_{i}-\frac{(1+r)}{2}(v_{i}-v_{j})\cdot\omega_{i,j}\omega_{i,j},\\\
v_{j}^{\prime}&=v_{j}+\frac{(1+r)}{2}(v_{i}-v_{j})\cdot\omega_{i,j}\omega_{i,j},\end{array}\right.$
(2.3)

where $r\in[0,1]$ is the _restitution coefficient_ , measuring the
“inelasticity” of the collision, and $\omega_{i,j}$ is the normalized vector
$\left(x_{i}(t_{1})-x_{j}(t_{1})\right)/|x_{i}(t_{1})-x_{j}(t_{1})|$ joining
the centers of the two colliding particles $i$ and $j$, at the time of the
collision $t_{1}$. After the first collision happened, and the velocities of
the colliding particles are changed according to the laws (2.3), the particles
will again move in straight line, with constant velocities, until the next
collision happens, and so on. Such a process defines the inelastic hard sphere
dynamics.

######  Definition 1 ($r$-inelastic hard sphere flow).

Let $r\in\ ]0,1[$ be a positive number smaller than $1$, and let us consider
the mapping $\mathcal{D}_{3}\times\mathbb{R}_{+}\rightarrow\mathcal{D}_{3}$,
$\left(Z_{0},t\right)\mapsto Z(t)$, where $Z(t)$ is defined recursively using
the free transport (2.1.1) as long as $Z(t)$ belongs to
$\overset{\circ}{\mathcal{D}}_{3}$, and the collision mapping (2.3) for the
times $t_{n}$ such that $Z(t)\in\partial\mathcal{D}_{3}$.
Such a mapping is called the _$r$ -inelastic hard sphere flow_.

####  2.1.2 Dissipation of the kinetic energy

In the rest of this work, it will be useful to decompose the relative
velocities into their projections on the lines of contact
$\text{span}\\{\omega_{i,j}\\}$, and their projections on the orthogonal of
these lines.

######  Definition 2 (Normal and tangential components of the relative
velocities).

For two particles i and j in contact, with respective positions $x_{i}$ and
$x_{j}$, and respective velocities $v_{i}$ and $v_{j}$, we will denote by
$W_{i,j}$ the relative velocity $v_{j}-v_{i}$, and often by $W_{j}$ when i is
the particle 0.
We define the _normal component of the relative velocity of the pair_ i\- j as
the orthogonal projection $\eta_{i,j}$ of the relative velocity $v_{j}-v_{i}$
on the line of contact
$\omega_{i,j}=\frac{(x_{j}-x_{i})}{|x_{j}-x_{i}|}=x_{j}-x_{i}$, that is:

$\displaystyle\eta_{i,j}=\left(v_{j}-v_{i}\right)\cdot\left(x_{j}-x_{i}\right).$
(2.4)

We define also the _tangential component of the relative velocity of the pair_
i\- j as the orthogonal projection $W_{i,j}^{\perp}$ of the relative velocity
$v_{j}-v_{i}$ on the orthogonal to the line of contact
$\omega_{i,j}=\frac{(x_{j}-x_{i})}{|x_{j}-x_{i}|}=x_{j}-x_{i}$, that is:

$\displaystyle
W_{i,j}^{\perp}=(v_{j}-v_{i})-\left(v_{j}-v_{i}\right)\cdot\left(x_{j}-x_{i}\right)\left(x_{j}-x_{i}\right)=(v_{j}-v_{i})-\left(\left(v_{j}-v_{i}\right)\cdot\omega_{i,j}\right)\omega_{i,j}$
(2.5)

######  Remark 1.

We defined the normal component $\eta$ as a scalar, and the tangential
component $W^{\perp}$ as a vector of $\mathbb{R}^{d}$, even though it
corresponds to a projection on a $(d-1)$-dimensional space.

######  Remark 2.

For a pair i and j of colliding particles, (2.3) can be rewritten as:

$\displaystyle\eta_{i,j}^{\prime}=-r\eta_{i,j}.$ (2.6)

A _fixed, constant_ proportion of the normal component of the relative
velocity is dissipated during each collision.

In the model we are considering, the momentum is clearly conserved along the
collisions, but we are prescribing a loss of kinetic energy during each
collision. The restitution coefficient is prescribing such a loss, quantified
in the following lemma.

######  Lemma 1.

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a pair of particles i and j, with respective velocities $v_{i}$ and
$v_{j}$, that collide according to the law (2.3).
Then the dissipation of the kinetic energy during the collision is quantified
as follows. If we denote by $\mathcal{E}_{c}$ the kinetic energy of the pair
before the collision, and $\mathcal{E}_{c}^{\prime}$ the kinetic energy after
the collision, we have:

$\mathcal{E}_{c}^{\prime}=\mathcal{E}_{c}-\frac{(1-r^{2})}{4}\eta_{i,j}^{2}=\frac{1}{4}|v_{i}+v_{j}|^{2}+\frac{1}{4}|W_{i,j}^{\perp}|^{2}+\frac{r^{2}}{4}\eta_{i,j}^{2}.$
(2.7)

By definition, the particles lose a certain ratio of their _normal_ relative
velocity in each collision, fixed by the positive coefficient $r$, as observed
in Remark 2. When $r=1$, we recover the case of the elastic hard sphere model.
In that case, the kinetic energy is preserved. When $r=0$, all the normal
relative velocity is dissipated. In particular, in the one dimensional case
$d=1$, the particles remain “glued” after such a collision, in the literature
authors often refer to the case $r=0$ as “sticky particles”. Note that there
is no reason for the particles to remain attached to each other after such a
collision in higher dimension: if the pre-collisional tangential relative
velocity is non zero, it will remain unchanged after the collision, and the
particles will then immediately separate.

####  2.1.3 The problem of the well-posedness of the inelastic hard sphere
flow

It is important to emphasize that this inelastic hard sphere flow is well-
defined only when binary collisions take place: if the three particles are in
contact at the same time, the laws of the dynamics (2.3) defining the post-
collisional velocities are ill-posed (in general, the system defining the
post-collisional velocities becomes over-determined). We have therefore to
restrict ourselves to study dynamics emerging from initial conditions that
never lead to triple collisions.
Another problem that can also appear in the sticky particles regime, linked to
the triple collision problem, is the following: if $r=0$, and if a pair of
colliding particles has no relative tangential velocity at the time of
collision, then these two particles will stay glued together after the
collision. Therefore, if one of these two particles collides with a third
particle in the future, the dynamics will be again ill posed, due to a triple
collision. In the case $r=0$, one has therefore also to restrict oneself to
collisions between particles such that the pre-collisional relative velocity
is not colinear to the direction between the two centers.
However, in the present work, we shall not discuss the elastic regime ($r=1$),
already addressed by Alexander (see [1], and Proposition 4.1.1, in Section
“The $N$-particle flow” in the more recent reference [12]), nor the sticky
particles, which is, to the best of our knowledge, left completely unstudied
so far.
In addition to the reasons of ill-posedness just described, a last one can
(and will) occur in the case we are interested in here ($0<r<1$): the
existence of accumulation points of the sequence of collision times between
the particles. In that case, there is no finite sequence of time intervals on
which it is possible to apply the free transport on the particles, between two
collisions, in order to reach any time beyond an accumulation point.
Therefore, the dynamics is not clearly globally well-defined. Such a
phenomenon is the central object of the present article.

###  2.2 From the dynamics of the particles to a discrete dynamical system

####  2.2.1 Reducing the number of variables

In the general case, our system of particles is described by a trajectory
$Z:t\mapsto Z(t)$, a time dependent function taking its values in the phase
space, which is a subset of $\mathbb{R}^{6d}$. With such a representation, we
keep track at any time of the positions and the velocities of the three
particles, each of these vectors being an element of $\mathbb{R}^{d}$. We have
defined a continuous dynamical system.
Nevertheless, it is possible to restrict the dimension of the dynamical system
we are studying. An approach, used in a fruitful way by Zhou and Kadanoff
([27]), consists in working in a frame attached to one of the particles. The
immediate drawback of this approach is that the momentum is not conserved in
this frame, which is not Galilean anymore, but we will be able to describe the
evolution of the dynamical system in a particularly efficient way. More
precisely, Zhou and Kadanoff focus on the case when the infinite sequence of
collisions between the particles is 0\- 1, then 0\- 2, and again 0\- 1, and so
on, over and over again. Therefore, in this case 0 has a central role: it will
collide infinitely many times with the two others, while both of the two other
particles will collide only with the central one.
As a consequence, it will make sense to choose this central particle as the
origin of the frame, and to measure quantities such as relative velocities and
distances from the velocity and position of the central particle. In other
words, we will assume that the central particle 0 is at rest, lying at the
center of our frame, and we will measure the relative positions and velocities
of the two other particles 1 and 2 in this frame.
On the other hand, when a collision occurs, the central particle is
systematically involved, one of the two other particles is of course a
colliding particle, while the third one remains “spectator” to this collision.
But when the next collision takes place, the role of the colliding and the
spectator particles are exchanged. Following Zhou and Kadanoff, this
observation motivates to consider another description of the dynamical system:
a first set of variables will describe the movement, with respect to the
central particle, of the colliding particle, whereas a second set of variables
will describe the behaviour of the spectator particle. Therefore, when the
sequence of collisions is the infinite repetition of the period 0\- 1, 0\- 2,
after each collision, the first set will describe, alternatively, the movement
of the particle 1 then the particle 2, and vice versa concerning the second
set of variables.
Finally, without any assumption on the order of the collisions, we see that we
can generalize the approach of Zhou and Kadanoff: in the end, there is no need
to have a central particle along the whole sequence of collisions. What
matters is the order of the collisions, that defines which colliding particle
will become spectator for the next collision, and which particle that just
collided will replace it. In other words, it is enough to describe a collision
between 0 and 2, assuming that it follows directly a collision between 0 and
1. The complete evolution of the system is then simply a composition of this
collision mapping, with the operation consisting only in relabelling of the
particles at each collision.

####  2.2.2 Writing the evolution laws of the dynamical system

Now that the general ideas in order to parametrize the system have been
introduced, let us now introduce the discrete dynamical system that we will
investigate.

##### The initial data.

Let us start with the initial data. We will assume that, at the initial time
$t=0$, the two particles 0 and 1 are in contact, and just collided, while the
particle 2 is at a positive distance from the two other particles 0 and 1.
Therefore, if we denote $v_{0}$ and $v_{1}$ the respective velocities of the
particles 0 and 1 just after this initial collision, and if we denote
$x_{0}=0$ and $x_{1}$ the respective positions of these particles at the
initial time, we have:

$\displaystyle\left|x_{1}-x_{0}\right|=1,$ (2.8)

translating the fact that 0 and 1 are in contact. Let us then denote by
$\omega_{1}$ the difference $x_{1}-x_{0}\in\mathbb{S}^{d-1}$. We have also

$\displaystyle\left(v_{1}-v_{0}\right)\cdot\left(x_{1}-x_{0}\right)=\left(v_{1}-v_{0}\right)\cdot\omega_{1}>0,$
(2.9)

describing the fact that the two particles 0 and 1 are in a post-collisional
configuration, that is, the distance between these two particles is increasing
for small positive times, since we have:

$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\left[t\mapsto\left|x_{1}(t)-x_{0}(t)\right|\right]$
$\displaystyle=\frac{2\left(\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\left(x_{1}(t)-x_{0}(t)\right)\right)\cdot\left(x_{1}(t)-x_{0}(t)\right)}{2\left|x_{1}(t)-x_{0}(t)\right|}$
$\displaystyle=\frac{\left(v_{1}-v_{0}\right)\cdot\left(x_{1}+tv_{1}-x_{0}-tv_{0}\right)}{\left|x_{1}(t)-x_{0}(t)\right|}\cdotp$

Introducing then the relative velocity $W_{1}=v_{1}-v_{0}$ of the particle 1
with respect to 0, (2.9) can be rewritten as

$\displaystyle W_{1}\cdot\omega_{1}>0.$ (2.10)

We assumed also that the particle 2 is at a positive distance from the two
others at the initial time, and in particular, we have

$\displaystyle\left|x_{2}-x_{0}\right|>1.$ (2.11)

If we denote by $\omega_{2}$ the vector
$\displaystyle{\frac{x_{2}-x_{0}}{|x_{2}-x_{0}|}}\in\mathbb{S}^{d-1}$, and
$1+d=|x_{2}-x_{1}|$, $x_{2}-x_{0}=(1+d)\omega_{2}$, and (2.11) can be
rewritten as

$\displaystyle d>0.$ (2.12)

Without loss of generality we can assume that 0 is also involved in the next
collision, together with 2, and
$\left(v_{2}-v_{0}\right)\cdot\left(x_{2}-x_{0}\right)<0$, or again, if we
denote by $W_{2}$ the relative velocity $v_{2}-v_{0}$:

$\displaystyle W_{2}\cdot\omega_{2}<0.$ (2.13)

In the end, the initial configuration of the system is described with the help
of the following collection of variables:

$\displaystyle\big{(}\underbrace{\omega_{1},W_{1}}_{\begin{subarray}{c}\text{position
and}\\\ \text{velocity of}\\\ \text{the particle}\\\ \leavevmode\hbox
to11.75pt{\vbox to11.75pt{\pgfpicture\makeatletter\hbox{\hskip
5.87407pt\lower-5.87407pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.67407pt}{0.0pt}\pgfsys@curveto{5.67407pt}{3.13374pt}{3.13374pt}{5.67407pt}{0.0pt}{5.67407pt}\pgfsys@curveto{-3.13374pt}{5.67407pt}{-5.67407pt}{3.13374pt}{-5.67407pt}{0.0pt}\pgfsys@curveto{-5.67407pt}{-3.13374pt}{-3.13374pt}{-5.67407pt}{0.0pt}{-5.67407pt}\pgfsys@curveto{3.13374pt}{-5.67407pt}{5.67407pt}{-3.13374pt}{5.67407pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{subarray}},\underbrace{d,\omega_{2},W_{2}}_{\begin{subarray}{c}\text{position
and}\\\ \text{velocity of}\\\ \text{the particle}\\\ \leavevmode\hbox
to11.75pt{\vbox to11.75pt{\pgfpicture\makeatletter\hbox{\hskip
5.87407pt\lower-5.87407pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.67407pt}{0.0pt}\pgfsys@curveto{5.67407pt}{3.13374pt}{3.13374pt}{5.67407pt}{0.0pt}{5.67407pt}\pgfsys@curveto{-3.13374pt}{5.67407pt}{-5.67407pt}{3.13374pt}{-5.67407pt}{0.0pt}\pgfsys@curveto{-5.67407pt}{-3.13374pt}{-3.13374pt}{-5.67407pt}{0.0pt}{-5.67407pt}\pgfsys@curveto{3.13374pt}{-5.67407pt}{5.67407pt}{-3.13374pt}{5.67407pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{subarray}}\big{)}\in\mathbb{S}^{d-1}\times\mathbb{R}^{d}\times\mathbb{R}_{+}^{*}\times\mathbb{S}^{d-1}\times\mathbb{R}^{d}.$
(2.14)

We need in total $4d-1$ real variables in order to describe the initial
configuration of the system.

##### The first collision after the initial configuration.

The next collision involves the particles 0 and 2. The condition (2.13) is
necessary, but not sufficient in order to have a collision between the
particles 0 and 2 at a positive time. Let us first discuss a characterization
for such a collision to happen in the future.
The distance, along time, between the two particles 0 and 2, if no collision
takes place on the time interval $]0,t[$, is given by

$\displaystyle\left|x_{2}(t)-x_{0}(t)\right|$
$\displaystyle=\left|(x_{2}-x_{0})+t(v_{2}-v_{0})\right|$
$\displaystyle=\left|(1+d)\omega_{2}+tW_{2}\right|$
$\displaystyle=\sqrt{(1+d)^{2}+2(1+d)\left(\omega_{2}\cdot
W_{2}\right)t+\left|W_{2}\right|^{2}t^{2}}.$

The collision between 0 and 2 will take place if and only if there exists a
positive time $\tau$ such that

$\displaystyle|W_{2}|^{2}\tau^{2}+2(1+d)\left(\omega_{2}\cdot
W_{2}\right)\tau+d^{2}+2d=0,$ (2.15)

and the particles 1 and 2 do not collide before ( 0 and 1 cannot, according to
(2.10)). The existence of a (real) solution to (2.15) is equivalent to
$\Delta\geq 0$, where:

$\displaystyle\Delta$ $\displaystyle=\left[2(1+d)\left(\omega_{2}\cdot
W_{2}\right)\right]^{2}-4|W_{2}|^{2}d(2+d)$
$\displaystyle=4\left[(1+d)^{2}\left(\omega_{2}\cdot
W_{2}\right)^{2}-d(2+d)|W_{2}|^{2}\right].$ (2.16)

In our case, we will require the positivity of the discriminant $\Delta$. In
that case, there exist two solutions $\tau$ to the equation (2.15). These two
solutions will be both positive, or both negative, and the sign of the
solutions will be the opposite of the sign of the coefficient of the first
degree, and according to the condition (2.13), the two roots will be positive.
Naturally, the only “physical” solution corresponds to the smallest solution
of the equation, describing the first time the two balls representing 0 and 2
intersect. The expression of the time of collision is then:

$\displaystyle\tau=-\frac{(1+d)\left(\omega_{2}\cdot
W_{2}\right)}{|W_{2}|^{2}}-\frac{\sqrt{(1+d)^{2}\left(\omega_{2}\cdot
W_{2}\right)^{2}-d(2+d)|W_{2}|^{2}}}{|W_{2}|^{2}}\cdotp$ (2.17)

Let us finally use the decomposition of the relative velocities introduced in
Definition 2, writing $W_{i}=\eta_{i}\omega_{i}+W_{i}^{\perp}$, with
$\omega_{i}\cdot W_{i}^{\perp}=0$. Note that the sign of $\eta_{i}$ is
describing if the pair of particles (if $i=1$, then we consider the pair 0\-
1, 0\- 2 if $i=2$) is in a pre- or in a post-collisional configuration.
Thanks to these notations, it will be possible to simplify the expressions of
the different quantities involved in the dynamics of the particle system. In
particular, we can rewrite the expression (2.17) of $\tau$ as

$\displaystyle\tau=-\frac{(1+d)\eta_{2}}{|W_{2}|^{2}}-\frac{\sqrt{(1+d)^{2}\eta_{2}^{2}-d(2+d)\left(\eta_{2}^{2}+|W_{2}^{\perp}|^{2}\right)}}{|W_{2}|^{2}}$
$\displaystyle=\frac{(1+d)(-\eta_{2})}{|W_{2}|^{2}}-\frac{\sqrt{(1+d)^{2}\eta_{2}^{2}-d(2+d)|W_{2}|^{2}}}{|W_{2}|^{2}}\cdotp$
(2.18)

Considering (2.2.2), let us introduce an important quantity, named after [27],
since this parameter allows to characterize the main regime studied in this
reference.

######  Definition 3 (Zhou-Kadanoff parameter).

Let $d$ be a positive number, and $W_{2}$ be a vector in $\mathbb{R}^{d}$. We
denote by $\zeta$ the positive quantity:

$\displaystyle\zeta=\frac{d(2+d)|W_{2}|^{2}}{(1+d)^{2}\eta_{2}^{2}}\geq 0.$
(2.19)

The number $\zeta$ will be called the _Zhou-Kadanoff parameter_ (in short, the
_ZK parameter_).

We will see that the evolution of the Zhou-Kadanoff parameter along the
different collisions is fundamental to understand the long time behaviour of
the particles. Using the last definition, (2.2.2) writes:

$\displaystyle\tau$
$\displaystyle=\frac{(1+d)(-\eta_{2})}{|W_{2}|^{2}}\left[1-\sqrt{1-\zeta}\right].$
(2.20)

In order to write completely the expressions defining the evolution of our
dynamical system, let us first notice that the distance at time $\tau$ between
the particle 0 and the spectator particle 1 is

$\displaystyle 1+d^{\prime}=|x_{1}(\tau)-x_{0}(\tau)|=|\omega_{1}+\tau
W_{1}|,$

which can be rewritten as

$\displaystyle d^{\prime}$ $\displaystyle=\sqrt{|\omega_{1}+\tau
W_{1}|^{2}}-1$ $\displaystyle=\sqrt{1+2\omega_{1}\cdot
W_{1}\tau+|W_{1}|^{2}\tau^{2}}-1$
$\displaystyle=\sqrt{1+2\eta_{1}\tau+|W_{1}|^{2}\tau^{2}}-1.$ (2.21)

Let us now rewrite the equations of the post-collisional velocities in a more
synthetic way. In order to do so, let us define more precisely the quantities
$\omega_{1}^{\prime}$ and $\omega_{2}^{\prime}$. These two unit vectors
represent, respectively, the directions from the central particle 0 to the
particle 1, respectively to the particle 2, at time $\tau$, that is:

$\omega_{1}^{\prime}=\frac{x_{1}(\tau)-x_{0}(\tau)}{|x_{1}(\tau)-x_{0}(\tau)|}=\frac{\omega_{1}+\tau
W_{1}}{(1+d^{\prime})},$ (2.22)

and

$\omega_{2}^{\prime}=\frac{x_{2}(\tau)-x_{0}(\tau)}{|x_{2}(\tau)-x_{0}(\tau)|}=(1+d)\omega_{2}+\tau
W_{2}.$ (2.23)

In particular, using these last notations, (2.3) can be rewritten as

$\displaystyle\left\\{\begin{array}[]{rl}v_{0}^{\prime}&=v_{0}-\frac{(1+r)}{2}\left(-W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime},\\\
v_{1}^{\prime}&=v_{1},\\\
v_{2}^{\prime}&=v_{2}+\frac{(1+r)}{2}\left(-W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime},\end{array}\right.$
(2.27)

which provides, in terms of _relative_ post-collisional velocities:

$\displaystyle\left\\{\begin{array}[]{rlll}W_{1}^{\prime}&=v_{1}^{\prime}-v_{0}^{\prime}&=(v_{1}-v_{0})+\frac{(1+r)}{2}\left(-W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime}&=W_{1}-\frac{(1+r)}{2}\left(W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime},\\\
W_{2}^{\prime}&=v_{2}^{\prime}-v_{0}^{\prime}&=(v_{2}-v_{0})+(1+r)\left(-W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime}&=W_{2}-(1+r)\left(W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime}.\end{array}\right.$
(2.30)

Since we have defined
$W_{i}^{\prime}=\eta_{i}^{\prime}\omega_{i}^{\prime}+\left(W_{i}^{\perp}\right)^{\prime}$,
we find, for the normal component of the first post-collisional relative
velocity:

$\displaystyle\eta_{1}^{\prime}=W_{1}^{\prime}\cdot\omega_{1}^{\prime}=\left(W_{1}-\frac{(1+r)}{2}\left(W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime}\right)\cdot\omega_{1}^{\prime}=W_{1}\cdot\omega_{1}^{\prime}-\frac{(1+r)}{2}\left(W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}.$

Using (2.22) and (2.23) we find

$\displaystyle W_{1}\cdot\omega_{1}^{\prime}=\frac{W_{1}\cdot(\omega_{1}+\tau
W_{1})}{(1+d^{\prime})}=\frac{\eta_{1}+\tau|W_{1}|^{2}}{(1+d^{\prime})},$

and

$\displaystyle
W_{2}\cdot\omega_{2}^{\prime}=W_{2}\cdot\left((1+d)\omega_{2}+\tau
W_{2}\right)=(1+d)\eta_{2}+\tau|W_{2}|^{2},$

therefore we have:

$\eta_{1}^{\prime}=\frac{1}{(1+d^{\prime})}\left(\eta_{1}+\tau|W_{1}|^{2}\right)-\frac{(1+r)}{2}\left(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}\right)\left((1+d)\eta_{2}+\tau|W_{2}|^{2}\right).$
(2.31)

In the same way, we have:

$\displaystyle\eta_{2}^{\prime}=W_{2}^{\prime}\cdot\omega_{2}^{\prime}=\left(W_{2}-(1+r)\left(W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime}\right)\cdot\omega_{2}^{\prime}=W_{2}\cdot\omega_{2}^{\prime}-(1+r)\left(W_{2}\cdot\omega_{2}^{\prime}\right)=-rW_{2}\cdot\omega_{2}^{\prime}.$
(2.32)

Here, we see how much the expression simplifies in this frame. Using again
(2.23), we find

$\eta_{2}^{\prime}=-rW_{2}\cdot\left((1+d)\omega_{2}+\tau
W_{2}\right)=-r(1+d)\eta_{2}-r\tau|W_{2}|^{2}.$ (2.33)

We can therefore write (using the intermediate expression we found for
$\eta_{1}^{\prime}$):

$\displaystyle\left(W_{1}^{\perp}\right)^{\prime}$
$\displaystyle=W_{1}^{\prime}-\eta_{1}^{\prime}\omega_{1}^{\prime}=W_{1}-\frac{(1+r)}{2}\left(W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime}-\left(W_{1}\cdot\omega_{1}^{\prime}-\frac{(1+r)}{2}(W_{2}\cdot\omega_{2}^{\prime})\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}\right)\omega_{1}^{\prime}$
$\displaystyle=W_{1}-W_{1}\cdot\omega_{1}^{\prime}\omega_{1}^{\prime}+\frac{(1+r)}{2}\left(W_{2}\cdot\omega_{2}^{\prime}\right)\big{[}(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime})\omega_{1}^{\prime}-\omega_{2}^{\prime}\big{]}$
$\displaystyle=W_{1}-\frac{(\eta_{1}+\tau|W_{1}|^{2})}{(1+d^{\prime})}\omega_{1}^{\prime}+\frac{(1+r)}{2}\left((1+d)\eta_{2}+\tau|W_{2}|^{2}\right)\big{[}(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime})\omega_{1}^{\prime}-\omega_{2}^{\prime}\big{]},$
(2.34)

and, similarly:

$\displaystyle\left(W_{2}^{\perp}\right)^{\prime}$
$\displaystyle=W_{2}^{\prime}-\eta_{2}^{\prime}\omega_{2}^{\prime}$
$\displaystyle=W_{2}-(1+r)\left(W_{2}\cdot\omega_{2}^{\prime}\right)\omega_{2}^{\prime}-\eta_{2}^{\prime}\omega_{2}^{\prime}$
$\displaystyle=W_{2}-(1+r)(W_{2}\cdot\omega_{2}^{\prime})\omega_{2}^{\prime}+rW_{2}\cdot\omega_{2}^{\prime}\omega_{2}^{\prime}$
$\displaystyle=W_{2}-(W_{2}\cdot\omega_{2}^{\prime})\omega_{2}^{\prime}$
$\displaystyle=W_{2}-\left((1+d)\eta_{2}+\tau|W_{2}|^{2}\right)\left((1+d)\omega_{2}+\tau
W_{2}\right).$ (2.35)

Equations (2.22), (2.23), (2.2.2) and (2.2.2) together with (2.2.2) provide
almost a complete description of the dynamical system involving only the
initial datum described in term of the variables $d$, $\omega_{1}$,
$\omega_{2}$, $W_{1}$ and $W_{2}$. It remains only the describe
$\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}$, which is equal to the cosine of
the angle between the particles 1 and 2 at time $\tau$. This can be done
writing:

$\displaystyle\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}$
$\displaystyle=\left(\frac{\omega_{1}+\tau
W_{1}}{(1+d^{\prime})}\right)\cdot\left((1+d)\omega_{2}+\tau W_{2}\right)$
$\displaystyle=\frac{(1+d)}{(1+d^{\prime})}\left(\omega_{1}\cdot\omega_{2}\right)+\left[\frac{1}{(1+d^{\prime})}\omega_{1}\cdot
W_{2}+\frac{(1+d)}{(1+d^{\prime})}\omega_{2}\cdot
W_{1}\right]\tau+\frac{W_{1}\cdot W_{2}}{(1+d^{\prime})}\tau^{2}\cdotp$ (2.36)

We can therefore convert the continuous dynamical system associated to the
$r$-inelastic hard sphere flow into a discrete dynamical system. More
precisely, let us write entirely the evolution of the variables
$\left(\omega_{1},W_{1},d,\omega_{2},W_{2}\right)$, describing a transition
from a post-collisional configuration just after a collision of type 0\- 1, to
a post-collisional configuration just after a collision of type 0\- 2.

######  Definition 4 (Complete single-collision mapping).

Let us consider three particles 0, 1, 2 in $\mathbb{R}^{d}$, of respective
positions $x_{i}$ and velocities $v_{i}$ ($x_{i},v_{i}\in\mathbb{R}^{d}\
\forall i\in\llbracket 1,3\rrbracket$) described by the configuration:

$\displaystyle\left(\omega_{1},W_{1},d,\omega_{2},W_{2}\right)\in\mathbb{S}^{d-1}\times\mathbb{R}^{d}\times\mathbb{R}_{+}^{*}\times\mathbb{S}^{d-1}\times\mathbb{R}^{d},$
(2.37)

where:

  * •

$\omega_{1}=x_{1}-x_{0}$ ( 0 and 1 are initially in contact),

  * •

$W_{1}=v_{1}-v_{0}$,

  * •

$1+d=|x_{2}-x_{0}|>0$ ( 0 and 2 are initially separated),

  * •

$(1+d)\omega_{2}=x_{2}-x_{0}$,

  * •

$W_{2}=v_{2}-v_{0}$.

Let us assume in addition that 0 and 1 are in a post-collisional
configuration, and that a collision of type 0\- 2 is the next collision that
will take place:

$\displaystyle\eta_{1}=W_{1}\cdot\omega_{1}>0,\hskip
14.22636pt\eta_{2}=W_{2}\cdot\omega_{2}<0,\hskip 5.69054pt\text{and}\hskip
5.69054pt\zeta=\frac{d(2+d)|W_{2}|^{2}}{(1+d)^{2}\eta_{2}^{2}}<1.$ (2.38)

We define then _complete single-collision mapping_ as the function:

$\displaystyle\mathfrak{C}:\left\\{\begin{array}[]{clc}\mathbb{S}^{d-1}\times\mathbb{R}^{d}\times\mathbb{R}_{+}^{*}\times\mathbb{S}^{d-1}\times\mathbb{R}^{d}&\rightarrow&\mathbb{S}^{d-1}\times\mathbb{R}^{d}\times\mathbb{R}_{+}^{*}\times\mathbb{S}^{d-1}\times\mathbb{R}^{d},\\\
\left(\omega_{1},W_{1},d,\omega_{2},W_{2}\right)&\mapsto&\left(\omega_{1}^{\prime},W_{1}^{\prime},d^{\prime},\omega_{2}^{\prime},W_{2}^{\prime}\right),\end{array}\right.$

where:

$\displaystyle\left\\{\begin{array}[]{rl}\omega_{1}^{\prime}&=\displaystyle{\frac{\omega_{1}+\tau
W_{1}}{(1+d^{\prime})}},\vspace{1mm}\\\
\eta_{1}^{\prime}&=\displaystyle{\frac{1}{(1+d^{\prime})}\left(\eta_{1}+\tau|W_{1}|^{2}\right)-\frac{(1+r)}{2}\left(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}\right)\left((1+d)\eta_{2}+\tau|W_{2}|^{2}\right)},\vspace{3mm}\\\
(W_{1}^{\perp})^{\prime}&=W_{1}-W_{1}\cdot\omega_{1}^{\prime}\omega_{1}^{\prime}+\displaystyle{\frac{(1+r)}{2}}\left(W_{2}\cdot\omega_{2}^{\prime}\right)\big{[}(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime})\omega_{1}^{\prime}-\omega_{2}^{\prime}\big{]}\vspace{1mm}\\\
&\hskip
56.9055pt=W_{1}-\displaystyle{\frac{(\eta_{1}+\tau|W_{1}|^{2})}{(1+d^{\prime})}}\omega_{1}^{\prime}+\displaystyle{\frac{(1+r)}{2}}\left((1+d)\eta_{2}+\tau|W_{2}|^{2}\right)\big{[}(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime})\omega_{1}^{\prime}-\omega_{2}^{\prime}\big{]},\vspace{3mm}\\\
d^{\prime}&=\displaystyle{\sqrt{1+2\eta_{1}\tau+|W_{1}|^{2}\tau^{2}}-1},\vspace{3mm}\\\
\omega_{2}^{\prime}&=(1+d)\omega_{2}+\tau W_{2},\vspace{3mm}\\\
\eta_{2}^{\prime}&=-r(1+d)\eta_{2}-r\tau|W_{2}|^{2},\vspace{3mm}\\\
(W_{2}^{\perp})^{\prime}&=W_{2}-(W_{2}\cdot\omega_{2}^{\prime})\omega_{2}^{\prime}\vspace{2mm}\\\
&\hskip
56.9055pt=W_{2}-\left((1+d)\eta_{2}+\tau|W_{2}|^{2}\right)\left((1+d)\omega_{2}+\tau
W_{2}\right),\end{array}\right.$ (2.48)

with $\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}$ defined in (2.2.2), and
$\tau$ is defined as

$\displaystyle\tau=\frac{(1+d)(-\eta_{2})}{|W_{2}|^{2}}\left[1-\sqrt{1-\zeta}\right],\text{
and }\zeta=\frac{d(2+d)|W_{2}|^{2}}{(1+d)^{2}\eta_{2}^{2}}\geq 0.$

The mapping $\mathfrak{C}$ introduced in the previous definition encodes
completely the dynamics of the system of three inelastic particles, describing
the evolution of the particles between a collision of type 0\- 1 and a
collision of type 0\- 2. All the distances and velocities are measured from
the position $x_{0}$ and velocity $v_{0}$ of the particle 0, central for the
pair of collisions 0\- 1, 0\- 2.

######  Remark 3.

$\mathfrak{C}$ defines a mapping from
$\mathbb{S}^{d-1}\times\mathbb{R}^{d}\times\mathbb{R}_{+}^{*}\times\mathbb{S}^{d-1}\times\mathbb{R}^{d}$
into itself: we have a $(4d-1)$-dimensional dynamical system, $7$-dimensional
for $d=2$, $11$-dimensional for $d=3$.
Nevertheless, we mentioned already that $W_{i}^{\perp}$ belong by definition
to $\omega_{i}^{\perp}$, so it would be possible to reduce the dimension of
the dynamical system by $2$ in theory. In any case, from the expressions of
(2.48), it is clear that we are facing a complicated dynamical system.

##  3 General properties of the inelastic collapse of three particles

This section is devoted to establish and list elementary properties that are
satisfied by the system of particles when a collapse takes place. In
particular, we will concentrate on obtaining qualitative behaviours of the
sequences of variables used to describe the system of inelastic particles.
The thorough discussion of the present section sets a mathematically rigorous
framework to study inelastic collapses. To the best of our knowledge, this is
the first study of this sort in the literature.

###  3.1 Definition of the inelastic collapse

######  Definition 5 (Inelastic collapse).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three particles 0, 1 and 2 evolving according to the
$r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, with $\tau^{*}>0$.
We say that the system undergoes an inelastic collapse at the time $\tau^{*}$,
called the _time of the collapse_ , or the _collapsing time_ , if there exists
an increasing sequence of positive times $(t_{n})_{n\in\mathbb{N}}$, with
$t_{0}=0$, such that $t_{n}\xrightarrow[n\rightarrow+\infty]{}\tau^{*}$, when
the times $t_{n}$ correspond exactly to the times of collisions between the
particles, and such that $\sup_{n}t_{n}$ is the only accumulation point of the
sequence $\left(t_{n}\right)_{n\in\mathbb{N}}$.
In particular, for any $t\in\ ]t_{n},t_{n+1}[$, the system of particles
evolves according to the free flow.

######  Remark 4.

It is important here to notice that we considered only _well-defined_
trajectories of particles. In particular, we implicitly assume that no triple
collision takes place on the time interval $[0,\tau^{*}[$, nor grazing
collision. We also assumed that $\tau^{*}$ is the _first_ time of inelastic
collapse, by requiring that $\tau^{*}$ is the only accumulation point of times
of collision. Under the last assumption, requiring that the system does not
present triple of grazing collisions does not harm: such events, obtained
after a finite number of collisions, correspond to a set of initial data of
measure zero.

###  3.2 Converging quantities and vanishing quantities

We now dedicate ourselves to establish general properties, holding for all
kinds of collapsing systems of particles.

####  3.2.1 Convergence of the distances

In what follows, it will be convenient to consider the time intervals between
two collisions. In particular, we will denote $\tau_{n}=t_{n}-\tau_{n-1}$ the
times between two consecutive collisions. With these notations, the times
$\tau_{n}$ correspond to the time $\tau$ that is used in (2.48).
There is already a direct consequence of the definition, stated in the
Proposition that follows.

######  Proposition 1 (Convergence of the times between two collisions).

Let $r\in\ ]0,1[$ be a positive number smaller than $1$, and let us consider a
system of three particles 0, 1 and 2 evolving according to the $r$-inelastic
hard sphere flow (2.1.1), (2.3), on a time interval $[0,\tau^{*}[$, and
undergoing an inelastic collapse at time $\tau^{*}>0$.
Then the sequence $\left(\tau_{n}\right)_{n}$ of the intervals
$\tau_{n}=t_{n}-\tau_{n-1}$ between two consecutive collisions is summable,
that is, $\left(\tau_{n}\right)_{n}\in\ \ell^{1}$. In particular, we have:

$\displaystyle\tau_{n}\xrightarrow[n\rightarrow+\infty]{}0.$ (3.1)

Any collision is dissipating a positive amount of kinetic energy. Such a
dissipation was quantified in Lemma 1. In particular, the kinetic energy of
the system remains bounded, as all the velocities of the particles. Therefore,
since the positions of the particles at the times of the collisions $t_{n}$
are given by:

$\displaystyle
x_{i}(t_{n})=x_{i}(0)+\sum_{k=1}^{n}(t_{k}-t_{k-1})v_{i}(t_{k-1}),$

(for $i\in\ \\{1,2,3\\}$, corresponding to the index of the particle we are
considering), we clearly see that

$\displaystyle\left|x_{i}(t_{p})-x_{i}(t_{q})\right|\leq\sum_{k=p+1}^{q}\left|(t_{k}-t_{k-1})v_{i}(t_{k-1})\right|\leq\left(\sup_{n\in\mathbb{N}}|v_{i}(t_{n})|\right)\underbrace{\sum_{n\geq
p+1}\tau_{n}}_{\xrightarrow[p\rightarrow+\infty]{}0}$

for $p<q$, providing that the sequence of positions of the particles are
Cauchy sequences, hence converging at the time $\tau^{*}$ of the inelastic
collapse. As an immediate consequence, the directions between the particles
are also converging at the time $\tau^{*}$ of the inelastic collapse.
In summary, we can state the following result.

######  Proposition 2 (Convergence of the angular parameters $\omega$ and the
distance between colliding particles).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing an inelastic collapse at time $\tau^{*}>0$,
such that the particle 0 is implied in infinitely many collisions with the two
particles 1 and 2.
Then the sequences of the angular parameters
$\left(\omega_{1,n}\right)_{n\in\mathbb{N}}$ and
$\left(\omega_{2,n}\right)_{n\in\mathbb{N}}$, defined as
$\omega_{i,n}=x_{i}(t_{n})-x_{0}(t_{n})$ (for $i\in\ \\{1,2\\}$) and the
sequence of the distances $\left(d_{n}\right)_{n\in\ \mathbb{N}}$ between the
central particle 0 and the spectator particle are converging, and we have:

$\displaystyle d_{n}\xrightarrow[n\rightarrow+\infty]{}0.$ (3.2)

The limits of the sequences $\left(\omega_{1,n}\right)_{n\in\mathbb{N}}$ and
$\left(\omega_{2,n}\right)_{n\in\mathbb{N}}$ will be respectively denoted by
$\overline{\omega}_{1}$ and $\overline{\omega}_{2}$.

######  Remark 5.

The assumption in the previous proposition concerning the particle 0 is
necessary in order to be sure that the two pairs 0\- 1 and 0\- 2 are involved
in infinitely many collisions. Under this assumption, it is not possible to
deduce, a priori, anything about the number of collisions between the two
particles 1 and 2.
Let us also note that, up to rename the particles, such an assumption is
equivalent to assume that an infinite number of collisions take place: in this
case at least two pairs are involved in infinitely many collisions, so that at
least one particle is involved in infinitely many collisions.

We have then a clear geometrical image of the system at the final time
$\tau^{*}$ of the inelastic collapse: the pairs of particles that underwent an
infinite number of collisions are in contact.
It remains to study the limiting velocities at the time of collapse of such a
system of particles.

####  3.2.2 Convergence of the velocities

We can already establish, in full generality, the result that follows. Let us
start with introducing a convenient description of the order of collisions for
our purpose.
When the system of three inelastic particles 0, 1 and 2 experiences a collapse
at time $\tau^{*}>0$, infinitely many collisions take place, at the respective
times $t_{n}$, such that $t_{n}$ converges towards $\tau^{*}$. Each of these
collisions is either of type 0\- 1, 0\- 2 or 1\- 2. Therefore, considering any
of the three pairs $(i,j)=(0,1),(0,2)$ or $(1,2)$, we can associate a strictly
increasing function:

$\displaystyle\varphi^{(i,j)}:\mathbb{N}^{*}\rightarrow\mathbb{N}^{*}$

(or $\varphi^{(i,j)}:\\{1,\dots,N\\}\rightarrow\mathbb{N}^{*}$ if the system
experiences only $N$ collisions of type _ i\- j_), where $\varphi^{(i,j)}(n)$
is the index of the $n$-th collision of type _ i\- j_. Of course, the images
of the functions $\varphi^{(i,j)}$, for the three pairs $(i,j)$, form a
partition of $\mathbb{N}^{*}$. An important notion is the following.

######  Definition 6 (Counting collision function, maximal gaps between
collisions of type $(i,j)$).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing an inelastic collapse at time $\tau^{*}>0$.
For any of the pairs $(i,j)\in\ \left\\{(0,1),(0,2),(1,2)\right\\}$, we
consider the function
$\varphi^{(i,j)}:\mathbb{N}^{*}\rightarrow\mathbb{N}^{*}$ or
$\varphi^{(i,j)}:\\{1,\dots,N\\}\rightarrow\mathbb{N}^{*}$, where $N$ is the
number of collisions of type i\- j experienced by the system of particles
(with the convention that $\\{1,\dots,N\\}=\emptyset$ if $N=0$), and where
$\varphi^{(i,j)}(n)$ is defined as:

$\displaystyle\varphi^{(i,j)}(n)=k\ \forall n\in\ \mathbb{N}^{*}(\text{or
}n\in\ \\{1,\dots,N\\}),$ (3.3)

where $k$ is the index (counting all the collisions of the three possible
types) of the $n$-th collision of type i\- j. The function $\varphi^{(i,j)}$
is called the _counting collision function of type $i$\- $j$_.
In particular, $\varphi^{(i,j)}$ is a strictly increasing function.
Let $\varphi^{(i,j)}:\mathbb{N}^{*}\rightarrow\mathbb{N}^{*}$ or
$\varphi^{(i,j)}:\\{1,\dots,N\\}\rightarrow\mathbb{N}^{*}$ be the counting
collision function of type i\- j of the collapsing system of particles 0, 1
and 2. We call the _maximal gap between collisions of type $(i,j)$_, or the
_maximal gap of $\varphi^{(i,j)}$_, the positive number:

$\displaystyle\sup_{n\in\mathbb{N}^{*}}\left[\varphi^{(i,j)}(n+1)-\varphi^{(i,j)}(n)\right]\text{
or
}\max_{n\in\\{1,\dots,N-1\\}}\left[\varphi^{(i,j)}(n+1)-\varphi^{(i,j)}(n)\right],$
(3.4)

that we will denote by $K_{(i,j)}$.
For a system of collapsing particles, we say that the system has a _finite
maximal gap of type $(i,j)$_ if the counting collision function
$\varphi^{(i,j)}$ has a finite maximal gap.
Finally, we say that the collapsing system of particles _has finite maximal
gaps between collisions_ if all of its counting functions have a finite
maximal gap.

For example, if we consider the sequence of collisions: 0\- 1, 1\- 2, 0\- 1,
0\- 2, 0\- 1, 1\- 2, 0\- 1, 1\- 2, 0\- 1, 0\- 2, 0\- 1… and which goes on with
infinitely many repetitions of the period 0\- 2, 0\- 1, the counting functions
$\varphi^{(i,j)}$ are:

$\displaystyle\varphi^{(0,1)}(k)=1,3,5,7,9,11,\dots\forall
k\in\mathbb{N}^{*},\hskip 8.53581pt\varphi^{(0,2)}(k)=4,10,12,14,\dots\forall
k\in\mathbb{N}^{*},\varphi^{(1,2)}(k)=2,6,8\text{ for }k=1,2\text{ or }3,$

and the maximal gaps, respectively, between the collisions of type $(0,1)$,
$(0,2)$ and $(1,2)$, are $2$, $6$ and $4$.
In what follows, we will start with restricting ourselves to study collapsing
systems of particles with finite maximal gaps. But prescribing assumptions on
the maximal number of collisions of a certain type, we will actually show in
the two next sections that any collapsing system of three particles has finite
maximal gaps.

######  Proposition 3 (Summability and convergence of the normal components
$\eta$ of the relative velocities).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing an inelastic collapse at time $\tau^{*}>0$. Let
us also assume that, for some $(i,j)\in\ \left\\{(0,1),(0,2),(1,2)\right\\}$,
the pair of particles i\- j collides infinitely many times, and that the
system has a finite maximal gap of type $(i,j)$. Let us denote by
$\eta_{(i,j),n}$ the normal component of the relative velocity between the
particles i and j just before the $n$-th collision.
Then the sequences of normal components $\left(\eta_{(i,j),n}\right)_{n}$ of
the relative velocities are square summable, that is
$\left(\eta_{(i,j),n}\right)_{n}\in\ \ell^{2}$. In particular, for the pairs
of particles i\- j that collide infinitely many times, we have:

$\displaystyle\eta_{(i,j),n}\xrightarrow[n\rightarrow+\infty]{}0.$ (3.5)

###### Proof.

The result of Proposition 3 is a direct consequence of the quantification
(2.7) of the dissipation of the kinetic energy.
Indeed, let $(i_{p},j_{p})$ be the type of the $p$-th collision of the system.
Since the system can dissipate only a finite amount of kinetic energy, the
square of the normal component $\eta_{i_{p},j_{p},p}$ of the relative velocity
between the particles $i_{p}$ and $j_{p}$ at the time $t_{p}$ of this $p$-th
collision cannot be larger than any given constant, more than a finite number
of times, and even more, the sequence of these squares has to be summable,
that is:

$\displaystyle\sum_{p=1}^{+\infty}|\eta_{(i_{p},j_{p}),p}|^{2}<+\infty.$ (3.6)

Now, the order of the collisions being not prescribed, it is not clear a
priori how the normal component of the relative velocity of a prescribed pair
$(i,j)$ is subject to the same regularity described by (3.6) for
$\eta_{i_{p},j_{p}}$. But for a prescribed pair $(i,j)$, either the $p$-th
collision is such that the two particles $i$ and $j$ are involved in such a
collision, or only one of these two particles is colliding, say particle $i$,
while particle $j$ is not colliding at time $t_{p}$.
In the first case ( $i$ and $j$ colliding), we have:

$\displaystyle\eta_{(i,j),p}=\eta_{(i_{p},j_{p}),p},$ (3.7)

where $\left(\eta_{(i_{p},j_{p}),p}\right)_{p}$ is a square summable sequence.
In the second case ( $i$ colliding, $j$ not colliding), we have:

$\displaystyle
v_{j}^{\prime}-v_{i}^{\prime}=v_{j}(t_{p}^{-})-v_{i}(t_{p}^{-})+\frac{(1+r)}{2}\left(v_{i}(t_{p}^{-})-v_{k}(t_{p}^{-})\right)\cdot\omega_{(i,k),p}\omega_{(i,k),p},$

where $k\neq i,j$ is the second particle colliding, $v_{i}^{\prime}$ and
$v_{j}^{\prime}$ are the post-collisional velocities of the particles $i$ and
$j$, that is, the velocities just after the $p$-th collision, and
$\omega_{(i,k),p}=x_{i}(t_{p})-x_{k}(t_{p})$ (of norm $1$ at $t_{p}$, since
$i$ and $k$ are in contact at $t_{p}$). Therefore, we find:

$\displaystyle\eta_{(i,k),p}^{\prime}=\eta_{(i,k),p+1}=\eta_{(i,k),p}+\frac{(1+r)}{2}\eta_{(i_{p},j_{p}),p}\omega_{(i_{p},j_{p}),p}\cdot\omega_{(i,k),p},$
(3.8)

so, in all the cases, at time $t_{p}^{+}$, either we have:

$\displaystyle\eta_{(i,j),n+1}=a_{n}$ (3.9)

or we have

$\displaystyle\eta_{(i,j),n+1}=\eta_{(i,j),n}+\alpha_{n}a_{n},$ (3.10)

where $\left(a_{n}\right)_{n}$ and $\left(\alpha_{n}\right)_{n}$ are two
sequences of real numbers such that

$\displaystyle\sum_{n\geq 0}a_{n}^{2}<+\infty\hskip 14.22636pt\text{and}\hskip
14.22636pt0\leq\alpha_{n}\leq\alpha_{0}<1\hskip 8.53581pt\forall n\in\
\mathbb{N}^{*}.$ (3.11)

Now, by definition of the counting collision function (see Definition 6
above), (3.9) holds if and only if $p=\varphi^{(i,j)}(n)$, and by assumption
the system has a finite maximal gap of type $(i,j)$, that is, there exists
$K_{(i,j)}\in\ \mathbb{N}^{*}$ such that

$\displaystyle 0\leq\varphi^{(i,j)}(n+1)-\varphi^{(i,j)}(n)\leq K_{(i,j)}.$

Therefore, we can write:

$\displaystyle\sum_{n\geq 0}\eta_{(i,j),n}^{2}$ $\displaystyle=\sum_{n\geq
0}\left[\eta_{(i,j),\varphi^{(i,j)}(n)}^{2}+\sum_{k=\varphi^{(i,j)}(n)+1}^{\varphi^{(i,j)}(n+1)-1}\eta_{(i,j),k}^{2}\right]+\sum_{n=0}^{\varphi^{(i,j)}(0)-1}\eta_{(i,j),n}^{2}$
$\displaystyle=\sum_{n\geq
0}\left[a^{2}_{\varphi^{(i,j)}(n)-1}+\sum_{k=1}^{\varphi^{(i,j)}(n+1)-\varphi^{(i,j)}(n)-1}a^{2}_{\varphi^{(i,j)}(n)-1}(1+k\alpha_{n})^{2}\right]+\sum_{n=0}^{\varphi^{(i,j)}(0)-1}\eta_{(i,j),n}^{2}$
$\displaystyle\leq\sum_{n\geq
0}\left(\sum_{k=0}^{K_{i,j}-1}(1+k\alpha_{n})^{2}\right)a^{2}_{\varphi^{(i,j)}(n)-1}\leq\left(\sum_{k=0}^{K_{i,j}-1}(1+k\alpha_{n})^{2}\right)\left(\sum_{n\geq
0}a^{2}_{n}\right)<+\infty,$

which concludes the proof of the proposition. ∎

###  3.3 Asymptotic behaviour of the variables

Let us carry on the study of general collapsing systems of three particles.
The purpose of this section is to compare the asymptotic behaviours of the
different variables of the dynamical system (2.48) when an inelastic collapse
takes place. These additional results will be useful in order to identify the
leading order terms in the (quite complicated) expression (2.48) of the
dynamical system.
Let us emphasize that in this section neither, we will not assume any order on
the sequence of the collisions.

Around the regime of the collapse, the variables $\tau$ and $\eta_{i}$ (for
the pairs of particles involved in infinitely many collisions) are vanishing.
We will then compare the decay of the vanishing variables, namely:

$\displaystyle d,d^{\prime},\eta_{1}^{\prime}\text{ and
}\eta_{2}^{\prime},\left(\omega_{1}^{\prime}-\omega_{1}\right),\left(\omega_{2}^{\prime}-\omega_{2}\right),\left(\left(W_{1}^{\perp}\right)^{\prime}-W_{1}^{\perp}\right)\text{
and }\left(\left(W_{2}^{\perp}\right)^{\prime}-W_{2}^{\perp}\right)$

with the reference variables $\tau$ and $\eta$. Let us start with introducing
some useful notations.

####  3.3.1 Notations

In the rest of this section, concerning a variable $y$, we will denote by

$\displaystyle y=\mathcal{O}\left(x_{1},\dots,x_{n}\right)$

when this variable $y$ is bounded by the $n$ other variables
$x_{1},\dots,x_{n}$, that is, if there exist $n$ positive constants
$C_{1},\dots,C_{n}>0$ such that $|y|\leq C_{1}|x_{1}|+\dots C_{n}|x_{n}|$. If
the two variables $y$ and $z$ are bounded by each other, that is, if there
exist two positive constants $C_{1},C_{2}>0$ such that $|y|\leq C_{1}|z|\hskip
14.22636pt$ and $|z|\leq C_{2}|y|$, such a situation will be denoted by

$\displaystyle y\lessgtr z.$

In the same way, we will denote by

$\displaystyle y=o\left(x_{1},\dots,x_{n}\right)$

when the variable $y$ is negligible with respect to the $n$ variables
$x_{1},\dots,x_{n}$, that is, if there exist $n$ functions
$\varphi_{1},\dots,\varphi_{n}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$
vanishing at $0$ and such that
$|y|\leq\varphi_{1}\left(|x_{1}|\right)+\dots+\varphi_{n}\left(|x_{n}|\right)$.

####  3.3.2 Comparison of the variables I: the general case

In the case of a general collapsing system of three collapsing, we have the
following result.

######  Proposition 4 (Asymptotic comparison of the variables in the
collapsing regime).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing an inelastic collapse at time $\tau^{*}>0$. Let
us assume that the pair of consecutive collisions 0\- 1, 0\- 2 takes places
infinitely many times, and that the system has finite maximal gaps of types
$(0,1)$ and $(0,2)$. Let us denote by $t_{\varphi(n)}$ the different times
when a collision of type 0\- 1 takes place, and when such a collision is
immediately followed at times $t_{\varphi(n)+1}=t_{\varphi(n)}+\tau_{n}$ by a
collision of type 0\- 2. At time $t_{\varphi(n)}$, let us denote by $d_{n}$
the distance between the particles 0 and 2, by $\omega_{1,n}$ and
$\omega_{2,n}$ the respective angular parameters between the pairs 0 and 1,
and 0 and 2 respectively, by $\eta_{1,n}$ and $\eta_{2,n}$ the respective
normal components of the relative velocities, and by $W_{1,n}^{\perp}$ and
$W_{2,n}^{\perp}$ the respective tangential components of the relative
components of the pairs 0 and 1, and 0 and 2 respectively. At time
$t_{\varphi(n)+1}=t_{\varphi(n)}+\tau_{n}$, let us denote by $d^{\prime}_{n}$
the distance between the particles 0 and 1, and by $\omega_{1,n}^{\prime}$ and
$\omega_{2,n}^{\prime}$ the angular parameters, by $\eta_{1,n}^{\prime}$ and
$\eta_{2,n}^{\prime}$ the normal components, and by
$(W_{1,n}^{\perp})^{\prime}$ and $(W_{2,n}^{\perp})^{\prime}$ the respective
tangential components of the relative components of the pairs 0 and 1, and 0
and 2 respectively.
Then, we have the following asymptotic relations, as $n\rightarrow+\infty$:

$\displaystyle\tau_{n}\lessgtr\frac{d_{n}}{(-\eta_{2,n})},\hskip
1.42262pt\text{which implies:}\hskip
5.69054ptd_{n}=\mathcal{O}\left(\tau_{n}(-\eta_{2,n})\right),\hskip
1.42262pt\text{and in particular:}\hskip 5.69054ptd_{n}=o(\tau_{n})\text{ and
}d_{n}=o(-\eta_{2,n}),$ (3.12) $\displaystyle
d^{\prime}_{n}=\mathcal{O}\left(\eta_{1,n}\tau_{n},\tau_{n}^{2}|W_{1,n}^{\perp}|^{2}\right)=\mathcal{O}\left(\tau_{n}\left(\eta_{1,n}+\tau_{n}\right)\right)=o(\tau_{n}),$
(3.13)
$\displaystyle|\eta_{1,n}^{\prime}|=\mathcal{O}\left(\eta_{1,n},-\eta_{2,n},\tau_{n}|W_{1,n}^{\perp}|^{2}\right),\hskip
2.84526pt\text{and}\hskip
14.22636pt\eta_{2,n}^{\prime}=\mathcal{O}(-\eta_{2,n}),$ (3.14)
$\displaystyle\left|\omega_{1,n}^{\prime}-\omega_{1,n}\right|=\mathcal{O}\left(\tau_{n}\eta_{1,n},\tau_{n}|W_{1,n}^{\perp}|\right),\hskip
2.84526pt\text{and}\hskip
14.22636pt\left|\omega_{2,n}^{\prime}-\omega_{2,n}\right|=\mathcal{O}\left(\tau_{n}(-\eta_{2,n}),\tau_{n}|W_{2,n}^{\perp}|\right)$
(3.15)

and finally:

$\displaystyle\left|(W_{1,n}^{\perp})^{\prime}-W_{1,n}^{\perp}\right|$
$\displaystyle=\mathcal{O}\left(\tau_{n}\eta_{1,n}^{2},\tau_{n}^{2}\eta_{1,n}|W_{1,n}^{\perp}|,\tau_{n}|W_{1,n}^{\perp}|^{2},-\eta_{2,n}\right),$
$\displaystyle\hskip 85.35826pt\text{and}\hskip
5.69054pt\left|(W_{2,n}^{\perp})^{\prime}-W_{2,n}^{\perp}\right|=\mathcal{O}\left(\tau_{n}\eta^{2}_{2,n},\tau_{n}|W_{2,n}^{\perp}|^{2}\right).$
(3.16)

In addition, the relations:

$\displaystyle|W_{2,n}^{\perp}|=o(-\eta_{2,n})\hskip 5.69054pt\text{or}\hskip
5.69054pt|W_{2,n}^{\perp}|=\mathcal{O}(-\eta_{2,n})$ (3.17)

cannot hold.

###### Proof.

All the results we will obtain are consequences of the equations (2.48). Let
us observe that the assumption stating that the system has finite maximal gaps
of types $(0,1)$ and $(0,2)$ ensures that the two sequences
$\left(\eta_{1,n}\right)_{n}$ and $\left(\eta_{2,n}\right)_{n}$ are both
vanishing as $n$ goes to infinity.
Let us start with the general conditions relative to the collision time
$\tau_{n}$. First, this time $\tau_{n}$ has to vanish at the limiting time of
the collapse, so that either

$\displaystyle\frac{(-\eta_{2,n})}{|W_{2,n}|^{2}}=\frac{(-\eta_{2,n})}{\eta_{2,n}^{2}+|W_{2,n}^{\perp}|^{2}}{\xrightarrow[n\rightarrow+\infty]{}0},$
(3.18)

or

$\displaystyle 1-\sqrt{1-\zeta_{n}}{\xrightarrow[n\rightarrow+\infty]{}0},$
(3.19)

that is $\zeta_{n}{\xrightarrow[n\rightarrow+\infty]{}0}$, where
$\zeta_{n}=\displaystyle{\frac{d_{n}\left(2+d_{n}\right)\left|W_{2,n}\right|^{2}}{\left(1+d_{n}\right)^{2}\eta_{2,n}^{2}}}\cdotp$
If (3.18) holds, we can deduce that (3.17) cannot hold. Besides, $\zeta_{n}$
has always to be smaller than $1$, so for $n$ large enough
$d_{n}\leq\eta_{2,n}^{2}/(2|W_{2,n}|^{2})$. Since $d_{n}$ vanishes
asymptotically, we recover that $|W_{2,n}^{\perp}|=o(-\eta_{2,n})$ or
$|W_{2,n}^{\perp}|=\mathcal{O}(-\eta_{2,n})$ cannot hold. The estimate (3.17)
is proved in full generality.
Since for an argument $x\in\ [0,1]$ we have always $x/2\leq 1-\sqrt{1-x}\leq
x$, we deduce that we have

$\displaystyle\frac{(1+d_{n})(-\eta_{2,n})}{|W_{2,n}|^{2}}\cdot\frac{d_{n}(d_{n}+2)|W_{2,n}|^{2}}{2(1+d_{n})^{2}\eta_{2,n}^{2}}=\frac{d_{n}(2+d_{n})}{2(1+d_{n})(-\eta_{2,n})}\leq\tau_{n}\leq\frac{d_{n}(2+d_{n})}{(1+d_{n})(-\eta_{2,n})}\cdotp$
(3.20)

In particular, in the regime of the collapse we deduce (3.12).
We can now turn to the consequence of the evolution equations (2.48).
Concerning the distance $d^{\prime}_{n}$ between 0 and 1 at the time
$t_{\varphi(n)}+\tau_{n}$ of collision between 0 and 2, we have already:

$\displaystyle
d^{\prime}_{n}=\eta_{1,n}\tau_{n}+\frac{1}{2}|W_{1,n}|^{2}\tau_{n}^{2}+\mathcal{O}\left(\left(\eta_{1,n}\tau_{n}+|W_{1,n}|^{2}\tau_{n}^{2}\right)^{2}\right).$

As a direct consequence of the fact that $|W_{1}|$ remains bounded and that
$\eta_{1}$ is vanishing, we obtain (3.13).
Concerning now the normal components of the relative velocities, we find:

$\displaystyle|\eta_{1,n}^{\prime}|=\mathcal{O}\left(\eta_{1,n},-\eta_{2,n},\tau_{n}|W_{1,n}|^{2},\tau_{n}|W_{2,n}|^{2}\right)$
(3.21)

and

$\displaystyle|\eta_{2,n}^{\prime}|=\mathcal{O}\left(-\eta_{2,n},\tau_{n}|W_{2,n}|^{2}\right).$
(3.22)

Keeping in mind the expression (2.20) of the collision time $\tau_{n}$, we can
replace $\tau_{n}|W_{2,n}|^{2}$ by
$\mathcal{O}\left((-\eta_{2,n})\zeta_{n}\right)$ and then by
$\mathcal{O}(\eta_{2,n})$, because $\zeta_{n}$ is always smaller than $1$, so
that we can rewrite:

$\displaystyle|\eta_{1,n}^{\prime}|=\mathcal{O}\left(\eta_{1,n},-\eta_{2,n},\tau_{n}|W_{1,n}|^{2}\right)$
(3.23)

and

$\displaystyle|\eta_{2,n}^{\prime}|=\mathcal{O}\left(-\eta_{2,n}\right),$
(3.24)

so that (3.14) is proved.
Let us now turn to the difference of the converging variables. Concerning the
first angular parameters, we have
$\omega_{1,n}^{\prime}-\omega_{1,n}=-d_{n}^{\prime}\omega_{1,n}/(1+d_{n}^{\prime})+\tau_{n}W_{1,n}/(1+d_{n}^{\prime})$
so that
$\omega_{1,n}^{\prime}-\omega_{1,n}=\mathcal{O}\left(d_{n}^{\prime},\tau_{n}W_{1,n}\right)$,
which, combined with (3.13), provides

$\displaystyle\omega_{1,n}^{\prime}-\omega_{1,n}=\mathcal{O}\left(\tau_{n}\eta_{1,n},\tau_{n}|W_{1,n}^{\perp}|\right).$

In the same way, for the other angular parameter we find
$\omega_{2,n}^{\prime}-\omega_{2,n}=d_{n}\omega_{2,n}+\tau_{n}W_{2,n}$, so
that

$\displaystyle\omega_{2,n}^{\prime}-\omega_{2,n}=\mathcal{O}\left(d_{n},|\tau_{n}W_{2,n}|\right)=\mathcal{O}\left(\tau_{n}(-\eta_{2,n}),\tau_{n}|W_{2,n}^{\perp}|\right),$
(3.25)

and (3.15) is proved.
Concerning the differences between the consecutive tangential components of
the relative velocities, we obtain:

$\displaystyle\left(W_{1,n}^{\perp}\right)^{\prime}-W_{1,n}^{\perp}$
$\displaystyle=\eta_{1,n}\omega_{1,n}-\frac{(\eta_{1,n}+\tau_{n}|W_{1,n}|^{2})}{(1+d_{n}^{\prime})}\omega_{1,n}^{\prime}$
$\displaystyle\hskip
14.22636pt+\frac{(1+r)}{2}\left((1+d_{n})\eta_{2,n}+\tau_{n}|W_{2,n}|^{2}\right)\left[(\omega_{1,n}^{\prime}\cdot\omega_{2,n}^{\prime})\omega_{1,n}^{\prime}-\omega_{2,n}^{\prime}\right]$

so that
$\left(W_{1,n}^{\perp}\right)^{\prime}-W_{1,n}^{\perp}=\mathcal{O}\left(\eta_{1,n}(\omega_{1,n}^{\prime}-\omega_{1,n}),\eta_{1,n}d_{n}^{\prime},\tau_{n}|W_{1,n}|^{2},-\eta_{2,n},\tau_{n}|W_{2,n}|^{2}\right)$,
which can be simplified using (3.15), (3.13) and the explicit expression
(2.20) of $\tau_{n}$ as

$\displaystyle\left(W_{1,n}^{\perp}\right)^{\prime}-W_{1,n}^{\perp}$
$\displaystyle=\mathcal{O}\left(\tau_{n}\eta_{1,n}^{2},\tau_{n}^{2}\eta_{1,n}|W_{1,n}^{\perp}|,\tau_{n}^{2}\eta_{1,n}|W_{1,n}^{\perp}|^{2},\tau_{n}|W_{1,n}|^{2},-\eta_{2,n}\right)$
$\displaystyle=\mathcal{O}\left(\tau_{n}\eta_{1,n}^{2},\tau_{n}^{2}\eta_{1,n}|W_{1,n}^{\perp}|,\tau_{n}|W_{1,n}^{\perp}|^{2},-\eta_{2,n}\right),$
(3.26)

and from
$\left(W_{2,n}^{\perp}\right)^{\prime}-W_{2,n}^{\perp}=\eta_{2,n}\omega_{2,n}-\left((1+d_{n})\eta_{2,n}+\tau_{n}|W_{2,n}|^{2}\right)\left((1+d_{n})\omega_{2,n}+\tau_{n}W_{2,n}\right)$
we find

$\displaystyle\left(W_{2,n}^{\perp}\right)^{\prime}-W_{2,n}^{\perp}=\mathcal{O}\left(d_{n}(-\eta_{2,n}),\tau_{n}|W_{2,n}|^{2}\right)=\mathcal{O}\left(\tau_{n}\eta_{2,n}^{2},\tau_{n}|W_{2,n}^{\perp}|^{2}\right),$
(3.27)

which completes the proof of (4), and of Proposition 4. ∎

######  Remark 6.

Let us observe that we could also study in the same way the second order terms
of the normal components of the relative velocities. To be more explicit, we
have:

$\displaystyle\eta_{1,n}^{\prime}-\eta_{1,n}-\frac{(1+r)}{2}\cos\overline{\theta}\,\eta_{2,n}=\mathcal{O}\left(d_{n}^{\prime}\eta_{1,n},\tau_{n}|W_{1,n}|^{2},d_{n}\eta_{2,n},\tau_{n}|W_{2,n}|^{2}\right)=\mathcal{O}\left(\tau_{n}\eta_{1,n}^{2},\tau_{n}|W_{1,n}^{\perp}|^{2},-\eta_{2,n}\right)$

where $\overline{\theta}$ is the limiting angle between the particles 1 and 2,
measured from the particle 0, and

$\displaystyle\eta_{2,n}^{\prime}-r\eta_{2,n}=\mathcal{O}\left(d_{n}\eta_{2,n},\tau_{n}|W_{2,n}|^{2}\right).$

As a general comment concerning the estimates obtained in Proposition 4, let
us emphasize that all the variables are bounded only in terms of the normal
components $\eta_{1}$ and $\eta_{2}$, and the time of collision $\tau$.
Therefore, understanding the asymptotic behaviour of these last variables will
enable to understand completely the full dynamical system. On the other hand,
it does not seem possible to compare, in full generality, the asymptotic
behaviour of the normal components $\eta$ with the time of collision $\tau$.
In other words, a natural problem appears: we need to compare the asymptotics
of the variables $\eta$ and $\tau$, that govern completely the evolution
equations (2.48).

####  3.3.3 Comparison of the variables II: the generic case, with non
vanishing tangential velocities

Let us now revisit the inequalities comparing the asymptotics of the variables
obtained in Proposition 4, considering that none of the tangential velocities
vanish asymptotically. To be more accurate, we will assume that the norms of
the relative velocities are bounded from below by a positive constant, say
$\overline{w}>0$, after a sufficiently large number of collisions.
A first consequence of the assumption $|W_{2,n}^{\perp}|\geq\overline{w}>0$,
and perhaps the most important, is that in this case the Zhou-Kadanoff
parameter $\zeta_{n}$ satisfies

$\displaystyle\zeta_{n}\lessgtr\frac{d_{n}}{\eta_{2,n}^{2}}\cdotp$ (3.28)

Therefore, since in order to have a collapse this parameter has always to
remain below $1$, we deduce that

$\displaystyle d_{n}=\mathcal{O}(\eta_{2,n}^{2}),$ (3.29)

which refines the estimate $d_{n}=o\left(-\eta_{2,n}\right)$ of (3.12). Now,
considering again the formula (2.20) of the time of collision $\tau_{n}$, we
find:

$\displaystyle\tau_{n}\leq\frac{2d_{n}}{(-\eta_{2,n})}=\underbrace{\frac{2d_{n}}{\eta_{2,n}^{2}}}_{\text{bounded}}\cdot(-\eta_{2,n}),$
(3.30)

so that in particular we have the important estimate:

$\displaystyle\tau_{n}=\mathcal{O}(\eta_{2,n}).$ (3.31)

As a consequence, in the “generic” case
$|W_{1,n}|,|W_{2,n}|\geq\overline{w}>0$, we can assert that the time of
collision $\tau_{n}$ is bounded by the normal components $\eta_{n}$, and
therefore, when the relative velocities do not vanish at the final time of the
collapse, only the normal components of the relative velocities govern the
evolution of the dynamical system (2.48).

######  Proposition 5 (Asymptotic comparison of the variables in the
collapsing regime, assuming that the relative velocities do not vanish).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing an inelastic collapse at time $\tau^{*}>0$. Let
us assume that the pair of consecutive collisions 0\- 1, 0\- 2 takes places
infinitely many times, and that the system has finite maximal gaps of types
$(0,1)$ and $(0,2)$. Let us assume also that there exists $\overline{w}>0$
such that $|W_{2,n}|\geq\overline{w}$ for all $n$ large enough.
Then, with the same notations as in Proposition 4, we have the following
asymptotic relations, as $n\rightarrow+\infty$:

$\displaystyle\tau_{n}=\mathcal{O}(-\eta_{2,n}),\hskip
8.53581pt\text{and}\hskip 14.22636ptd_{n}=\mathcal{O}(\eta_{2,n}^{2}),$ (3.32)
$\displaystyle d^{\prime}_{n}=\mathcal{O}\left(\eta_{1,n}\tau_{n}\right),$
(3.33)
$\displaystyle|\eta_{1,n}^{\prime}|=\mathcal{O}\left(\eta_{1,n},-\eta_{2,n}\right),\hskip
8.53581pt\text{and}\hskip
14.22636pt\eta_{2,n}^{\prime}=\mathcal{O}(-\eta_{2,n}),$ (3.34)
$\displaystyle\left|\omega_{1,n}^{\prime}-\omega_{1,n}\right|=\mathcal{O}\left(\tau_{n}\right),\hskip
8.53581pt\text{and}\hskip
14.22636pt\left|\omega_{2,n}^{\prime}-\omega_{2,n}\right|=\mathcal{O}\left(\tau_{n}\right)$
(3.35)

and finally:

$\displaystyle\left|(W_{1,n}^{\perp})^{\prime}-W_{1,n}^{\perp}\right|$
$\displaystyle=\mathcal{O}\left(-\eta_{2,n}\right),\hskip
8.53581pt\text{and}\hskip
14.22636pt\left|(W_{2,n}^{\perp})^{\prime}-W_{2,n}^{\perp}\right|=\mathcal{O}\left(\tau_{n}\right).$
(3.36)

######  Remark 7.

Even under the assumption on the positive bound from below for the norm of the
relative velocities, it seems that one cannot conclude a priori that $\tau$
will be eventually negligible with respect to the normal components: it might
perfectly remain of the same order asymptotically.
Note also that, in order to establish the estimates of Proposition 5, we
needed _only_ the information that the _second_ relative velocity $|W_{2}|$
does not vanish, which is not surprising, since the equations (2.48) are
describing a collision of type 0\- 2.

####  3.3.4 Comparison of the variables III: the consequences of the regime
$\tau\ll\eta$

In this section, let us make a formal observation concerning a further
simplification of the full dynamical dynamical. We observed in Proposition 5
that the regime $\tau=\mathcal{O}(\eta)$ provides much simpler estimates on
the variables of the dynamical system (2.48) than in the general case,
addressed in Proposition 4. In this section, let us assume $\tau\ll\eta$.
Actually, this setting is also the one studied by Zhou and Kadanoff in [27].
These authors call this regime the “flat surface approximation”.
So, considering (2.48), if the time $\tau$ of the next collision becomes small
with respect to the normal components $\eta_{1}$ and $\eta_{2}$ of the
relative velocities, we find:

$\displaystyle\eta_{1}^{\prime}\simeq\frac{1}{(1+d^{\prime})}\eta_{1}-\frac{(1+r)}{2}\left(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}\right)(1+d)\eta_{2},$

neglecting in a first time the terms involving $\tau$ in the sums
$\eta_{1}+\tau|W_{1}|^{2}$ and $(1+d)\eta_{2}+\tau|W_{2}|^{2}$ (of course, the
norms $|W_{1}|$ and $|W_{2}|$ remain bounded along the evolution of the
system), and then keeping only the leading order terms (since $d$ and
$d^{\prime}$ converge to zero in the case of a collapse):

$\displaystyle\eta_{1}^{\prime}\simeq\eta_{1}-\frac{(1+r)}{2}\left(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}\right)\eta_{2},$

and, in the same way, for the second relative velocity:

$\displaystyle\eta_{2}^{\prime}\simeq-r\eta_{2}.$

Considering only the leading order for the cosine
$\left(\omega_{1}^{\prime}\cdot\omega_{2}^{\prime}\right)$, we find in the
end:

$\displaystyle\left\\{\begin{array}[]{rl}\eta_{1}^{\prime}&\simeq\displaystyle{\eta_{1}-\frac{(1+r)}{2}\cos\overline{\theta}\eta_{2}},\\\
\eta_{2}^{\prime}&\simeq-r\eta_{2},\end{array}\right.$ (3.39)

where $\overline{\theta}$ corresponds to the limiting angle between the two
particles 1 and 2 at the final time of the collapse.
In particular, the evolution of the normal components $\eta_{1}$ and
$\eta_{2}$ are completely determined without using any other variables. We
recovered formally, the system studied by Zhou and Kadanoff in [27]. In turn,
since the evolution of all the other variables are driven by the normal
components, we can deduce the complete evolution of the dynamical system
(2.48).

####  3.3.5 Comparison of the variables IV: sufficient condition for the
regime $\tau\ll\eta$

We saw in the previous section that the idealized regime $\tau\ll\eta$ enables
to simplify very much the full, complicated, dynamical system (2.48).
Therefore, a natural and important question consists in determining when such
a regime takes place. It turns out that this is the main difficulty.
Our main observations concerning the expression (2.20) of $\tau$ are the
following: first, in the collapse regime, since in particular $d$ vanishes, we
can write for the Zhou-Kadanoff parameter:

$\displaystyle\zeta=\frac{d(2+d)|W_{2}|^{2}}{(1+d)^{2}\eta_{2}^{2}}\simeq
2\frac{d|W_{2}|^{2}}{\eta_{2}^{2}}\cdotp$

Second, and it is the crucial observation, when the ZK-parameter $\zeta$ is
small, the expression of the collision time $\tau$ becomes:

$\displaystyle\tau\sim\frac{(-\eta_{2})}{|W_{2}|^{2}}\left(\frac{\zeta}{2}\right)\sim\left(\frac{(-\eta_{2})}{|W_{2}|^{2}}\right)\frac{\left(\displaystyle{2\frac{d|W_{2}|^{2}}{\eta_{2}^{2}}}\right)}{2}=\frac{d}{(-\eta_{2})}\cdotp$
(3.40)

Note that we made no assumption concerning bounds from below on $|W_{2}|$
while obtaining (3.40). The relation (3.40) refines the estimate (3.12)
obtained in Proposition 4 on the one hand, and (3.30) on the other hand, for
this time we obtained better than an upper bound on $\tau$. In the regime
$\zeta\rightarrow 0$, the expression of $\tau$ takes then a particularly
simple (approximated) form. Moreover, assuming that the tangential velocities
do not vanish at the time of the collapse, that is, assuming that
$|W_{2}|^{2}$ converges towards a strictly positive limit, we see that the
assumption $\zeta\rightarrow 0$ is equivalent to $d\ll\eta_{2}^{2}$. But then,
in that case, since we found $\tau\simeq d/(-\eta_{2})$, we deduce that:

$\displaystyle\tau\sim\frac{d}{(-\eta_{2})}=(-\eta_{2})\frac{d}{\eta_{2}^{2}}\sim
C(-\eta_{2})\zeta,$

where $C$ is a positive constant, which implies that $\tau$ is negligible with
respect to $\eta_{2}$ in the regime $\zeta\rightarrow 0$. Therefore, the
regime $\zeta\rightarrow 0$ implies the regime $\tau\ll\eta$ briefly studied
in the last section. It is now clear that the estimate of the asymptotic
behaviour of the quantity $\zeta$ is crucial in order to understand the
dynamics of the particles. Then, let us devote a definition to this crucial
regime.

######  Definition 7 (Zhou-Kadanoff regime).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing an inelastic collapse at time $\tau^{*}>0$.
We say that the _Zhou-Kadanoff regime takes place_ (or _ZK-regime_ , in short)
takes place, if the sequence $\left(\zeta_{n}\right)_{n}$ of the $n$-th Zhou-
Kadanoff parameter, defined for the $n$-th collision, for any $n\in\
\mathbb{N}^{*}$, is converging towards $0$, that is, if for any $n\in\
\mathbb{N}^{*}$, considering that the collision is of type $i_{n}$\- $k_{n}$,
we have:

$\displaystyle\zeta_{n}=\frac{d_{n}(2+d_{n})\left|W_{n}\right|^{2}}{(1+d_{n})^{2}\eta_{n}^{2}}{\xrightarrow[n\rightarrow+\infty]{}0},$
(3.41)

where $d_{n}=d_{(i_{n},k_{n}),n}$ is the distance between the particles
$i_{n}$ and $k_{n}$, $W_{2,n}=W_{2,(i_{n},k_{n}),n}$ is the relative velocity
between the particles $i_{n}$ and $k_{n}$, and
$\eta_{n}=\eta_{(i_{n},k_{n}),n}$ is the normal component of this relative
velocity, when the collision $i_{n}$\- $j_{n}$ takes place.

In addition, according to the estimate (3.12), holding in full generality,
since $\tau$ behaves asymptotically as $d/(-\eta_{2})$, the condition
$d/\eta_{2}^{2}\rightarrow 0$ is not only a sufficient condition to have
$\tau\ll\eta_{2}$, but it is also a necessary condition. Therefore, the ratio

$\displaystyle\varphi_{2}=\frac{d}{\eta_{2}^{2}}$

turns out to be a natural measure of the defect from the regime $\tau\ll\eta$.

However, there are several limitations and difficulties: on the one hand, we
will see that the regime $\zeta\rightarrow 0$ is not always true when a
collapse occurs. Another natural question that arises is then: is this regime
at least the only one we can observe, that is, the only one associated to a
set of initial data of positive measure? On the other hand, we proved that
$\tau\ll\eta$ under the assumption that $\zeta\rightarrow 0$ _and_ that
$|W_{2}|^{2}$ is not vanishing at the final time of the collapse. The last
assumption cannot be true in full generality, although one might expect that
it is almost always true. When such an assumption does not hold, it implies in
particular that at least two particles of the system remain attached after the
final time of the collapse. This configuration is preventing to get well-
posedness of the system of particles, and is a major obstruction to obtain an
Alexander theorem for the system of three inelastic particles. As a
consequence, understanding the cases when one or two relative velocities
completely vanish at the final time of the collapse is of particular interest
for the study of collapsing systems of particles.
In the companion paper [10], we study the two-dimensional system obtained by
keeping only the leading order terms of (2.48), using the information of
Proposition 5. In the end, the system reduces in studying only the two
variables $\varphi_{1}=\eta_{1}/(-\eta_{2})$ and $\varphi_{2}$. We investigate
in particular when the ZK-regime (equivalent to $\varphi_{2}\rightarrow 0$ for
non vanishing tangential velocities) takes place. We prove in this paper that
the regime $\varphi_{2}$ is stable in a non trivial region of the plane
$(\varphi_{1},\varphi_{2})$. We also conjecture the existence of a separatrix
in this plane, that enables to characterize the ZK-regime.

##  4 Sequence of collisions involving only two pairs of particles: the
nearly-linear collapse

In this section, we consider a collapsing system of particles colliding
(eventually) with the following sequence of collisions: the infinite
repetitions of the pairs of collisions 0\- 1, 0\- 2. For reasons that will
become clear later in this section, such a collapse will be called a _nearly-
linear collapse_. This is the configuration already investigated in [27]. In
the first part of this section, we perform a rigorous study of the convergence
of the variables of the dynamical system (2.48) in the case of the linear
collapse, which was, to the best of our knowledge, still not done in the
literature. In a second part, we present the results already existing in the
literature, namely, in [27].
Let us note that for a collapsing system of three particles, there are only
two possible cases: either only two pairs are involved in infinitely many
collisions, or the three pairs are all involved in infinitely many collisions.
The latter case is discussed in Section 5. We study in this section the
former.
Finally, let us observe that the present situation covers the one-dimensional
case, which is completely understood for $3$ particles (see [9]). Indeed, if
the particles evolve along a line, the external particles cannot collide with
each other, so if the system experiences infinitely many collisions, it has to
be between the two pairs formed, respectively, by the two external particles,
and the central one. In this work we do not restrict ourselves to the one-
dimensional case, but we will see that the results we can gather concerning
the linear collapse can be seen as a spatial perturbation of the one-
dimensional case, especially when considering the results of [27].

###  4.1 Converging quantities in the case of the nearly-linear collapse

######  Definition 8 (Nearly-linear inelastic collapse).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing an inelastic collapse at time $\tau^{*}>0$.
We say that the system experiences a _nearly-linear collapse_ if the sequence
of collisions becomes eventually the infinite repetition of the pairs of
collisions 0\- 1, 0\- 2.
The particles 0, involved in infinitely collisions with the two other
particles 1 and 2, will be called the _central particle_. The particles 1 and
0, involved in infinitely many collisions only with the particle 0, will be
called, alternatively, the _colliding_ , and the _spectator_ particles.

######  Remark 8.

The nomenclature of Definition 8, concerning the name of the particles
according to their respective roles along the different collisions, is taken
from [27]. Let us note that this definition is not arbitrary, up to rename the
particles.

Let us start with the geometrical description of the nearly-linear collapse,
following as an immediate corollary of Proposition 2.

######  Proposition 6 (Geometry of the nearly-linear collapse).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing a nearly-linear inelastic collapse at time
$\tau^{*}>0$.
Then, the particles 0 and 1, resp. 0 and 2, are in contact at the collapsing
time $\tau^{*}$, that is, we have:

$\displaystyle|\overline{\omega}_{1}|=1\hskip 14.22636pt\text{and}\hskip
14.22636pt|\overline{\omega}_{2}|=1.$ (4.1)

The particles are then in contact in the limit, forming a linear structure, in
the sense that the particles 1\- 0\- 2 form a sort of a queue, possibly
curved, hence the name of this collapse. Such a structure is typical from the
collapsing systems of inelastic particles, as it was first observed by [21].
The interested reader may also consult [24] for more intriguing observations
about this phenomenon.

We can also describe the convergence of the normal components of the relative
velocities. By assumption the nearly-linear collapse has finite maximal gaps
(see Definition 6). In particular, the results of Proposition 3 hold true for
the linear collapse. We can improve this result in the present case and
provide an _explicit_ rate of convergence.

######  Proposition 7 (Convergence of the normal components of the relative
velocities for the nearly-linear collapse).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing a nearly-linear inelastic collapse at time
$\tau^{*}>0$.
Then we have:

$\displaystyle\eta_{1,n},\eta_{2,n}\xrightarrow[n\rightarrow+\infty]{}0,$
(4.2)

where $\eta_{i,n}$ denotes the normal component of the relative velocity
between the particles 0 and i, that is
$\eta_{i,n}=W_{i}\cdot\omega_{i}=(v_{i}-v_{0})\cdot(x_{i}-x_{0})$.
In addition, the normal components $\eta_{1,n}$ and $\eta_{2,n}$ converge
exponentially fast to zero, at a rate at least equal to

$\displaystyle\max\left(\frac{(1+r)}{2}\big{|}\cos\overline{\theta}\big{|},r\right),$

where $\overline{\theta}$ denotes the angle between the angular parameters
$\omega_{1}=x_{1}-x_{0}$ and $\omega_{2}=x_{2}-x_{0}$ at the limiting time
$\tau^{*}$ of the collapse. As a consequence, the series of the normal
components $\sum_{n\geq 0}|\eta_{1,n}|$ and $\sum_{n\geq 0}|\eta_{2,n}|$ are
both converging.

###### Proof.

When the sequence of collisions is prescribed and given by the infinite
repetition of 0\- 1, 0\- 2, we can compute explicitly the normal components of
the relative velocities: first we compute the post-collisional velocities
according to (2.48), (the system describes a collision between 0 and 2, while
1 remains spectator), and then we have to exchange the roles between the
particles 1 and 2, in order to perform the next collision. We can then repeat
the process in order to obtain the velocities after an arbitrary number of
collisions. This process writes explicitly, if we denote by $\eta_{c}$ the
normal component of the relative velocity with the colliding particle, while
$\eta_{s}$ corresponds to the spectator particle, the particles 1 and 2 being
alternatively colliding and spectator:

$\left\\{\begin{array}[]{rl}\eta_{s,n+1}&=-r\left(1+\varepsilon_{1,n}\right)\eta_{c,n}+\mathcal{O}_{1,n}\tau_{n},\\\
\eta_{c,n+1}&=\left(1+\varepsilon_{2,n}\right)\eta_{s,n}-\frac{(1+r)}{2}\cos\overline{\theta}\left(1+\varepsilon_{3,n}\right)\eta_{c,n}+\mathcal{O}_{2,n}\tau_{n},\end{array}\right.$
(4.3)

where $\left(\varepsilon_{i,n}\right)_{n}$ are vanishing sequences,
$\left(\mathcal{O}_{i,n}\right)_{n}$ are bounded sequences, and
$\cos\overline{\theta}$ is the limit of the sequence
$\left(\omega_{1,n}\cdot\omega_{2,n}\right)_{n}$ according to Proposition 4.
Since the eigenvalues of the limiting matrix

$\displaystyle A=\begin{pmatrix}0&-r\\\
1&-\frac{(1+r)}{2}\cos\overline{\theta}\end{pmatrix}$ (4.4)

are given, when they are real, by the expression:

$\displaystyle\frac{\text{Tr}(A)\pm\sqrt{\text{Tr}^{2}(A)-4\det(A)}}{2}$

and so these eigenvalues are bounded in absolute value by

$\displaystyle\lambda_{\pm}=\frac{|\text{Tr}(A)|+\sqrt{\text{Tr}^{2}(A)-4\det(A)}}{2}\leq|\text{Tr}(A)|=\left|\frac{(1+r)}{2}\cos\overline{\theta}\right|<1$

(because $\det(A)=r>0$). In the case when they are complex, that is when
$\text{Tr}^{2}(A)-4\det(A)<0$, they are given by

$\displaystyle\lambda_{\pm}=\frac{\text{Tr}(A)\pm
i\sqrt{4\det(A)-\text{Tr}^{2}(A)}}{2},$

so that their squared modulus is

$\displaystyle|\lambda_{\pm}|^{2}=\frac{1}{4}\left[\text{Tr}^{2}(A)+4\det(A)-\text{Tr}^{2}(A)\right]=\det(A)=r<1.$

In all the cases, we observe that the spectral radius of the limiting matrix
$A$ is strictly smaller than $1$. Therefore, there exists a norm on
$\mathbb{C}^{2}$ such that the induced operator norm of all the matrices
$A_{n}$, for $n$ large enough, is strictly smaller than $1$, where

$\displaystyle A_{n}=\begin{pmatrix}0&-r(1+\varepsilon_{1,n})\\\
1+\varepsilon_{2,n}&-\frac{(1+r)}{2}(1+\varepsilon_{3,n})\cos\overline{\theta}\end{pmatrix}.$

As a consequence, since (4.3) implies by immediate recursion that

$\displaystyle\begin{pmatrix}\eta_{s,n}\\\
\eta_{c,n}\end{pmatrix}=\left[\prod_{k=0}^{n-1}A_{k}\right]\begin{pmatrix}\eta_{s,0}\\\
\eta_{c,0}\end{pmatrix}+\sum_{k=0}^{n-1}\tau_{k}\left[\prod_{j=k+1}^{n-1}A_{j}\right]\begin{pmatrix}\mathcal{O}_{1,k}\\\
\mathcal{O}_{2,k}\end{pmatrix},$

we can deduce that the normal components $\eta_{s,n}$ and $\eta_{c,n}$ both
converge to zero as $n$ goes to infinity, exponentially fast. ∎

######  Remark 9.

Note that in the proof above we did not need to require any extra assumption
on the spectrum of the matrix $A$ (such as the realness of the eigenvalues).

Thanks to Proposition 7, we are now in position to conclude the study of the
convergence of the different variables of the dynamical system (2.48), in the
case of a nearly-linear collapse.

######  Proposition 8 (Convergence of the tangential components of the
relative velocities for the nearly-linear collapse).

Let $r\in\ ]0,1[$ be a positive real number smaller than $1$, and let us
consider a system of three inelastic particles 0, 1 and 2 evolving according
to the $r$-inelastic hard sphere flow (2.1.1), (2.3), on a time interval
$[0,\tau^{*}[$, and undergoing a nearly-linear inelastic collapse at time
$\tau^{*}>0$.
Then the sequences of the tangential components
$\left(W_{1}^{\perp}\right)_{n\in\mathbb{N}}$ and
$\left(W_{2}^{\perp}\right)_{n\in\mathbb{N}}$ of the relative velocities
$W_{1}=v_{1}-v_{0}$ and $W_{2}=v_{2}-v_{0}$ are converging.

###### Proof.

Let us prove the result for the first relative velocity. According to (2.48),
when the collision between the particles 0 and 2 takes place, the tangential
component of the first relative velocity is modified according to:

$\displaystyle W_{1,n+1}^{\perp}$
$\displaystyle=W_{1,n}-\frac{\eta_{1,n}+\tau_{n}|W_{1,n}|^{2}}{(1+d_{n+1})}+\frac{(1+r)}{2}\left((1+d_{n})\eta_{2,n}+\tau_{n}|W_{2,n}|^{2}\right)\left[(\omega_{1,n+1}\cdot\omega_{2,n+1})\omega_{1,n+1}-\omega_{2,n+1}\right]$
$\displaystyle=W^{\perp}_{1,n}+\eta_{1,n}\omega_{1,n}-\frac{\eta_{1,n}+\tau_{n}|W_{1,n}|^{2}}{(1+d_{n+1})}$
$\displaystyle\hskip
113.81102pt+\frac{(1+r)}{2}\left((1+d_{n})\eta_{2,n}+\tau_{n}|W_{2,n}|^{2}\right)\left[(\omega_{1,n+1}\cdot\omega_{2,n+1})\omega_{1,n+1}-\omega_{2,n+1}\right].$

According to Proposition 1, the series of the time intervals $\tau_{n}$
between two consecutive collisions is converging. On the other hand, according
to Proposition 7, the series of the normal components are also summable.
Finally, according to Proposition 2, we know that the sequence of the
distances $\left(d_{n}\right)_{n}$ is vanishing and that the sequences of the
angular parameters $\left(\omega_{i,n}\right)_{n}$ are converging. Therefore,
we can write the iteration satisfied by the tangential component of the first
relative velocity as:

$\displaystyle W_{1,n+1}^{\perp}=W_{1,n}^{\perp}+\delta_{n},$

where $\sum_{n\geq 0}\delta_{n}$ is a summable series since we have:

$\displaystyle|\delta_{n}|\leq|\eta_{1,n}|+|\eta_{1,n}|+\tau_{n}\mathcal{E}_{0}+(1+r)\left(2|\eta_{2,n}|+\tau_{n}\mathcal{E}_{0}\right),$

where $\mathcal{E}_{0}$ denoted the kinetic energy at the initial time.
Therefore, we conclude that the series $\sum_{n\geq
0}\left(W_{1,n+1}^{\perp}-W_{1,n}^{\perp}\right)$ is summable, that is, the
sequence $\left(W_{1,n}^{\perp}\right)_{n\geq 0}$ is converging.
The argument is exactly the same for the tangential component of the second
relative velocity, and the conclusion follows accordingly. ∎

###  4.2 The results of Zhou and Kadanoff concerning the nearly-linear
collapse

For the sake of completeness, let us recall the results in the literature
concerning the nearly-linear collapse in dimension $d\geq 2$. In [27], Zhou
and Kadanoff obtained two necessary conditions concerning the nearly-linear
collapse, that write as follows. Let $r\in\ ]0,1[$ be a positive real number
smaller than $1$, and let us consider a system of three inelastic particles 0,
1 and 2 evolving according to the $r$-inelastic hard sphere flow (2.1.1),
(2.3), on a time interval $[0,\tau^{*}[$, undergoing a nearly-linear inelastic
collapse at time $\tau^{*}>0$, and such that the angle between the pairs of
particles 0\- 1 and 0\- 2 converges to the limit value $\theta$ at the
collapsing time $\tau^{*}$.

  * •

The existence of such a nearly-linear inelastic collapse is possible only if:

$\displaystyle-\cos\theta\geq\frac{4\sqrt{r}}{1+r}\cdotp$ (4.5)

  * •

In addition, such a nearly-linear inelastic collapse is stable (with respect
to perturbations of the initial data leading to such a collapse) only if:

$\displaystyle-\cos\theta>\frac{2r^{1/3}(1+r^{1/3})}{1+r}\cdotp$ (4.6)

In particular, according to (4.5), in the limit $r\rightarrow 0$ of the very
small restitution parameter (that is, in the case when the system dissipates a
lot of kinetic energy), nearly-linear inelastic collapses such that the
particles form any obtuse angle in the limit can a priori exist. On the other
hand, considering the constraint $\theta=\pi$, which corresponds to an
inelastic collapse such that the particles are perfectly aligned (or, in other
words, such that the particles, at the final time of the inelastic collapse,
all lie in a one dimensional space), provides a bound from above on $r$,
namely $4\sqrt{r}\leq 1+r$, or again $r^{2}-14r+1\geq 0$. Since $r\leq 1$,
this is equivalent to $r\leq r_{\text{exist.}}$ with:

$\displaystyle r_{\text{exist.}}=7-4\sqrt{3}\simeq 0.07179677.$ (4.7)

Concerning the second condition (4.6), note that the function $r\mapsto
2r^{1/3}(1+r^{1/3})/(1+r)$ is increasing on $[0,1]$, is equal to $0$ at $r=0$,
and equal to $2$ as $r=1$. Therefore, there exists a unique critical value
$r_{c}$ of the restitution parameter such that for $r\geq r_{c}$,
$2r^{1/3}(1+r^{1/3})/(1+r)\geq 1$. For such restitution parameters, there
exists no angle $\theta$ that could fulfill the inequality (4.6), so that no
elastic collapse can be stable for that range of restitution coefficients. The
critical value is given by $1=2r_{c}^{1/3}(1+r^{1/3})/(1+r_{c})$, or again
$1-r_{c}^{1/3}+r_{c}^{2/3}=2r_{c}^{1/3}$ so that
$r_{\text{stabi.}}^{1/3}=(3-\sqrt{5})/2$, and then in the end:

$\displaystyle
r_{\text{stabi.}}=\frac{1}{8}\left(27-27\sqrt{5}+45-5\sqrt{5}\right)=\frac{72-32\sqrt{5}}{8}=9-4\sqrt{5}\simeq
0.05572809.$ (4.8)

The collapses in [27] are obtained as self-similar solutions of a simplified,
one-dimensional version of the complete dynamical system (2.48), corresponding
to fixed points of some mapping, which turns out to be an homography (4.5)
provides the existence of such fixed points, and (4.6) ensures that the Ansatz
$\tau\ll\eta$ holds true, which allows in turn to obtain formally the one-
dimensional version of the complete dynamical system.

###  4.3 Explicit construction of a stable set of initial configurations
leading to the collapse

The previous results do not describe the basin of attraction of the collapsing
final configurations, with a given angle. In particular, it seems particularly
delicate to deduce, considering an initial data leading to a nearly-linear
collapse, what will be the limiting angle of the three particles at the
collapsing time $\tau^{*}$. Let us address a weaker version of this problem by
constructing explicitly a set of the phase space that has a non trivial
interior, composed with initial configurations of systems that all experience
a nearly-linear collapse, with a final angle that is prescribed, up to an
error under control.
To the best of our knowledge, this is the first explicit description of a
stable set of initial data leading to collapse, in dimension strictly larger
than $1$.

######  Theorem 1.

Let $r$ be a positive number such that

$\displaystyle 0<r<r_{\text{stabi}}$ (4.9)

with $r_{\text{stabi}}=9-4\sqrt{5}$, let $\theta_{0}\in\,]\pi/2,\pi]$ be an
angle such that:

$\displaystyle 0<-\cos\theta_{0}<\frac{2r^{1/3}(1+r^{1/3})}{1+r},$ (4.10)

and let $\delta_{\theta}$ be any positive real number.
Then, there exists a set $\mathcal{ZK}$ of initial configurations of three
inelastic particles 0, 1, 2, with a non trivial interior, such that the
trajectory $Z(t)$ issued from any initial configuration $Z_{0}$ of the set
$\mathcal{ZK}$ experiences a nearly-linear collapse in finite time, with 0 as
a central particle. In addition, the final angle $\overline{\theta}(Z_{0})$,
at the time of the inelastic collapse, formed by the external particles 1 and
2 around 0, is such that:

$\displaystyle\left|(-\cos\overline{\theta})-(-\cos\theta_{0})\right|\leq\delta_{\theta}.$
(4.11)

######  Remark 10.

It is possible to describe explicitly the set of initial configurations
mentioned in the theorem.

###### Proof of Theorem 1.

Before starting the construction of an initial datum leading to a collapse,
let us first make some preliminary observations.
We define $\alpha_{0}$ as the positive quantity:

$\displaystyle\alpha_{0}=\frac{(1+r)}{2}\left(-\cos\theta_{0}\right),$ (4.12)

smaller than $1$. The assumption (4.9) ensures that the mapping:

$\displaystyle\varphi_{1}\mapsto\frac{r}{\alpha_{0}-\varphi_{1}}$ (4.13)

has the two positive fixed points

$\displaystyle
F_{1}:\varphi_{1,\alpha_{0}}^{\pm}=\frac{\alpha_{0}\pm\sqrt{\alpha_{0}^{2}-4r}}{2},$
(4.14)

where only $\varphi_{1,\alpha_{0}}^{-}$, the smaller, is stable, that is, we
have:

$\displaystyle\left|F_{1}^{\prime}\left(\varphi_{1,\alpha_{0}}^{-}\right)\right|=\left|\frac{r}{\left(\alpha_{0}-\varphi_{1,\alpha_{0}}^{-}\right)^{2}}\right|<1.$
(4.15)

By definition, $\varphi_{1,\alpha_{0}}^{-}<\alpha_{0}/2$, let us denote

$\displaystyle\delta_{1}=\frac{\alpha_{0}}{2}-\varphi_{1,\alpha_{0}}^{-}>0.$
(4.16)

Since $\alpha_{0}<1$ and $r>0$, let us observe in particular that

$\displaystyle-1<\varphi_{1,\alpha_{0}}^{-}-\alpha_{0}<0.$ (4.17)

We denote by $\delta_{2}$ and $\delta_{3}$ the two positive quantities defined
as:

$\displaystyle\delta_{2}=\delta_{2}(r,\theta_{0})=\frac{1}{2}\left(\alpha_{0}-\varphi_{1,\alpha_{0}}^{-}\right)\hskip
8.53581pt\text{and}\hskip
8.53581pt\delta_{3}=\delta_{3}(r,\theta_{0})=\frac{1}{2}\left(1-\left(\alpha_{0}-\varphi_{1,\alpha_{0}}^{-}\right)\right).$
(4.18)

We introduce also the positive real number $0<C_{\eta}<1$, defined by means
of:

$\displaystyle C_{\eta}=1-\delta_{3}.$ (4.19)

According to (4.15), let us denote by $h_{4}$ the positive quantity defined
as:

$\displaystyle
h_{4}=h_{4}(r,\theta_{0})=\frac{1}{2}\left(1-\frac{r}{\left(\alpha_{0}-\varphi_{1,\alpha_{0}}^{-}\right)^{2}}\right),$
(4.20)

and let us define the positive quantity $\delta_{4}=\delta_{4}(r,\theta_{0})$
as:

$\displaystyle\frac{r}{\left(\alpha_{0}-(\varphi_{1,\alpha_{0}}^{-}+\delta_{4})\right)^{2}}=1-h_{4}.$
(4.21)

In the same way, the second assumption (4.10) ensures that the positive
quantity:

$\displaystyle\frac{\varphi_{1,\alpha_{0}}^{-}}{\left(\alpha_{0}-\varphi_{1,\alpha_{0}}^{-}\right)^{2}}$
(4.22)

is strictly smaller than $1$. Let us denote by $h_{5}$ the positive quantity
defined as:

$\displaystyle
h_{5}=h_{5}(r,\theta_{0})=\frac{1}{2}\left(1-\frac{\varphi_{1,\alpha_{0}}^{-}}{\left(\alpha_{0}-\varphi_{1,\alpha_{0}}^{-}\right)^{2}}\right),$
(4.23)

and let us define the positive quantity $\delta_{5}=\delta_{5}(r,\theta_{0})$
as:

$\displaystyle\frac{\varphi_{1,\alpha_{0}}^{-}}{\left(\alpha_{0}-(\varphi_{1,\alpha_{0}}^{-}+\delta_{5})\right)^{2}}=1-h_{5}.$
(4.24)

Finally, we choose two positive numbers $0<V_{0}<V_{1}$ satisfying:

$\displaystyle\frac{V_{1}}{V_{0}}=\frac{1-h_{5}/2}{1-h_{5}}\cdotp$ (4.25)

Now, let us introduce some notation. We assume that the initial configuration
of the system is such that the particles 0 and 1 are in contact, and just
collided. The next collision to take place is between 0 and 2, and between 0
and 1 right after, and so on. Therefore 0 is the central particle (that is,
involved in each collision), while the respective roles of 1 and 2 are
alternating between spectator and colliding.
If we write the configuration of the system at the time of the $n$-th
collision (assuming that the first collision to take place at a positive time
is between 0 and 2), such that the time between the $n$-th and the $(n+1)$-th
collision is denoted by $\tau_{n}$, and if we use the subscripts $*_{s,n}$ to
denote the particle that just collided at the $n$-th collision (so that this
particle will be spectator for the next collision) and $*_{c,n}$ to denote the
particle that will collide just after the $n$-th collision, we have for the
initial configuration:

$\displaystyle\left\\{\begin{array}[]{rl}W_{1,0}&=W_{s,0},\\\
W_{2,0}&=W_{c,0},\\\ x_{1,0}-x_{0,0}&=\omega_{1,0}=\omega_{s,0},\\\
x_{2,0}-x_{0,0}&=(1+d_{0})\omega_{2,0}=(1+d_{0})\omega_{c,0}.\end{array}\right.$
(4.30)

and we can rewrite the equations (2.48) of the dynamics as:

$\displaystyle\left\\{\begin{array}[]{rl}W_{c,n+1}&=W_{s,n}-\frac{(1+r)}{2}W_{c,n}\cdot\omega_{s,n+1}\omega_{s,n+1},\\\
W_{s,n+1}&=W_{c,n}-(1+r)W_{c,n}\cdot\omega_{s,n+1}\omega_{s,n+1},\\\
\omega_{c,n+1}&=\frac{1}{1+d_{n+1}}\left(\omega_{s,n}+\tau_{n}W_{s,n}\right),\\\
\omega_{s,n+1}&=(1+d_{n})\omega_{c,n}+\tau_{n}W_{c,n},\\\
d_{n+1}&=\sqrt{1+2\eta_{s,n}\tau_{n}+|W_{s,n}|^{2}\tau_{n}^{2}}-1,\end{array}\right.$
(4.36)

with

$\displaystyle\tau_{n}=\frac{(1+d_{n})}{|W_{c,n}|^{2}}(-\eta_{c,n})\left[1-\sqrt{1-\zeta_{n}}\right]\hskip
8.53581pt\text{and}\hskip
8.53581pt\zeta_{n}=\frac{d_{n}(2+d_{n})}{(1+d_{n})^{2}}\frac{|W_{c,n}|^{2}}{\eta_{c,n}^{2}},$
(4.37)

and where $\eta_{c,n}$ and $\eta_{s,n}$ denote respectively:

$\displaystyle\eta_{c,n}=W_{c,n}\cdot\omega_{c,n}\hskip
8.53581pt\text{and}\hskip 8.53581pt\eta_{s,n}=W_{s,n}\cdot\omega_{s,n}.$
(4.38)

With these notations, the necessary and sufficient conditions for the collapse
to take place are:

$\displaystyle\forall n\in\mathbb{N},\hskip 8.53581pt\eta_{c,n}<0\hskip
8.53581pt\text{and}\hskip 8.53581pt\zeta_{n}<1$ (4.39)

(which ensures that the particles are initially approaching after the $n$-th
collision, and that they will indeed collide), together with:

$\displaystyle\forall n\in\mathbb{N},\forall t\in[0,\tau_{n}],\hskip
8.53581pt|(1+d_{n})\omega_{c,n}-\omega_{s,n}+t\left(W_{c,n}-W_{s,n}\right)|>1,$
(4.40)

which ensures that just after a collision between 0 and 1 (respectively 0 and
2), a collision between 1 and 2 cannot happen before the next scheduled
collision, between 0 and 2 (respectively 0 and 1).
We need now to compute some evolution laws. Let us start with the normal
component $\eta_{c,n}$ of the colliding pair. For all $n\geq 1$ we have:

$\displaystyle\eta_{c,n+1}$ $\displaystyle=W_{c,n+1}\cdot\omega_{c,n+1}$
$\displaystyle=\frac{1}{1+d_{n+1}}\eta_{s,n}-\frac{(1+r)}{2}(1+d_{n})\sqrt{1-\zeta_{n}}\left(-\omega_{s,n+1}\cdot\omega_{c,n+1}\right)(-\eta_{c,n})+\frac{1}{1+d_{n+1}}\tau_{n}|W_{s,n}|^{2}.$
(4.41)

For the spectator pair we have:

$\displaystyle\eta_{s,n+1}$ $\displaystyle=W_{s,n+1}\cdot\omega_{s,n+1}$
$\displaystyle=r(1+d_{n})\sqrt{1-\zeta_{n}}(-\eta_{c,n}).$ (4.42)

The square of the norms of the relative velocities evolve as:

$\displaystyle|W_{c,n+1}|^{2}$
$\displaystyle=|W_{s,n}|^{2}-(1+r)\left(W_{c,n}\cdot\omega_{s,n+1}\right)\left(W_{s,n}\cdot\omega_{s,n+1}\right)+\frac{(1+r)^{2}}{2}\left(W_{c,n}\cdot\omega_{s,n+1}\right)^{2},$
(4.43)

and

$\displaystyle|W_{s,n+1}|^{2}$
$\displaystyle=|W_{c,n}|^{2}+(r^{2}-1)\left(W_{c,n}\cdot\omega_{s,n+1}\right)^{2}.$
(4.44)

Finally, let us write the evolution of the angle $\theta_{n}$ between
$\omega_{c,n}$ and $\omega_{s,n}$. We have:

$\displaystyle-\cos\theta_{n+1}$
$\displaystyle=\frac{1+d_{n}}{1+d_{n+1}}\left(-\cos\theta_{n}\right)$
$\displaystyle\hskip
14.22636pt-\frac{1}{1+d_{n+1}}\tau_{n}\left(\omega_{s,n}\cdot
W_{c,n}\right)-\frac{1+d_{n}}{1+d_{n+1}}\tau_{n}\left(\omega_{c,n}\cdot
W_{s,n}\right)-\frac{1}{1+d_{n+1}}\tau_{n}^{2}\left(W_{s,n}\cdot
W_{c,n}\right).$ (4.45)

Let us now describe explicitly initial data leading to the collapse. We define
first:

$\displaystyle\delta_{x}=\delta_{y}=\text{min}\left(\delta_{1},\delta_{2},\delta_{3},\delta_{4},\delta_{5}\right),$
(4.46)

then we introduce:

$\displaystyle|x_{0}|=\text{min}\left((1-h_{4})\delta_{x},\frac{V_{0}}{V_{1}}\frac{\delta_{2}^{2}h_{5}}{12}\right),$
(4.47)

and for any positive real number $\delta_{\theta}>0$, which will be used to
measure the maximal variation of the angles during the collapse, we define:

$\displaystyle|\Delta\theta|=\text{min}\left(\frac{|\cos\theta_{0}|}{2},\frac{h_{4}}{8}\delta_{x},\delta_{\theta}\right).$
(4.48)

We are now in position to describe the initial data. We introduce:

$\displaystyle\overline{\eta}=\text{min}\left(1,\frac{(\sqrt{2}-1)}{16}\frac{V_{0}}{V_{1}^{1/2}},\frac{V_{0}}{33V_{1}^{1/2}}|\Delta\theta|,\frac{(1-C_{\eta})(V_{1}-V_{0})}{12(V_{1}^{1/2}+2)},\frac{V_{0}}{V_{1}^{1/2}}\left(\frac{2}{15\alpha_{0}}+\frac{h_{4}\delta_{2}}{6}+\frac{h_{4}}{12\alpha_{0}}+\frac{h_{4}}{24}\right)\delta_{x}\right),$
(4.49)
$\displaystyle\overline{\zeta}=\text{min}\left(2(\sqrt{2}-1),\frac{V_{0}h_{5}}{3V_{1}(\alpha_{0}+8V_{1}/V_{0})}\delta_{2}^{2},\frac{1}{16}\left(1+4\delta_{2}+\frac{V_{0}}{V_{1}}\right)h_{4}\delta_{x}\right),$
(4.50)

and finally:

$\displaystyle\overline{d}=\overline{d}\left((-\eta_{c,0})\right)=\text{min}\left(\frac{\sqrt{2}-1}{4},\frac{|\Delta\theta|}{3},\frac{1}{5}(1+\delta_{2}h_{4})\delta_{x},\frac{1}{4}\left(1+2\delta_{2}\right)h_{4}\delta_{x},\frac{\overline{\zeta}}{2V_{1}}(\eta_{c,0})^{2}\right),$
(4.51)

noting the bound $\overline{d}$ is a function of the initial normal component
$-\eta_{c,0}$, which corresponds to a uniform bound on the initial Zhou-
Kadanoff parameter $\zeta_{0}$.
We will then consider initial data such that:

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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} are in contact, in a
post-collisional configuration},\\\ &\eta_{2,0}=\eta_{c,0}<0,\\\
&|\eta_{2,0}|\leq\overline{\eta},\\\
&\left|\frac{\eta_{s,0}}{(-\eta_{c,0})}-\varphi_{1,\alpha_{0}}^{-}\right|\leq|x_{0}|,\\\
&d_{0}\leq\overline{d}\left((-\eta_{c,0})\right),\\\
&V_{0}+\frac{1}{3}(V_{1}-V_{0})\leq|W_{c,0}|^{2},|W_{s,0}|^{2}\leq
V_{1}-\frac{1}{3}(V_{1}-V_{0}),\\\
&\omega_{c,0}\cdot\omega_{s,0}=\cos\theta_{0}.\end{array}\right.$ (4.59)

Let us now show that such initial configurations generate trajectories that
eventually collapse, proceeding by recursion. More precisely, we will show
that the following properties:

  * •

the negativity of the normal component of the relative velocity of the
colliding pair:

$\displaystyle\eta_{c,n}<0,$ (4.60)

  * •

for all $k\leq n$:

$\displaystyle|\eta_{c,k}|\leq\overline{\eta}\hskip 8.53581pt\text{and
if}\hskip 8.53581ptn\geq 1,\hskip 8.53581pt|\eta_{c,k}|\leq
C_{\eta}|\eta_{c,k-1}|,$ (4.61)