# The sticky particle dynamics of the 1D pressureless Euler-alignment system
as a gradient flow

Sondre Tesdal Galtung  Department of Mathematical Sciences, NTNU – Norwegian
University of Science and Technology, 7491 Trondheim, Norway
<EMAIL_ADDRESS>
###### Abstract.

We show how the sticky dynamics for the one-dimensional pressureless Euler-
alignment system can be obtained as an $L^{2}$-gradient flow of a convex
functional. This is analogous to the Lagrangian evolution introduced by Natile
and Savaré for the pressureless Euler system, and by Brenier et al. for the
corresponding system with a self-interacting force field.

Our Lagrangian evolution can be seen as the limit of sticky particle
Cucker–Smale dynamics, similar to the solutions obtained by Leslie and Tan
from a corresponding scalar balance law, and provides us with a uniquely
determined distributional solution of the original system in the space of
probability measures with quadratic moments and corresponding square-
integrable velocities. Moreover, we show that the gradient flow also provides
an entropy solution to the balance law of Leslie and Tan, and how their
results on cluster formation follow naturally from (non-)convexity properties
of the so-called natural velocity of the flow.

###### Key words and phrases:

Euler-alignment system, gradient flows, sticky particles, Wasserstein
distance, entropy solutions, cluster formation

###### 1991 Mathematics Subject Classification:

35Q35, 35L67, 35Q92, 49J40, 76N10, 82C22

##  1\. Introduction

We consider the one-dimensional pressureless Euler–alignment system for
density $\rho_{t}$ and scalar velocity $v_{t}$ depending on time $t\geq 0$ and
position $x\in\mathbb{R}$, which reads

(1.1a) $\displaystyle\partial_{t}\rho_{t}+\partial_{x}(\rho_{t}v_{t})$
$\displaystyle=0,$ (1.1b)
$\displaystyle\partial_{t}(\rho_{t}v_{t})+\partial_{x}(\rho_{t}v^{2}_{t})$
$\displaystyle=\rho_{t}(\phi\ast(\rho_{t}v_{t}))-\rho_{t}v_{t}(\phi\ast\rho_{t}).$

Here $\phi$ is a communication protocol appearing in convolutions with the
density and momentum, which we assume to be symmetric, radially nonincreasing
and locally integrable, i.e., $\phi\in L^{1}_{\mathrm{loc}}(\mathbb{R})$. In
particular, $\phi$ may be weakly singular of order $\beta\in(0,1)$, that is,
there exist constants $R>0$ and $c>0$ such that

(1.2) $\phi(r)\geq cr^{-\beta}\quad\text{for all }r\in(0,R).$

A different facet of the communication protocol concerns its asymptotic
behavior. Here $\phi$ can have a thin or fat tail, respectively defined by

(1.3)
$\int_{1}^{\infty}\phi(x)\mathop{}\\!\mathrm{d}x<\infty\qquad\text{or}\qquad\int_{1}^{\infty}\phi(x)\mathop{}\\!\mathrm{d}x=\infty.$

That is, the tail of $\phi$ is either integrable or not. A thin tail implies
that there is only weak or no communication over larger distances, while a fat
tail means there is strong communication.

The Euler-alignment system originates from the theory of collective behavior,
see [32] for a comprehensive overview of such models. It has been studied
under various regularity assumptions, and one may even consider measure-valued
solutions. A formal integration of (1.1) shows that both the total mass and
momentum are conserved, the latter by virtue of the symmetry of the
communication protocol.

The corresponding particle dynamics, which can be obtained by considering
(1.1) for the empirical measures

$\rho_{t}=\sum_{i=1}^{N}m_{i}\delta(x-x_{i}),\qquad\rho_{t}v_{t}=\sum_{i=1}^{N}m_{i}v_{i}\delta(x-x_{i}),$

where $\delta$ denotes a Dirac measure, is described by the following system
of ordinary differential equations,

(1.4) $\dot{x}_{i}=v_{i},\qquad\dot{v}_{i}=-\sum_{\begin{subarray}{c}j=1\\\
x_{j}\neq x_{i}\end{subarray}}^{N}m_{j}\phi(x_{i}-x_{j})(v_{i}-v_{j})$

for masses $\\{m_{i}\\}_{i=1}^{N}$, positions $\\{x_{i}\\}_{i=1}^{N}$ and
velocities $\\{v_{i}\\}_{i=1}^{N}$ of $N$ particles. This corresponds to the
well-known Cucker–Smale model for swarming [12].

Such alignment models, especially those with Cucker–Smale-type communication
protocols, have been studied intensively in both one and more dimensions,
covering both hydrodynamic and particle systems like (1.1) and (1.4), as well
as kinetic equations [11, 28]. Topics of interest include well-posedness of
solutions and critical thresholds for blow-up [33, 9, 4], and asymptotic
behavior [17, 15, 19]. An important notion in this context is flocking, that
the support of the solutions remain bounded in time; it has been shown in [16,
33] that fat-tailed protocols lead to flocking for both the particle system
(1.4) and strong solutions of (1.1).

When the communication protocol is too singular, i.e., not locally integrable,
concentration of mass cannot happen [9]; we are interested in this case, which
is why we here assume $\phi\in L^{1}_{\text{loc}}(\mathbb{R})$. Moreover, as
detailed below, we will study the connection between (1.1) and (1.4) through a
gradient flow structure. On this note, we mention recent the works [28, 27]
which concern a gradient flow framework for kinetic alignment systems.

###  1.1. Reduced systems

Here we give the ideas for how the second-order Euler-alignment system has
been studied through reduction to first-order systems, both in the particle
and continuum settings.

Leveraging the fact that we are on the line, the alignment force driving the
acceleration in (1.4) can be written as an exact derivative, involving the
primitive of the communication kernel $\phi$,

(1.5) $\Phi(x)=\int_{0}^{x}\phi(y)\mathop{}\\!\mathrm{d}y.$

From our assumptions it follows that this is a continuous and odd function. An
extreme case is the so-called all-to-all communication $\phi\equiv K$ for some
positive constant $K>0$, cf. [1], in which case $\Phi(x)=Kx$. For future
reference, we list some useful properties of $\Phi$.

######  Lemma 1.1.

The function $\Phi$ in (1.5) has the following properties:

  1. (a)

It is continuous, nondecreasing and odd, $\Phi(-x)=-\Phi(x)$.

  2. (b)

It is concave and subadditive for $x\geq 0$, and so by (i), it is convex and
superadditive for $x\leq 0$,

$\Phi(\lvert x\rvert+\lvert y\rvert)\leq\Phi(\lvert x\rvert)+\Phi(\lvert
y\rvert).$

In particular, if $\phi\neq 0$, then $\Phi$ is a modulus of continuity.

  3. (c)

It is pointwise linearly bounded, i.e.,

(1.6) $\lvert\Phi(x)\rvert\leq\Phi(1)\max\\{1,\lvert
x\rvert\\}\leq\Phi(1)(1+\lvert x\rvert).$

  4. (d)

By (ii) and the (reverse) triangle inequality

$\lvert\lvert\Phi(x)\rvert-\lvert\Phi(y)\rvert\rvert\leq\Phi(\lvert\lvert
x\rvert-\lvert y\rvert\rvert)\leq\Phi(\lvert x-y\rvert).$

###### Proof.

Property (a) follows from the definition (1.5) and $\phi\geq 0$. Property (b)
is a consequence of $\phi$ being nonincreasing for $x\geq 0$. Since $\phi(x)$
is nonincreasing for $x\geq 0$, we furthermore have that $\Phi(x)$ is concave
and subadditive for $x\geq 0$. Observe that for $x\in[0,1]$,
$\Phi(x)\leq\Phi(1)=\lVert\phi\rVert_{L^{1}(0,1)}$. On the other hand, for
$x>1$, $\Phi(x)-\Phi(1)\leq\Phi(1)(x-1)$ since the integrand $\phi$ is
nonincreasing and must be less than its average $\Phi(1)$ on $[0,1]$. Hence,
by symmetry, $\lvert\Phi(x)\rvert\leq\Phi(1)\max\\{1,\lvert x\rvert\\}$. ∎

The function $\Phi$ in (1.5) plays a central role in several works, e.g., [17,
20] where one takes advantage of working on the line in order to reduce the
second order system (1.4) to a first-order particle system, and similarly in
[21] where (1.1) is studied using a scalar balance law.

####  1.1.1. The particle case

In the discrete, or particle case, the initial value problem for (1.4) can be
rewritten as

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\left(\dot{x}_{i}+\sum_{j=1}^{N}m_{j}\Phi(x_{i}-x_{j})\right)=0,\quad
x_{i}(0)=\bar{x}_{i},\quad v_{i}(0)=\bar{v}_{i},$

which in turn can be integrated to yield the Kuramoto-type equation

$\dot{x}_{i}+\sum_{j=1}^{N}m_{j}\Phi(x_{i}-x_{j})=\bar{v}_{i}+\sum_{j=1}^{N}m_{j}\Phi(\bar{x}_{i}-\bar{x}_{j})\eqqcolon\bar{\psi}_{i}.$

Observe how in the reduction to a first-order system, $\bar{v}_{i}$ goes from
an initial condition to a parameter through the quantity $\bar{\psi}_{i}$,
known as the natural velocity, cf. [17]. This is likely in analogy to the
corresponding quantity for the original Kuramoto system [18], which models
coupled, nonlinear oscillators and reads

(1.7)
$\dot{\theta}_{i}=\omega_{i}-\frac{\nu}{N}\sum_{j=1}^{N}\sin(\theta_{i}-\theta_{j}),\quad
i\in\\{1,\dots,N\\},\quad\nu>0.$

Here $\theta_{i}$ is the phase of the $i$th oscillator, while the constant
$\omega_{i}$ is often called its natural frequency. There is an extensive
literature on the Kuramoto model, where one has studied equilibria and
phenomena like phase-locking and synchronization, as well as extensions to
kinetic models like the Kuramoto–Sakaguchi equation [23]; we will merely refer
to the aforementioned recent work and the references therein. It was observed
in [34] that (1.7) has a gradient flow structure; however, the corresponding
energy functional is not convex. Applying this idea to the reduction of the
Cucker–Smale system, it can be seen as a gradient flow for the scalar
potential

(1.8) $\frac{1}{2}\sum_{j=1}^{N}\sum_{k=1}^{N}m_{j}m_{k}\int_{0}^{x_{j}-x_{k}}
\Phi(y)\mathop{}\\!\mathrm{d}y-\sum_{j=1}^{N}m_{j}\bar{\psi}_{j}x_{j},$

and the study of clustering in [17, 15, 38] shows that the corresponding
trajectories are well-defined between collisions. In their analysis, the
trajectories are allowed to cross, and the natural velocities
$\\{\bar{\psi}_{i}\\}_{i=1}^{N}$ provide an ordering of the asymptotic
trajectories. However, Leslie and Tan [22] argue that this crossing of
trajectories is not well-suited for hydrodynamic limits of the particle
system, and so they advocate for a different point of view, which we describe
next.

####  1.1.2. An associated scalar balance law

Returning to the continuum setting, we introduce the quantity

(1.9) $\psi_{t}\coloneqq v_{t}+\Phi\ast\rho_{t},$

which can be seen as the continuum version of the discrete conserved variable
from the previous section. Then one can formally rewrite (1.1) as

(1.10a) $\displaystyle\partial_{t}\rho_{t}+\partial_{x}(\rho_{t}v_{t})$
$\displaystyle=0,$ (1.10b)
$\displaystyle\partial_{t}(\rho_{t}\psi_{t})+\partial_{x}(\rho_{t}\psi_{t}v_{t})$
$\displaystyle=0.$

Based on the above system and drawing inspiration from [7], Leslie and Tan
[21] derive an associated scalar balance law for
$M_{t}=\rho_{t}((-\infty,x])$,

(1.11) $\partial_{t}M_{t}+\partial_{x}A(M_{t})=(\phi\ast
M_{t})\partial_{x}M_{t},$

where the flux function $A$ depends only on the initial data. The initial
density $\bar{\rho}$ is assumed to be a compactly supported probability
measure, i.e., $\bar{\rho}\in\mathscr{P}_{c}(\mathbb{R})$, while the initial
velocity $\bar{v}$ is essentially bounded with respect to $\bar{\rho}$, i.e.,
$\bar{v}\in L^{\infty}(\mathbb{R},\bar{\rho})$. After establishing existence
and uniqueness of entropy solutions of (1.11) in the sense of Kružkov, they
show, using the $BV$-calculus of Vol’pert [36], that these solutions provide
distributional solutions of (1.1) exhibiting “sticky dynamics”. Furthermore,
with this framework they study clustering for the Euler-alignment system (1.1)
in [22], and show that the global clustering behavior can be deduced from the
initial data through the flux function $A$.

###  1.2. The aim of this paper

Before stating the aim and main results of this paper, we present the
motivation for our study.

####  1.2.1. Gradient flows and entropy solutions for pressureless Euler
systems

Pressureless Euler systems similar to (1.1) have previously been studied using
gradient flows in Hilbert spaces. Suppose $\rho_{t}$ is a probability measure
for $t\geq 0$, denoted $\rho_{t}\in\mathscr{P}(\mathbb{R})$, and that $v_{t}$
is such that the momentum $v_{t}\rho_{t}$ is a Radon measure, denoted
$v_{t}\rho_{t}\in\mathscr{M}(\mathbb{R})$. In particular, we are interested in
probability measures $\rho_{t}$ with finite second moment and velocities
$v_{t}$ which are square-integrable with respect to $\rho_{t}$. That is,
$\rho_{t}\in\mathscr{P}_{2}(\mathbb{R})$ and $v_{t}\in
L^{2}(\mathbb{R},\rho_{t})$, meaning

$\int_{\mathbb{R}}\lvert
x\rvert^{2}\mathop{}\\!\mathrm{d}\rho_{t}<\infty\quad\text{and}\quad\int_{\mathbb{R}}\lvert
v_{t}\rvert^{2}\mathop{}\\!\mathrm{d}\rho_{t}<\infty.$

Then it turns out that $\mathscr{P}_{2}(\mathbb{R})$ is isometric to the
convex cone $\mathscr{K}\subset L^{2}(0,1)$ of nondecreasing functions. That
is, any $\rho_{t}\in\mathscr{P}_{2}(\mathbb{R})$ can be associated with an
$X_{t}\in\mathscr{K}$, and vice versa.

This framework was used by Natile and Savaré to study the pressureless Euler
system [24], corresponding to (1.1) with $\phi\equiv 0$. The idea is to use a
similar reduction to a first-order system to see the initial (natural)
velocity as the gradient of a linear functional. To ensure that solutions
remain in $\mathscr{K}$, this functional is augmented with the indicator
function $I_{\mathscr{K}}$, which in turn replaces the gradient with a set-
valued subdifferential. The convexity of the functional makes the
subdifferential a maximally monotone operator. Hence, by the theory of Brézis
[8], there is a unique solution $X_{t}$, with velocity $V_{t}$, evolving
according to the minimal element of the subdifferential. This gives rise to a
distributional solution of the original system which turns out to be globally
sticky, that is, mass that clusters will remain clustered. In particular, they
recover results of [7] obtained with a conservation law-approach.

In what can be seen as an extension of the above framework, called the
Lagrangian evolution, Brenier et al. [6] study the pressureless Euler system
with a self-interacting force field $f[\rho_{t}]$ driving the momentum
equation. Under certain assumptions on $f$, the resulting force can be seen as
a Lipschitz perturbation of the subdifferential in [24], which is still
covered by the results of Brézis, i.e., [8, Theoreme 3.17, Remarque 3.14]. A
specific case covered by their theory is the Euler–Poisson system, which is
also shown to have globally sticky solutions in the case of an attractive
force. Note that this form of differential inclusion can be framed as an
evolution variational inequality, and both [24] and [6] make use of additional
results from [31], see also [30], to have better estimates for the velocity
$V_{t}$.

On a slightly different note, [5] concerns a singular, nonlocal interaction
equation, which can be regarded as a first-order version of the Euler–Poisson
system. For this equation one can also formulate $L^{2}$-gradient flow
solutions, as well as, in the spirit of [7], entropy solutions of an
associated conservation law. The authors prove that these notions of solution
are in fact equivalent, by means of passing to a limit in particle
approximations. Based on an observation relating the minimal evolution of the
gradient flow and the Oleĭnik E condition for conservation laws, [10] extended
the equivalence from [5] to pressureless Euler systems with appropriate
forcing terms $f$. Specifically, one finds that the solutions of Euler–Poisson
obtained in [6] using gradient flows and solutions obtained from entropy
solutions of a conservation law in [25] are equivalent. We also mention the
work [3], where [5] and [21] inspired the use of a balance law to study the
mean-field limit of a second-order particle system for opinion dynamics.

####  1.2.2. Aim and main results

We mentioned before how sticky solutions of (1.1) can be obtained from entropy
solutions of the balance law (1.11) for initial data
$(\bar{\rho},\bar{v})\in\mathscr{P}_{\mathrm{c}}(\mathbb{R})\times
L^{\infty}(\mathbb{R},\bar{\rho})$. Considering the aforementioned
equivalences between gradient flows and entropy solutions, it is then tempting
to follow [24, 6] in extending to
$(\bar{\rho},\bar{v})\in\mathscr{P}_{2}(\mathbb{R})\times
L^{2}(\mathbb{R},\bar{\rho})$ by formulating an associated gradient flow
solution. However, there are some apparent obstacles. For one, (1.1) is not
covered directly by the theory in [6], as the forcing term is of the form
$f[\rho_{t},v_{t}]$ rather than $f[\rho_{t}]$. Moreover, in [21] they allow
for weakly singular $\phi$, for which the corresponding alignment force cannot
be seen as a Lipschitz perturbation of the subdifferential in [24].

We take a different approach to overcome this problem, regarding instead the
alignment force as part of the subdifferential. Indeed, unlike for the
Kuramoto model, the functional (1.8) is in fact convex and its $L^{2}$-version
can then be augmented with the indicator function $I_{\mathscr{K}}$ to form a
convex functional.

Yet another indication that the framework of $L^{2}$-gradient flows is
suitable for the problem at hand is found in [22]. There the authors use the
flux function $A$ from (1.11), which can be regarded as the primitive of the
$L^{2}$-version of the natural velocities, and its lower convex envelope
$A^{**}$ to predict the clustering behavior of the sticky solutions. The
suggestion comes from a relation in [24] which connects the projection of a
function onto the convex cone $\mathscr{K}$ and the lower convex envelope of
its primitive. In this sense, $A^{**}$ encodes the projection of the natural
velocity onto this cone.

With this in mind, we seek the following type of solutions to the Euler-
alignment system.

######  Definition 1.2 (Distributional solutions of the Euler-alignment
system).

The pair $(\rho_{t},v_{t})\in\mathscr{P}_{2}(\mathbb{R})\times
L^{2}(\mathbb{R},\rho_{t})$ is a distributional solution of the initial value
problem for the Euler-alignment system if it satisfies (1.1) in the
distributional sense, and for initial values $(\bar{\rho},\bar{v})$ we have

(1.12) $\lim\limits_{t\to 0+}\rho_{t}=\bar{\rho}\quad\text{in}\
\mathscr{P}_{2}(\mathbb{R})\qquad\text{and}\qquad\lim\limits_{t\to
0+}v_{t}\rho_{t}=\bar{v}\bar{\rho}\quad\text{in}\ \mathscr{M}(\mathbb{R}).$

The paper is organized as follows. Section 2 provides a motivation of the
gradient flow solution for the particle case. Some auxiliary results on
optimal transport and convex analysis are presented in Section 3, leading to
the associated Lagrangian solutions in Section 4. Section 5 details how the
Lagrangian solution yields a solution of the original Euler-alignment system,
as well as for the associated balance law. Finally, in Section 6 we show how
the Lagrangian solutions can be used to derive the clustering properties of
the sticky solutions. The main results are

  * •

There is a Lagrangian, or $L^{2}$-gradient flow, solution concept associated
with the Euler-alignment system (1.1), see Theorem 4.6.

  * •

This Lagrangian solution can be realized as the limit of particle solutions,
cf. Theorem 4.16, from which we deduce that it is globally sticky and features
projection formulas and a semigroup property, see Corollary 4.17.

  * •

The globally sticky Lagrangian solution gives rise to a distributional
solution of (1.1) and an entropy solution of (1.11), both uniquely defined,
see Theorems 5.1 and 5.3.

  * •

We can deduce asymptotic clustering properties from the Lagrangian solution,
see Theorem 6.2.

We emphasize that the techniques we use are inherently one-dimensional in
nature, and so these ideas are not directly applicable in higher dimensions.

##  2\. Motivation in particle case

We will follow [6] in first motivating the $L^{2}$-gradient flow using the
particle system.

###  2.1. Deriving the particle dynamics

We denote $\boldsymbol{x}=(x_{1},\dots,x_{N})\in\mathbb{R}^{N}$,
$\boldsymbol{v}=(v_{1},\dots,v_{N})\in\mathbb{R}^{N}$ and
$\boldsymbol{m}=(m_{1},\dots,m_{N})\in\mathbb{R}^{N}_{+}$, where
$\mathbb{R}^{N}_{+}$ denotes the elements of $\mathbb{R}^{N}$ with strictly
positive entries. Let us introduce the auxiliary functions

(2.1) $\psi_{i}(t)\coloneqq
v_{i}(t)+\sum_{j=1}^{N}m_{j}\Phi\left(x_{i}(t)-x_{j}(t)\right),\qquad\bar{\psi}_{i}\coloneqq\bar{v}_{i}+\sum_{j=1}^{N}m_{j}\Phi\left(\bar{x}_{i}-\bar{x}_{j}\right),\quad
i\in\\{1,\dots,N\\},$

which we collect in vectors
$\boldsymbol{\psi}=\boldsymbol{v}+\boldsymbol{\Phi}^{\ast}_{\boldsymbol{m}}(\boldsymbol{x})$
and
$\bar{\boldsymbol{\psi}}=\bar{\boldsymbol{v}}+\boldsymbol{\Phi}^{\ast}_{\boldsymbol{m}}(\bar{\boldsymbol{x}})$.
Then the particle dynamics (1.4) can be rephrased as the following system of
differential equations,

(2.2) $\dot{x}_{i}=\bar{\psi}_{i}-\sum_{j=1}^{N}m_{j}\Phi(x_{i}-x_{j}),\quad
x_{i}(0)=\bar{x}_{i},\quad i\in\\{1,\dots,N\\}.$

We do not want trajectories of this system to cross, and so we require the
particles to retain their initial ordering. To this end we introduce the
closed, convex cone

(2.3) $\mathbb{K}^{N}\coloneqq\left\\{\boldsymbol{x}\in\mathbb{R}^{N}\colon
x_{1}\leq x_{2}\leq\dots\leq x_{N}\right\\},$

and assume our initial data belongs to this cone, that is,
$\bar{\boldsymbol{x}}\in \mathbb{K}^{N}$. Note that for
$\boldsymbol{x}\in\mathbb{K}^{N}$ and $\boldsymbol{m}\in\mathbb{R}_{+}^{N}$ we
can define the $\boldsymbol{m}$-weighted Euclidean $p$-norm

(2.4)
$\lVert\boldsymbol{x}\rVert_{\boldsymbol{m},p}\coloneqq\left(\sum_{i=1}^{N}m_{i}\lvert
x_{i}\rvert^{p}\right)^{1/p},\qquad\lVert\boldsymbol{x}\rVert_{\boldsymbol{m},\infty}=\lVert\boldsymbol{x}\rVert_{\infty}.$

For $p=2$ we can define an associated $\boldsymbol{m}$-weighted inner product

$\langle\boldsymbol{y},\boldsymbol{x}\rangle_{\boldsymbol{m}}\coloneqq\sum_{i=1}^{N}m_{i}y_{i}x_{i},$

and for convenience we write
$\lVert\cdot\rVert_{\boldsymbol{m}}\coloneqq\lVert\cdot\rVert_{\boldsymbol{m},2}$.

####  2.1.1. Short-time existence and uniqueness of solutions

We note that the right-hand side of (2.2) is a continuous function in $x_{i}$,
and so existence of solutions $\boldsymbol{x}$ in
$C^{1}([0,\infty);\mathbb{R}^{N})$ is guaranteed. On the other hand, we cannot
expect Lipschitz-continuity of the right-hand side, e.g., if $\phi$ is weakly
singular in the sense of (1.2). However, owing to the monotonicity of
$\Phi^{\ast}_{\boldsymbol{m}}$ we can obtain uniqueness anyway, at least until
particles meet, i.e., when $\boldsymbol{x}$ hits the boundary
$\partial\mathbb{K}^{N}$ given by

(2.5)
$\partial\mathbb{K}^{N}=\\{\boldsymbol{x}\in\mathbb{K}^{N}\colon\Omega_{\boldsymbol{x}}\neq\emptyset\\},\qquad\Omega_{\boldsymbol{x}}\coloneqq\\{j\colon
x_{j}=x_{j+1},j\in\\{1,\dots,N-1\\}\\}.$

Let us specify what is meant by monotonicity here. For
$\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{N}$ we compute

$\displaystyle\langle\boldsymbol{x}-\boldsymbol{y},\Phi^{\ast}_{\boldsymbol{m}}(\boldsymbol{x})-\Phi^{\ast}_{\boldsymbol{m}}(\boldsymbol{y})\rangle_{\boldsymbol{m}}$
$\displaystyle=\sum_{i=1}^{N}m_{i}(x_{i}-y_{i})\sum_{j=1}^{N}m_{j}\left[\Phi(x_{i}-x_{j})-\Phi(y_{i}-y_{j})\right]$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}m_{i}m_{j}[(x_{i}-x_{j})-(y_{i}-y_{j})]\left[\Phi(x_{i}-x_{j})-\Phi(y_{i}-y_{j})\right]\geq
0,$

where the second equality follows from $\Phi$ being odd, and the final
inequality from $\Phi$ being nondecreasing. Hence, this monotonicity yields a
one-sided Lipschitz condition, and we can show a stability estimate. For
$i\in\\{1,2\\}$, let $\boldsymbol{x}_{i}(t)$ be a solution of (2.2) with
initial data
$(\boldsymbol{x}_{i}(0),\boldsymbol{v}_{i}(0))=(\bar{\boldsymbol{x}}_{i},\bar{\boldsymbol{v}}_{i})$
so that one has
$\bar{\boldsymbol{\psi}}_{i}=\bar{\boldsymbol{v}}_{i}+\boldsymbol{\Phi}^{\ast}_{\boldsymbol{m}}(\boldsymbol{\bar{x}}_{i})$.
Then we have

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\frac{1}{2}\lVert\boldsymbol{x}_{1}-\boldsymbol{x}_{2}\rVert_{\boldsymbol{m}}^{2}=\langle\boldsymbol{x}_{1}-\boldsymbol{x}_{2},\dot{\boldsymbol{x}}_{1}-\dot{\boldsymbol{x}}_{2}\rangle_{\boldsymbol{m}}\leq\langle\boldsymbol{x}_{1}-\boldsymbol{x}_{2},\bar{\boldsymbol{\psi}}_{1}-\bar{\boldsymbol{\psi}}_{2}\rangle_{\boldsymbol{m}}\leq\lVert\boldsymbol{x}_{1}-\boldsymbol{x}_{2}\rVert_{\boldsymbol{m}}\lVert\bar{\boldsymbol{\psi}}_{1}-\bar{\boldsymbol{\psi}}_{2}\rVert_{\boldsymbol{m}},$

which implies

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\lVert\boldsymbol{x}_{1}-\boldsymbol{x}_{2}\rVert_{\boldsymbol{m}}\leq\lVert\bar{\boldsymbol{\psi}}_{1}-\bar{\boldsymbol{\psi}}_{2}\rVert_{\boldsymbol{m}},$

and in turn

$\lVert\boldsymbol{x}_{1}(t)-\boldsymbol{x}_{2}(t)\rVert_{\boldsymbol{m}}\leq\lVert\boldsymbol{x}_{1}(s)-\boldsymbol{x}_{2}(s)\rVert_{\boldsymbol{m}}+(t-s)\lVert\bar{\boldsymbol{\psi}}_{1}-\bar{\boldsymbol{\psi}}_{2}\rVert_{\boldsymbol{m}}.$

Suppose we start with distinct initial particles, i.e.,
$\bar{x}_{1}<\bar{x}_{2}<\dots<\bar{x}_{N}$, meaning
$\bar{\boldsymbol{x}}\in\mathrm{int}(\mathbb{K}^{N})$, the interior of
$\mathbb{K}^{N}$. Then we have a unique solution
$\boldsymbol{x}(t)\in\mathbb{K}^{N}$ at least for a short time, until
$\boldsymbol{x}(t)\in\partial\mathbb{K}^{N}$. To ensure that the solution
remains in $\mathbb{K}^{N}$ after particles meet, we will in the following
provide a well-defined procedure for resolving the dynamics in this case.

####  2.1.2. Collision dynamics and the differential inclusion

For $\boldsymbol{x}\in\mathbb{K}^{N}$, the tangent cone to $\mathbb{K}^{N}$ at
$\boldsymbol{x}$, cf. [2, Definition 6.38], is defined as the following
closure,

(2.6)
$T_{\boldsymbol{x}}\mathbb{K}^{N}\coloneqq\mathrm{cl}\\{\vartheta(\boldsymbol{y}-\boldsymbol{x})\colon\boldsymbol{y}\in\mathbb{K}^{N},\vartheta\geq
0\\}.$

This is the cone in which the velocity of $\boldsymbol{x}$ should belong for
$\boldsymbol{x}$ to remain within $\mathbb{K}^{N}$, and in our setting (2.6)
is equivalent to

$T_{\boldsymbol{x}}\mathbb{K}^{N}=\\{\boldsymbol{v}\in\mathbb{R}^{N}\colon
v_{j}\leq v_{j+1}\ \text{for all}\ j\in\Omega_{\boldsymbol{x}}\\},$

where we recall $\Omega_{\boldsymbol{x}}$ from (2.5). In particular, we see
that if $\Omega_{\boldsymbol{x}}=\emptyset$, i.e.,
$\boldsymbol{x}\in\mathrm{int}(\mathbb{K}^{N})$, then
$T_{\boldsymbol{x}}\mathbb{K}^{N}=\mathbb{R}^{N}$. Assume a subset of
particles $\mathcal{J}_{i}(t)\subset\\{1,\dots,N\\}$ collides at time $t$,
where we define

$\mathcal{J}_{i}(t)\coloneqq\left\\{j\colon x_{j}(t)=x_{i}(t)\right\\},\qquad
i_{*}(t)=\min\mathcal{J}_{i}(t),\qquad
i^{*}(t)\coloneqq\max\mathcal{J}_{i}(t).$

Furthermore we assume the completely inelastic collision rule

(2.7)
$\boldsymbol{v}(t+)=\mathop{}\mathrm{P}_{T_{\boldsymbol{x}(t)}\mathbb{K}^{N}}\boldsymbol{v}(t-)$

leading to

$v_{i}(t+)=\frac{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}v_{j}(t-)}{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}},$

which is natural in the sense that it conserves momentum. Now, as noted in
[21], by continuity of the trajectory $\boldsymbol{x}(t)$ and the continuity
of $\Phi$ it follows that $\psi_{j}(t-)-v_{j}(t-)=\psi_{i}(t-)-v_{i}(t-)$ for
all $j\in\mathcal{J}_{i}(t)$. This in turn leads to

(2.8)
$\psi_{j}(t+)=\frac{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}\psi_{j}(t-)}{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}}=\frac{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}\bar{\psi}_{j}}{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}}$

where the final identity comes from $\psi_{i}(t)$ being constant between
collisions by virtue of (2.2). For convenience we take the velocities
$v_{i}(t)$, and then consequently also $\psi_{i}(t)$, to be right-continuous
functions of $t$. Afterwards and until the next collision, the clustered
particle will only be affected by particles not belonging to the cluster.
Indeed, for $k\in\mathcal{J}_{i}(t)$ we have

$\dot{v}_{k}(t)=-\sum_{\begin{subarray}{c}j=1\\\
x_{j}(t)\notin\mathcal{J}_{i}(t)\end{subarray}}^{N}m_{j}\phi(x_{k}(t)-x_{j}(t))(v_{k}(t)-v_{j}(t))$

or, since $\Phi(0)=0$,

$v_{k}(t)+\sum_{j=1}^{n}m_{j}\Phi(x_{k}(t)-x_{j}(t))=\frac{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}\bar{\psi}_{j}}{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}}.$

Moreover, amassed particles will not break apart again, since every particle
in labeled by $\mathcal{J}_{i}(t)$ moves with the same velocity. As pointed
out in [6], this sticky evolution can equivalently be defined by relabeling
the amassed particles as a single new particle with mass given by the sum of
the previous masses. At every collision, the number of particles in the system
would then decrease from the initial $N$. Recalling (2.8), the collision
dynamics can therefore be seen as a projection of the initial vector
$\bar{\boldsymbol{\psi}}$ of natural velocities onto the tangent cone of
$\boldsymbol{x}(t)$. That is,

(2.9)
$\boldsymbol{\psi}(t+)=\mathop{}\mathrm{P}_{T_{\boldsymbol{x}(t)}\mathbb{K}^{N}}\boldsymbol{\psi}(t-)=\mathop{}\mathrm{P}_{T_{\boldsymbol{x}(t)}\mathbb{K}^{N}}\bar{\boldsymbol{\psi}}.$

From this we see how the collision, or clustering, dynamics only depends on
$\bar{\boldsymbol{\psi}}$, as expected from the particle dynamics in [22].
From the above arguments, we have the following result.

######  Lemma 2.1.

Let
$(\bar{\boldsymbol{x}},\bar{\boldsymbol{v}})\in\mathrm{int(\mathbb{K}^{N})\times\mathbb{R}^{N}}$
and define $\bar{\boldsymbol{\psi}}\in\mathbb{R}^{N}$ as in (2.1). Then, for
$t\geq 0$ there is a uniquely defined, globally sticky solution
$(\boldsymbol{x}(t),\boldsymbol{v}(t))\in\mathbb{K}^{N}\times
T_{\boldsymbol{x}(t)}\mathbb{K}^{N}$ of (2.2) satisfying
$(\boldsymbol{x}(0),\boldsymbol{v}(0))=(\bar{\boldsymbol{x}},\bar{\boldsymbol{v}})$.

Following [6, Section 1.2], we then argue that the instantaneous force that
changes the velocity on impact at $\boldsymbol{x}\in\partial\mathbb{K}^{N}$
should belong to the normal cone111A cone very similar to $\mathbb{K}^{N}$,
and its polar cone, is treated in [2, Exercise 6.16].
$N_{\boldsymbol{x}}\mathbb{K}^{N}$. Then we can incorporate the collision
dynamics in (1.4) by rephrasing it as the second-order differential inclusion

(2.10)
$\dot{x}_{i}=v_{i},\qquad\dot{v}_{i}+N_{\boldsymbol{x}}\mathbb{K}^{N}\ni-\sum_{\begin{subarray}{c}j=1\\\
x_{j}\neq x_{i}\end{subarray}}^{N}m_{j}\phi(x_{i}-x_{j})(v_{i}-v_{j}).$

or equivalently, using (2.1),

(2.11)
$\dot{\boldsymbol{x}}=\boldsymbol{v},\qquad\dot{\boldsymbol{\psi}}+N_{\boldsymbol{x}}\mathbb{K}^{N}\ni\boldsymbol{0}.$

As observed in [24], if $\boldsymbol{x}\colon[0,\infty)\to\mathbb{K}^{N}$
satisfies a global stickiness condition, there is a monotonicity property for
the normal cones, namely

$N_{\boldsymbol{x}(s)}\mathbb{K}^{N}\subset
N_{\boldsymbol{x}(t)}\mathbb{K}^{N}\quad\text{for all }s<t.$

Then, for $\boldsymbol{\xi}\colon[0,\infty)\to\mathbb{R}^{N}$ satisfying
$\boldsymbol{\xi}(t)\in N_{\boldsymbol{x}(t)}\mathbb{K}^{N}$, e.g.,
$-\dot{\boldsymbol{\psi}}(t)$ in (2.11), we obtain

$\int_{s}^{t}\boldsymbol{\xi}(r)\mathop{}\\!\mathrm{d}r\in
N_{\boldsymbol{x}(t)}\mathbb{K}^{N}\quad\text{for all }s<t.$

Combining the above with a formal integration of the second equation of (2.11)
then yields

$\boldsymbol{\psi}(t)+N_{\boldsymbol{x}(t)}\mathbb{K}^{N}\ni\bar{\boldsymbol{\psi}},$

which we in turn rephrase as the first-order differential inclusion

(2.12)
$\dot{\boldsymbol{x}}+\boldsymbol{\Phi}^{\ast}_{\boldsymbol{m}}(\boldsymbol{x})+N_{\boldsymbol{x}}\mathbb{K}^{N}\ni\bar{\boldsymbol{\psi}}.$

This relation lays the foundation for our study of the problem in the
continuum case.

####  2.1.3. The barycentric lemma

Consider again a subset of particles $\mathcal{J}_{i}(t)$ colliding at time
$t$; their velocities must necessarily satisfy

$v_{i_{*}(t)}(t-)\geq v_{i_{*}(t)+1}(t-)\geq\dots\geq v_{i^{*}(t)-1}(t-)\geq
v_{i^{*}(t)}(t-),$

or else they would not have collided in the first place. However, by the
continuity of the trajectories $x_{i}$ and the continuity of $\Phi$ we also
have

$\psi_{i_{*}(t)}(t-)\geq\psi_{i_{*}(t)+1}(t-)\geq\dots\geq\psi_{i^{*}(t)-1}(t-)\geq\psi_{i^{*}(t)}(t-).$

Combining the above chain of inequalities with the collision dynamics (2.7),
we obtain

(2.13)
$\frac{\sum_{j=k}^{i^{*}(t)}m_{j}\psi_{j}(t-)}{\sum_{j=k}^{i^{*}(t)}m_{j}}\leq\psi_{i}(t+)=\frac{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}\psi_{j}(t-)}{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}}\leq\frac{\sum_{j=i_{*}(t)}^{k}m_{j}\psi_{j}(t-)}{\sum_{j=i_{*}(t)}^{k}m_{j}},$

and, relying once more on $\psi_{i}(t)$ being constant between collisions, we
can write this as

(2.14)
$\frac{\sum_{j=k}^{i^{*}(t)}m_{j}\bar{\psi}_{j}}{\sum_{j=k}^{i^{*}(t)}m_{j}}\leq\psi_{i}(t+)=\frac{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}\bar{\psi}_{j}}{\sum_{j\in\mathcal{J}_{i}(t)}m_{j}}\leq\frac{\sum_{j=i_{*}(t)}^{k}m_{j}\bar{\psi}_{j}}{\sum_{j=i_{*}(t)}^{k}m_{j}}.$

This is the barycentric lemma used in [7] for the case $\phi\equiv 0$, and
which was shown in [21] to still hold in our current case. It turns out that
the barycentric lemma is a particular, in fact, particle case of the Oleĭnik E
condition [26] for the flux function $A$ from the balance law (1.11). We
return to this matter in Section 5.

##  3\. Auxiliary results

In order to define Lagrangian, or $L^{2}$-gradient flow, solutions for the
Euler-alignment system, we will need some auxiliary results which play a
central role in [24, 6].

###  3.1. Some results from optimal transport

Consider a probability measure $\rho\in\mathscr{P}(\mathbb{R})$, for which we
define its right-continuous cumulative distribution function

(3.1) $M_{\rho}(x)=\rho((-\infty,x]),\quad x\in\mathbb{R}.$

Then $\partial_{x}M_{\rho}=\rho$ in $\mathscr{D}^{\prime}(\mathbb{R})$, i.e.,
in the distributional sense. We can then define its right-continuous monotone
rearrangement, or generalized inverse [37], through

(3.2) $X_{\rho}\coloneqq\inf\left\\{x\colon M_{\rho}(x)>m\right\\},\quad
m\in\Omega,$

where $\Omega\coloneqq(0,1)$. We denote the restriction of the one-dimensional
Lebesgue measure $\mathcal{L}^{1}$ to $\Omega$ by
$\mathfrak{m}\coloneqq\mathcal{L}^{1}|_{\Omega}$, such that we have the push-
forward relations

(3.3)
$X_{\rho\\#}\mathfrak{m}=\rho,\qquad\int_{\mathbb{R}}\varphi(x)\mathop{}\\!\mathrm{d}\rho(x)=\int_{\Omega}\varphi(X_{\rho}(m))\mathop{}\\!\mathrm{d}m$

for any Borel function $\varphi\colon\mathbb{R}\to[0,\infty]$.

The $p$-Wasserstein distance, or Kantorovich–Rubinstein metric, between two
measures is defined as

(3.4)
$W_{p}^{p}(\rho_{1},\rho_{2})\coloneqq\min\left\\{\int_{\mathbb{R}\times\mathbb{R}}\lvert
x-y\rvert^{p}\mathop{}\\!\mathrm{d}\varrho(x,y)\colon\varrho\in\mathscr{P}(\mathbb{R}\times\mathbb{R}),\varpi^{i}_{\\#}\varrho=\rho_{i}\right\\},$

where $\varpi^{i}$ is the projection on the $i$th coordinate, i.e.,
$\varpi^{i}(x_{1},x_{2})=x_{i}$. In one dimension, the unique optimal coupling
of measures can be explicitly found using the monotone rearrangement. Indeed,
by the Hoeffding–Frechét theorem [35, Theorem 2.18] the optimal coupling is
given by

(3.5) $\varrho=X_{\rho_{1},\rho_{2}\\#}\mathfrak{m},\qquad
X_{\rho_{1},\rho_{2}}=(X_{\rho_{1}},X_{\rho_{2}}).$

A direct consequence of this is the identity

$W_{p}(\rho_{1},\rho_{2})=\left(\int_{\Omega}\lvert
X_{\rho_{1}}-X_{\rho_{2}}\rvert^{p}\mathop{}\\!\mathrm{d}\omega\right)^{1/p}=\lVert
X_{\rho_{1}}-X_{\rho_{2}}\rVert_{L^{p}(\Omega)}.$

That is, the $p$-Wasserstein distance of two probability measures equals the
$L^{p}$-distance of their monotone rearrangements. Let us then for
$p\in[1,\infty)$ introduce the space

(3.6)
$\mathscr{T}_{p}\coloneqq\left\\{(\rho,v)\colon\rho\in\mathscr{P}_{p}(\mathbb{R}),v\in
L^{p}(\mathbb{R},\rho)\right\\},$

for which we can define a semidistance $U_{p}$ as follows,

(3.7)
$U_{p}^{p}((\rho_{1},v_{1}),(\rho_{2},v_{2}))\coloneqq\int_{\mathbb{R}\times\mathbb{R}}\lvert
v_{1}(x)-v_{2}(y)\rvert^{p}\mathop{}\\!\mathrm{d}\varrho(x,y)=\lVert
v_{1}\circ X_{1}-v_{2}\circ X_{2}\rVert_{L^{p}(\Omega)}^{p},$

where the final identity again follows from (3.5). In turn, we can define a
metric $D_{p}$ through

(3.8) $D_{p}^{p}((\rho_{1},v_{1}),(\rho_{2},v_{2}))\coloneqq
W_{p}^{p}(\rho_{1},\rho_{2})+U_{p}^{p}((\rho_{1},v_{1}),(\rho_{2},v_{2})),$

so that $(\mathscr{T}_{p},D_{p})$ is a metric, but not complete, space, see
[24, Proposition 2.1].

###  3.2. Some results from convex analysis

Let $f\colon\mathscr{X}\to\mathbb{R}\cup\\{+\infty\\}$ be an extended real-
valued function for some set $\mathscr{X}$. Its effective domain, or just
domain, $\operatorname*{dom}(f)$ is the set of points where $f$ is finite,
i.e.,

$\operatorname*{dom}(f)\coloneqq\\{x\in\mathscr{X}\colon f(x)<+\infty\\}.$

If $\operatorname*{dom}(f)\neq\emptyset$, then $f$ is called proper.
Throughout the paper we will consider the real-valued Hilbert space
$L^{2}(\Omega)$, where we write
$\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{L^{2}(\Omega)}$ for its
associated inner product and, sometimes for brevity,
$\lVert\cdot\rVert=\lVert\cdot\rVert_{L^{2}(\Omega)}$ for its norm.

The metric projection onto a nonempty, closed, convex subset $\mathscr{C}$ of
a Hilbert space is a well-defined Lipschitz map
$\mathop{}\mathrm{P}_{\mathscr{C}}\colon L^{2}(\Omega)\to\mathscr{C}$, and for
all $X\in L^{2}(\Omega)$ it is characterized by

(3.9) $Y=\mathop{}\mathrm{P}_{\mathscr{C}}X\iff Y\in\mathscr{C},\quad\langle
X-Y,Z-Y\rangle\leq 0\ \text{for all}\ Z\in\mathscr{C},$

see, e.g., [2, Theorem 3.16] or [29, Proposition 1.37]. Denoting by
$\mathscr{C}^{\circ}$ the element of minimal norm in a closed and convex set
$\mathscr{C}\subset L^{2}(\Omega)$, (3.9) is equivalent to

$\lVert X-Y\rVert_{L^{2}(\Omega)}=\min_{Z\in\mathscr{C}}\lVert
X-Z\rVert_{L^{2}(\Omega)}\iff Y=\left(X-\mathscr{C}\right)^{\circ}.$

Let us now consider what can be seen as a generalization of $\mathbb{K}^{N}$
from (2.3), namely

(3.10) $\mathscr{K}\coloneqq\\{X\in L^{2}(\Omega)\colon X\text{ is
nondecreasing and right-continuous}\\},$

which is a closed, convex cone in $L^{2}(\Omega)$. The indicator function
$I_{\mathscr{K}}$ of $\mathscr{K}$ is then given by

(3.11) $I_{\mathscr{K}}(X)=\begin{cases}0,&X\in\mathscr{K},\\\
+\infty,&X\notin\mathscr{K}.\end{cases}$

Since the set $\mathscr{K}$ is convex, $I_{\mathscr{K}}$ is a convex
functional.

For a given proper and convex functional $\mathcal{F}$ on $L^{2}(\Omega)$, its
subdifferential at $X\in L^{2}(\Omega)$ is defined as the set

$\partial\mathcal{F}(X)=\left\\{Y\in
L^{2}(\Omega)\colon\mathcal{F}(Z)-\mathcal{F}(X)\geq\int_{\Omega}Y(Z-X),\
\forall\,Z\in L^{2}(\Omega)\right\\}$

For $X\in\mathscr{K}$, the subdifferential $\partial I_{\mathscr{K}}(X)$
coincides with the normal cone $N_{X}\mathscr{K}$ of $\mathscr{K}$ at $X$,
which can be defined as

(3.12) $N_{X}\mathscr{K}\coloneqq\left\\{W\in
L^{2}(\Omega)\colon\int_{\Omega}W(Y-X)\mathop{}\\!\mathrm{d}m\leq 0\
\forall\,Y\in\mathscr{K}\right\\}.$

Comparing (3.9) and (3.12) we find the following equivalence. For any $X,Y\in
L^{2}(\Omega)$ we have

(3.13) $Y=\mathop{}\mathrm{P}_{\mathscr{K}}X\iff X-Y\in\partial
I_{\mathscr{K}}(Y).$

The tangent cone $T_{X}\mathscr{K}$ of $\mathscr{K}$ at $X\in\mathscr{K}$ can
then be defined similarly to the particle case as

(3.14) $T_{X}\mathscr{K}\coloneqq\mathrm{cl}\left\\{\vartheta(Y-X)\colon
Y\in\mathscr{K},\vartheta\geq 0\right\\}.$

Alternatively, it can be characterized as the as the polar cone of the normal
cone, that is,

(3.15) $T_{X}\mathscr{K}\coloneqq\left\\{U\in
L^{2}(\Omega)\colon\int_{\Omega}U(m)W(m)\mathop{}\\!\mathrm{d}m\leq 0\
\forall\,W\in N_{X}\mathscr{K}\right\\}.$

It is convenient to introduce the following set, generalizing
$\Omega_{\boldsymbol{x}}$ in (2.5),

(3.16) $\Omega_{X}\coloneqq\\{m\in\Omega\colon X\text{ is constant in a
neighborhood of }m\\},$

which can be thought of as the set of “concentrated mass” for
$\rho\in\mathscr{P}(\mathbb{R})$ corresponding to $X_{\rho}=X$. Note that this
set can be written as a countable union of disjoint intervals,
$\Omega_{X}=\sqcup_{i}(\alpha_{i},\beta_{i})$. Then we may equivalently
characterize the tangent cone as follows, cf. [6, Lemma 2.4],

(3.17) $T_{X}\mathscr{K}=\left\\{U\in L^{2}(\Omega)\colon U\text{ is
nondecreasing on each maximal interval
}(\alpha,\beta)\subset\Omega_{X}\right\\}.$

The following characterization of $N_{X}\mathscr{K}$ can be found in [6, Lemma
2.3] and builds upon [24, Theorem 3.9].

######  Lemma 3.1 (Characterization of the normal cone $N_{X}\mathscr{K}$).

Let $X\in\mathscr{K}$ and $W\in L^{2}(\Omega)$ be given, and write

(3.18)
$\Xi_{W}(m)\coloneqq\int_{0}^{m}W(\omega)\mathop{}\\!\mathrm{d}\omega\quad\text{for
all }m\in[0,1].$

Then $W\in N_{X}\mathscr{K}$ if and only if $\Xi_{W}\in\mathscr{N}_{X}$, where
$\mathscr{N}_{X}$ is the convex cone defined as

$\mathscr{N}_{X}\coloneqq\left\\{\Xi\in C([0,1])\colon\Xi\geq
0\enspace\mathrm{in}\enspace[0,1]\enspace\mathrm{and}\enspace\Xi=0\enspace\mathrm{in}\enspace[0,1]\setminus\Omega_{X}\right\\}.$

In particular, for every $X_{1},X_{2}\in\mathscr{K}$ we have

(3.19) $\Omega_{X_{1}}\subset\Omega_{X_{2}}\implies
N_{X_{1}}\mathscr{K}\subset N_{X_{2}}\mathscr{K}.$

Based on (3.16) we introduce the closed subspace $\mathscr{H}_{X}\subset
L^{2}(\Omega)$ given by

(3.20) $\mathscr{H}_{X}\coloneqq\left\\{U\in L^{2}(\Omega)\colon U\text{ is
constant on each interval }(m_{l},m_{r})\subset\Omega_{X}\right\\},$

for which we have the implications

(3.21)
$\Omega_{X_{1}}\subset\Omega_{X_{2}}\implies\mathscr{H}_{X_{2}}\subset\mathscr{H}_{X_{1}},\qquad
X_{1},X_{2}\in\mathscr{K}$

and

$U\in\mathscr{H}_{X}\iff\pm U\in T_{X}\mathscr{K}.$

We can define the projection of $f\in L^{2}(\Omega)$ onto the subspace (3.20)
as

(3.22)
$\mathop{}\mathrm{P}_{\mathscr{H}_{X}}f=\begin{cases}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha}^{\beta}f(\omega)\mathop{}\\!\mathrm{d}\omega\coloneqq\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}f(\omega)\mathop{}\\!\mathrm{d}\omega,&m\in(\alpha,\beta)\text{
a maximal interval of }\Omega_{X},\\\ f,&\text{for a.e.
}m\in\Omega\setminus\Omega_{X}.\end{cases}$

From [6, Lemma 2.6] we have

(3.23) $X\in\mathscr{K},\quad U\in
T_{X}\mathscr{K}\implies\mathop{}\mathrm{P}_{\mathscr{H}_{X}}U-U\in
N_{X}\mathscr{K}$

The characterization (3.17) of the tangent cone $T_{X}\mathscr{K}$ leads us to
consider the closed, convex cone defined by

(3.24) $\mathscr{K}_{(\alpha,\beta)}\coloneqq\left\\{X\in
L^{2}((\alpha,\beta))\colon X\text{ is nondecreasing and right-
continuous}\right\\}$

for $(\alpha,\beta)\subset(0,1)$, such that $\mathscr{K}_{(0,1)}=\mathscr{K}$.
In [24, Theorem 3.1], a characterization of the projection of $f\in
L^{2}(\Omega)$ onto $\mathscr{K}$ is given in terms of the right-derivative of
its primitive’s lower convex envelope on $\Omega=(0,1)$. We recall that the
lower convex envelope $F^{**}_{(\alpha,\beta)}$ of a function $F\in
C([\alpha,\beta],\mathbb{R})$ is defined as

(3.25) $F^{**}_{(\alpha,\beta)}(m)=\sup\left\\{a+bm\colon a,b\in\mathbb{R},\
a+b\omega\leq F(\omega)\ \forall\>\omega\in[\alpha,\beta]\right\\},\quad
m\in[\alpha,\beta].$

This is the greatest bounded, (lower semi-)continuous and convex function $G$
satisfying $G\leq F$ in $[\alpha,\beta]$; hence it is left- and right-
differentiable for every $m\in(\alpha,\beta)$, and its right-derivative
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}m}F^{**}$ is
nondecreasing and right-continuous. We note that the notation involving $**$
alludes to the fact that the lower convex envelope in this case coincides with
the biconjugate, or twice the Legendre–Fenchel transform, of $F$. However,
there is nothing in the proof of [24, Theorem 3.1] which relies on the domain
being $(0,1)$; hence, with the appropriate changes we have the following
result.

######  Proposition 3.2 (Projection on $\mathscr{K}_{(\alpha,\beta)}$).

Let $f\in L^{2}((\alpha,\beta))$ for $(\alpha,\beta)\subset(0,1)$ and let
$F(m)=\int_{\alpha}^{m}f(\omega)\mathop{}\\!\mathrm{d}\omega$. Then

(3.26)
$\mathop{}\mathrm{P}_{\mathscr{K}_{(\alpha,\beta)}}f=\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}m}F^{**}_{(\alpha,\beta)},$

where $F^{**}_{(\alpha,\beta)}$ is the lower convex envelope in (3.25).
Moreover, for any convex, lower semi-continuous function
$\varphi\colon\mathbb{R}\to(-\infty,+\infty]$ and $f,g\in
L^{2}((\alpha,\beta))$ we have

$\int_{\alpha}^{\beta}\varphi\left(\mathop{}\mathrm{P}_{\mathscr{K}_{(\alpha,\beta)}}f-\mathop{}\mathrm{P}_{\mathscr{K}_{(\alpha,\beta)}}g\right)\mathop{}\\!\mathrm{d}m\leq\int_{\alpha}^{\beta}\varphi(f-g)\mathop{}\\!\mathrm{d}m.$

In particular, $\mathop{}\mathrm{P}_{\mathscr{K}_{(\alpha,\beta)}}$ is a
contraction in every space $L^{p}((\alpha,\beta))$, $p\in[1,\infty]$.

A useful auxiliary result for proving the above result is [24, Lemma 3.2],
which below is appropriately modified to match our setting.

######  Lemma 3.3.

For any $f\in L^{2}((\alpha,\beta))$ and $F\in C([\alpha,\beta],\mathbb{R})$
as defined in Proposition 3.2, $F^{**}_{(\alpha,\beta)}$ is continuous on
$[\alpha,\beta]$, locally Lipschitz on $(\alpha,\beta)$, and coincides with
$F$ for $m=\alpha,\beta$. If moreover $f\in L^{\infty}((\alpha,\beta))$, then
$F$ and $F^{**}_{(\alpha,\beta)}$ are Lipschitz-continuous on
$[\alpha,\beta]$.

As a consequence of the above results, for any $f\in L^{2}(\Omega)$ we can
consider its restriction to $(\alpha,\beta)\subset\Omega$, namely
$f|_{(\alpha,\beta)}\in L^{2}((\alpha,\beta))$, and slightly abusing notation
we will still denote this by $f$. Hence, when we write
$\mathop{}\mathrm{P}_{\mathscr{K}_{(\alpha,\beta)}}f\in
L^{2}((\alpha,\beta))$, we mean the projection of $f|_{(\alpha,\beta)}$ on
$\mathscr{K}_{(\alpha,\beta)}$ from (3.24). Then, if we identify
$\mathscr{K}=\mathscr{K}_{(0,1)}$ and $F^{**}=F^{**}_{(0,1)}$ in the above, we
recover the original results.

Now, combining the characterization (3.17), (3.24) and Proposition 3.2, we see
that for $X\in\mathscr{K}$ and $f\in L^{2}(\Omega)$, the projection of $f$ on
$T_{X}\mathscr{K}$ can be written as

(3.27)
$\mathop{}\mathrm{P}_{T_{X}\mathscr{K}}f=\begin{cases}\mathop{}\mathrm{P}_{\mathscr{K}_{(\alpha,\beta)}}f,&m\in(\alpha,\beta)\text{
a maximal interval of }\Omega_{X},\\\ f,&\text{for a.e.
}m\in\Omega\setminus\Omega_{X}.\end{cases}$

Let us write $F(m)=\int_{0}^{m}f(\omega)\mathop{}\\!\mathrm{d}\omega$,
$F_{(\alpha,\beta)}=F(\alpha)+\int_{\alpha}^{m}f(\omega)\mathop{}\\!\mathrm{d}\omega$
and let $F_{(\alpha,\beta)}^{**}$ be the lower convex envelope of
$F_{(\alpha,\beta)}$, as in (3.25), such that $F_{(\alpha,\beta)}^{**}\leq
F_{(\alpha,\beta)}$. Then it follows from (3.26) and Lemma 3.3 that

$F_{(\alpha,\beta)}(\beta)-F_{(\alpha,\beta)}(\alpha)=F_{(\alpha,\beta)}^{**}(\beta)-F_{(\alpha,\beta)}^{**}(\alpha)=\int_{\alpha}^{\beta}\mathop{}\mathrm{P}_{\mathscr{K}_{(\alpha,\beta)}}f\mathop{}\\!\mathrm{d}m.$

Consequently, since $\Omega_{X}=\sqcup_{i}(\alpha_{i},\beta_{i})$, it follows
from (3.27) that

$F(m)\geq\int_{0}^{m}\mathop{}\mathrm{P}_{T_{X}\mathscr{K}}f\mathop{}\\!\mathrm{d}\omega=\begin{cases}F_{(\alpha,\beta)}^{**}(m),&m\in(\alpha,\beta)\text{
a maximal interval of }\Omega_{X},\\\
F(m),&m\in[0,1]\setminus\Omega_{X}.\end{cases}$

In particular, recalling (3.22), we find

$\int_{\Omega}\mathop{}\mathrm{P}_{\mathscr{K}}f\mathop{}\\!\mathrm{d}m=\int_{\Omega}\mathop{}\mathrm{P}_{T_{X}\mathscr{K}}f\mathop{}\\!\mathrm{d}m=\int_{\Omega}\mathop{}\mathrm{P}_{\mathscr{H}_{X}}f\mathop{}\\!\mathrm{d}m=\int_{\Omega}f\mathop{}\\!\mathrm{d}m.$

That is, although the projections onto the convex cones $\mathscr{K}$,
$T_{X}\mathscr{K}$ and $\mathscr{H}_{X}$ are contractions in the
$L^{2}(\Omega)$-norm, they do not change the average of the function being
projected. Such considerations will turn out useful when studying clustering
properties in Section 6.

######  Remark 3.4.

Let $\mathscr{C}$ be a closed, convex subset of $L^{2}(\Omega)$ and $X\in
L^{2}((0,T);L^{2}(\Omega))$ such that $X(t)\in\mathscr{C}$ for a.e.
$t\in(0,T)$. Then Jensen’s inequality yields

(3.28)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{T}X(t)\mathop{}\\!\mathrm{d}t\in\mathscr{C},\quad\text{and}\quad\int_{0}^{T}X(t)\mathop{}\\!\mathrm{d}t\in\mathscr{C}\quad\text{if
$\mathscr{C}$ is a cone.}$

##  4\. Lagrangian solutions of the Euler-alignment system

We recall the assumption $\bar{\rho}\in\mathscr{P}_{2}(\mathbb{R})$, for which
we use equations (3.1) and (3.2) to define the following monotone
rearrangement $\bar{X}\in L^{2}(\Omega)$ satisfying
$\bar{X}_{\\#}\mathfrak{m}=\bar{\rho}$. Further recalling $\bar{v}\in
L^{2}(\mathbb{R},\bar{\rho})$ as well as $\psi_{t}$ defined in (1.9), we
introduce the quantities $\bar{V}\in L^{2}(\Omega)$ and
$\bar{\Psi}\coloneqq\Psi[\bar{X},\bar{V}]$, where

(4.1) $\Psi[X,V]\coloneqq
V+\int_{\Omega}\Phi(X-X(\omega))\mathop{}\\!\mathrm{d}\omega=V+\Phi\ast\rho(X),\
\text{where}\ X_{\\#}\mathfrak{m}=\rho.$

Then it also follows, recall (1.6), that $\bar{\Psi}\in L^{2}(\Omega)$.
Mimicking the arguments in the particle case, we arrive at a first-order
differential inclusion corresponding to (2.12),

(4.2) $\dot{X}_{t}+\partial
I_{\mathscr{K}}(X_{t})\ni\bar{\Psi}-\Phi\ast\rho_{t}(X_{t}).$

We will always use the isometry between $\mathscr{K}\subset L^{2}(\Omega)$ and
$\mathscr{P}_{2}(\mathbb{R})$ to have $X_{t\\#}\mathfrak{m}=\rho_{t}$. The
relation (4.2) provides us with an associated “prescribed” velocity

(4.3)
$U[X_{t}]\coloneqq\bar{\Psi}-\int_{\Omega}\Phi(X_{t}-X_{t}(\omega))\mathop{}\\!\mathrm{d}\omega\in
L^{2}(\Omega),$

such that $U[\bar{X}]=\bar{V}$. Observe that $U[X_{t}]$ is the velocity
$X_{t}\in\mathscr{K}$ would have if $\partial I_{\mathscr{K}}(X_{t})=\\{0\\}$,
i.e., $T_{X_{t}}\mathscr{K}=L^{2}(\Omega)$; that is, in absence of mass
concentration.

The next result ensures that strong convergence of $X$ in $L^{2}(\Omega)$
yields strong convergence of $\Phi\ast\rho(X)$.

######  Lemma 4.1 (Uniform continuity of $\Phi\ast\rho$).

Assume $X_{i}\in\mathscr{K}$ and $\rho_{i}\in\mathscr{P}_{2}(\mathbb{R})$ with
$\rho_{i}=X_{i\\#}\mathfrak{m}$ for $i\in\\{1,2\\}$. Then there exists a
modulus of continuity $\omega_{\Phi}$, depending only on $\Phi$, such that

$\displaystyle\lVert(\Phi\ast\rho_{1})\circ X_{1}-(\Phi\ast\rho_{2})\circ
X_{2}\rVert_{L^{2}(\Omega)}$ $\displaystyle\leq\omega_{\Phi}(\lVert
X_{1}-X_{2}\rVert_{L^{2}(\Omega)}),$

or equivalently, using the semidistance $U_{2}$ (3.7) and the
Kantorovich–Rubinstein metric $W_{2}$ (3.4),

$U_{2}((\rho_{1},\Phi\ast\rho_{1}),(\rho_{2},\Phi\ast\rho_{2}))\leq\omega_{\Phi}(W_{2}(\rho_{1},\rho_{2})).$

###### Proof.

We will make use of the following estimate; let $x_{1},x_{2}\in\mathbb{R}$,
then for $\rho\in\mathscr{P}(\mathbb{R})$ we find

(4.4) $\displaystyle\lvert(\Phi\ast\rho)(x_{2})-(\Phi\ast\rho)(x_{1})\rvert$
$\displaystyle\leq\int_{\mathbb{R}}\left\lvert\int^{x_{2}}_{x_{1}}\phi(x-y)\mathop{}\\!\mathrm{d}x\right\rvert\mathop{}\\!\mathrm{d}\rho(y)$
$\displaystyle\leq\int_{\mathbb{R}}\left\lvert\int^{x_{2}}_{x_{1}}\phi\left(x-\frac{x_{1}+x_{2}}{2}\right)\mathop{}\\!\mathrm{d}x\right\rvert\mathop{}\\!\mathrm{d}\rho(y)=2\Phi\left(\frac{\lvert
x_{2}-x_{1}\rvert}{2}\right),$

where the second inequality follows from $\phi$ being radially decreasing. Now
we estimate

$\lVert(\Phi\ast\rho_{1})\circ X_{1}-(\Phi\ast\rho_{2})\circ
X_{2}\rVert_{L^{2}(\Omega)}^{2}\leq 2(I_{1}+I_{2})$

where

$I_{1}\coloneqq\lVert(\Phi\ast\rho_{1})\circ X_{2}-(\Phi\ast\rho_{2})\circ
X_{2}\rVert_{L^{2}(\Omega)}^{2},\qquad
I_{2}\coloneqq\lVert(\Phi\ast\rho_{1})\circ X_{1}-(\Phi\ast\rho_{1})\circ
X_{2}\rVert_{L^{2}(\Omega)}^{2},\qquad.$

Then, from (4.4), the concavity of $\Phi$ and Jensen’s inequality we obtain

$\displaystyle I_{1}$
$\displaystyle\leq\int_{\Omega}\left(\int_{\Omega}\left\lvert\Phi(X_{2}(m)-X_{1}(\omega))-\Phi(X_{2}(m)-X_{2}(\omega))\right\rvert\mathop{}\\!\mathrm{d}\omega\right)^{2}\mathop{}\\!\mathrm{d}m$
$\displaystyle\leq\int_{\Omega}\left(2\int_{\Omega}\Phi\left(\frac{\lvert
X_{1}(\omega)-X_{2}(\omega)\rvert}{2}\right)\mathop{}\\!\mathrm{d}\omega\right)^{2}\mathop{}\\!\mathrm{d}m\leq
4\Phi\left(\frac{1}{2}\lVert X_{1}-X_{2}\rVert_{L^{1}(\Omega)}\right)^{2}.$

On the other hand, defining $\Omega_{-}\coloneqq\\{m\in\Omega\colon\lvert
X_{1}-X_{2}\rvert\leq 2\\}$ and
$\Omega_{+}\coloneqq\Omega\setminus\Omega_{-}$, the same inequalities together
with (1.6) yield

$\displaystyle I_{2}$ $\displaystyle\leq
4\int_{\Omega}\Phi\left(\frac{1}{2}\lvert
X_{1}(m)-X_{2}(m)\rvert\right)^{2}\mathop{}\\!\mathrm{d}m$ $\displaystyle\leq
4\Phi(1)\int_{\Omega_{-}}\Phi\left(\frac{1}{2}\lvert
X_{1}(m)-X_{2}(m)\rvert\right)\mathop{}\\!\mathrm{d}m+4\int_{\Omega_{+}}\left(\Phi(1)\frac{1}{2}\lvert
X_{1}(m)-X_{2}(m)\rvert\right)^{2}\mathop{}\\!\mathrm{d}m$ $\displaystyle\leq
4\Phi(1)\Phi\left(\frac{1}{2}\lVert
X_{1}-X_{2}\rVert_{L^{1}(\Omega)}\right)+\left(\Phi(1)\lVert
X_{1}-X_{2}\rVert_{L^{2}(\Omega)}\right)^{2}.$

Since $\mathfrak{m}(\Omega)=1$, we have $\lVert
X\rVert_{L^{1}(\Omega)}\leq\lVert X\rVert_{L^{2}(\Omega)}$, and the result
follows from collecting all the bounds,

$\displaystyle\lVert(\Phi\ast\rho_{1})\circ X_{1}-(\Phi\ast\rho_{2})\circ
X_{2}\rVert_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
8\Phi\left(\max\left\\{1,\frac{1}{2}\lVert
X_{1}-X_{2}\rVert_{L^{2}(\Omega)}\right\\}\right)\Phi\left(\frac{1}{2}\lVert
X_{1}-X_{2}\rVert_{L^{2}(\Omega)}\right)$
$\displaystyle\quad+\left(\Phi(1)\lVert
X_{1}-X_{2}\rVert_{L^{2}(\Omega)}\right)^{2}.$

∎

###  4.1. The associated functional and gradient flow structure

Next we will see how (4.2) can be regarded as a gradient flow for a certain
convex and lower semi-continuous functional. Based on the gradient flow
structure in the particle case, cf. (1.8), we are led to consider the
following functional

(4.5)
$\mathcal{V}(X)=\mathcal{W}_{\Phi}(X)-\langle\bar{\Psi},X\rangle\coloneqq\frac{1}{2}\iint_{\Omega\times\Omega}W_{\Phi}(X(m)-X(\omega))\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}m-\int_{\Omega}\bar{\Psi}(m)X(m)\mathop{}\\!\mathrm{d}m,$

where $W_{\Phi}$ is the primitive of the odd function $\Phi$ from (1.5)
satisfying $W_{\Phi}(0)=0$, i.e., the even function

(4.6) $W_{\Phi}(x)\coloneqq\int_{0}^{x}\Phi(y)\mathop{}\\!\mathrm{d}y.$

This function is also convex on the whole of $\mathbb{R}$ due to $\Phi(x)$
being nondecreasing. We observe that $\mathcal{W}_{\Phi}$ is a proper
functional for $X\in L^{2}(\Omega)$, as it can be bounded with the pointwise
linear bound (1.6). The second term of $\mathcal{V}$ is linear in $X$, and
hence bounded by Cauchy–Schwarz and $\bar{\Psi}\in L^{2}(\Omega)$. We conclude
that $\mathcal{V}$ is a proper functional with
$\operatorname*{dom}(\mathcal{V})=L^{2}(\Omega)$. One can check that the
Gâteaux derivative $\nabla_{X}\mathcal{V}(X)$ of $\mathcal{V}$ is exactly the
negative of the right-hand side in (4.2), that is, $-U[X]$ from (4.3), owing
to the oddness of $\Phi$. That is,

$\nabla_{X}\mathcal{W}_{\Phi}(X)=\Phi\ast\rho(X),\qquad\nabla_{X}\mathcal{V}(X)=\Phi\ast\rho(X)-\bar{\Psi}=-U[X].$

We would like to show that $\mathcal{V}(X)$ is convex in $X\in\mathscr{K}$. To
this end, for $Y_{0},Y_{1}\in\mathscr{K}$ and $\vartheta\in(0,1)$ we introduce
the convex combination $Y_{\vartheta}\coloneqq(1-\vartheta)Y_{0}+\vartheta
Y_{1}$. Our goal is then to show

$\mathcal{V}(Y_{\vartheta})\leq(1-\vartheta)\mathcal{V}(Y_{0})+\vartheta\mathcal{V}(Y_{1}).$

The second term of (4.5) is already linear in $X$, and so we turn to the first
term. Observe next that we can write the argument of the inner integrand as

$Y_{\vartheta}(m)-Y_{\vartheta}(\omega)=(1-\vartheta)[Y_{0}(m)-Y_{0}(\omega)]+\vartheta[Y_{1}(m)-Y_{1}(\omega)].$

Then it follows directly from the convexity of $W_{\Phi}$ in (4.6) that also
the first term of (4.5) is convex in $X\in\mathscr{K}$, and we conclude that
the entire functional $\mathcal{V}(X)$ is convex in $X\in\mathscr{K}$.

Now, since $\mathcal{V}(X)$ is convex for $X\in\mathscr{K}$ and has Gâteaux
derivative $\nabla_{X}\mathcal{V}(X)=-U[X]$, it follows from [29, Proposition
3.20] that its subdifferential then reduces to a single element.

######  Proposition 4.2.

For $X\in\mathscr{K}$, the subdifferential $\partial\mathcal{V}(X)$ of
$\mathcal{V}$ defined in (4.5) consists of a single element. In particular, we
have $\partial\mathcal{V}(X)=\\{-U[X]\\}$ for $U[X]$ defined in (4.3).

In order to ensure that our flow remains in the cone $\mathscr{K}$, we have to
consider a modified version of the functional $\mathcal{V}$, namely

(4.7) $\bar{\mathcal{V}}(X)\coloneqq\mathcal{V}(X)+I_{\mathscr{K}}(X),$

where we recall the indicator function $I_{\mathscr{K}}$ from (3.11). We would
like to show that this is convex for $X\in L^{2}(\Omega)$; that is,
considering again the convex combination $Y_{\vartheta}$ from before, now with
$Y_{0},Y_{1}\in L^{2}(\Omega)$, we want to show

(4.8)
$\bar{\mathcal{V}}(Y_{\vartheta})\leq(1-\vartheta)\bar{\mathcal{V}}(Y_{0})+\vartheta\bar{\mathcal{V}}(Y_{1}).$

Suppose either of $Y_{0}$, $Y_{1}$ does not lie in the cone $\mathscr{K}$;
then according to (3.11) and (4.7), the right-hand side of (4.8) is infinite,
and the inequality is trivially satisfied. It remains to consider the case
when both $Y_{0}$ and $Y_{1}$ belong to $\mathscr{K}$. In this case the
indicator function vanishes due to the convexity of $\mathscr{K}$, such that
$\bar{\mathcal{V}}$ reduces to $\mathcal{V}$, which is the case we treated
before. Moreover, since $\mathcal{V}$ is continuous and $\mathscr{K}$ is
closed, the functional $\bar{\mathcal{V}}$ is lower semi-continuous. We have
therefore proved the following result, which will be useful for establishing
existence and uniqueness of solutions to (4.2).

######  Proposition 4.3.

The functional $\bar{\mathcal{V}}$ in (4.7) is convex and lower semi-
continuous in $L^{2}(\Omega)$.

Like in the proof of [5, Theorem 2.13], we will make use of that the
subdifferential
$\partial\bar{\mathcal{V}}=\partial(\mathcal{V}+I_{\mathscr{K}})$ is additive
in this case. Indeed, since $\mathcal{V}$ is proper, lower semi-continuous and
convex, as well as bounded on the whole domain of
$\operatorname*{dom}(I_{\mathscr{K}})=\mathscr{K}$, this follows from the
Moreau–Rockafellar theorem [29, Theorem 3.30], see also the the proof of [29,
Proposition 3.61].

######  Proposition 4.4.

For $X\in\mathscr{K}$, the subdifferential $\partial\bar{\mathcal{V}}(X)$ of
$\bar{\mathcal{V}}$ in (4.7) can be written as

$\partial\bar{\mathcal{V}}(X)=\partial I_{\mathscr{K}}(X)-U[X]=\partial
I_{\mathscr{K}}(X)-\bar{\Psi}+\int_{\Omega}\Phi(X-X(\omega))\mathop{}\\!\mathrm{d}\omega.$

Now we will provide the definition of an $L^{2}$-gradient flow solution
associated with the functional (4.7), cf. [5, Definition 2.11]. This will
justify the differential inclusion (4.2) for the Euler-alignment system.

######  Definition 4.5 ($L^{2}$-gradient flow).

An absolutely continuous curve $X_{t}\colon[0,\infty)\to L^{2}(\Omega)$ is an
$L^{2}$-gradient flow for the functional $\bar{\mathcal{V}}$ in (4.7) if it
satisfies the differential inclusion

(4.9) $-\dot{X}_{t}\in\partial\bar{\mathcal{V}}(X_{t})=-U[X_{t}]+\partial
I_{\mathscr{K}}(X_{t})\quad\text{for a.e.}\ t>0.$

Observe that (4.9) is exactly (4.2), which was derived heuristically from the
particle case in Section 2.

###  4.2. Lagrangian solutions

Since $L^{2}(\Omega)$ is a Hilbert space, we will, like the works [24, 6, 5],
rely on the theory of [8] to establish existence and uniqueness of a solution
$X_{t}$ evolving according to the minimal element of the subdifferential
$\partial\bar{\mathcal{V}}(X_{t})$. This element is sometimes called the
principal section, and we denote it by
$\partial^{\circ}\bar{\mathcal{V}}(X_{t})$. The subdifferential
$\partial\bar{\mathcal{V}}$ is a maximally monotone operator, cf. [8, Exemple
2.3.4], and we can apply [8, Theoreme 3.1] to deduce existence and uniqueness
of such solutions to (4.9). In fact, since $\bar{\mathcal{V}}$ is proper,
lower semi-continuous and convex we can directly apply [8, Theoreme 3.2] to
obtain the following result.

######  Theorem 4.6 (Lagrangian solution).

Let $\bar{X}\in\mathscr{K}$ and $\bar{V}\in L^{2}(\Omega)$, such that
$\bar{\Psi}=\Psi[\bar{X},\bar{V}]$ given by (4.1). Then there exists a unique
gradient flow solution $X_{t}\colon[0,\infty)\to\mathscr{K}$ in the sense of
Definition 4.5 with initial data $\bar{X}$ which evolves according to the
minimal selection of $\partial\bar{\mathcal{V}}$, that is,

$-\dot{X}_{t}\in\partial^{\circ}\bar{\mathcal{V}}(X_{t})\quad\text{for a.e.}\
t>0.$

We will call this solution the Lagrangian solution associated with the Euler-
alignment system.

From the same theory, the Lagrangian solution in Theorem 4.6 enjoys many
useful properties, and following [6, Theorem 3.5], we list some of them here.

######  Corollary 4.7 (Properties of the Lagrangian solution).

Let $X_{t}$ be the Lagrangian solution from Theorem 4.6, with corresponding
prescribed velocity $U[X_{t}]$ as defined in (4.3). Then it follows:

  1. (a)

The right-derivative
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X_{t}\eqqcolon
V_{t}$ exists for all $t\geq 0$.

  2. (b)

The velocity $V_{t}$ is the minimal element of the subdifferential, that is,

(4.10) $V_{t}=\left(U[X_{t}]-\partial I_{\mathscr{K}}(X_{t})\right)^{\circ}.$

In particular, (4.2) holds true for all $t\geq 0$ if we replace $\dot{X}_{t}$
with $V_{t}$.

  3. (c)

The velocity $V_{t}$ is the projection of $U[X_{t}]$ on the tangent cone
$T_{X_{t}}\mathscr{K}$:

(4.11)
$V_{t}=\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}U[X_{t}],\qquad\Psi_{t}\coloneqq
V_{t}+\Phi\ast\rho_{t}(X_{t})=\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}\bar{\Psi}\quad\text{for
all }t\geq 0.$

  4. (d)

Continuity of the velocity: $t\mapsto V_{t}$ is right-continuous for all
$t\geq 0$, in particular

(4.12) $\lim\limits_{t\to 0+}V_{t}=\bar{V}\quad\text{if and only
if}\quad\bar{V}\in T_{\bar{X}}\mathscr{K}.$

Moreover, the map $t\mapsto\lVert V_{t}\rVert_{L^{2}(\Omega)}$ is
nondecreasing, and so there is an at most countable set of times
$\mathcal{T}\subset(0,\infty)$ where it is discontinuous. Then $t\mapsto
X_{t}$ is continuously differentiable in $(0,\infty)\setminus\mathcal{T}$.
Defining $\rho_{t}\coloneqq
X_{t\\#}\mathfrak{m}\in\mathscr{P}_{2}(\mathbb{R})$, there exists a unique map
$v_{t}\in L^{2}(\mathbb{R},\rho_{t})$ such that

(4.13)
$\dot{X}_{t}=V_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}U[X_{t}]=v_{t}\circ
X_{t}\in\mathscr{H}_{X_{t}}\ \text{for every
}t\in(0,\infty)\setminus\mathcal{T}.$

  5. (e)

Dissipation identity:

(4.14)
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}\mathcal{V}(X_{t})=-\lVert
V_{t}\rVert_{L^{2}(\Omega)}^{2}\quad\text{for $t>0$,
implying}\quad\mathcal{V}(X_{s})-\mathcal{V}(X_{t})=\int_{s}^{t}\lVert
V_{r}\rVert_{L^{2}(\Omega)}^{2}\mathop{}\\!\mathrm{d}r$

  6. (f)

Stability estimates: Let $(X^{i}_{t},V_{t}^{i})$ be Lagrangian solutions of
(4.2) with respective initial data $(\bar{X}^{i},\bar{V}^{i})$, such that
$\bar{\Psi}^{i}=\Psi[\bar{X}^{i},\bar{V}^{i}]$. Then we have

(4.15) $\lVert X_{t}^{1}-X_{t}^{2}\rVert_{L^{2}(\Omega)}\leq\lVert
X_{s}^{1}-X_{s}^{2}\rVert_{L^{2}(\Omega)}+(t-s)\lVert\bar{\Psi}^{1}-\bar{\Psi}^{2}\rVert_{L^{2}(\Omega)}$

for $s\leq t$, as well as

(4.16) $\int_{0}^{T}\lVert
V_{t}^{1}-V_{t}^{2}\rVert^{2}_{L^{2}(\Omega)}\mathop{}\\!\mathrm{d}t\leq
C\left(\sum_{i=1}^{2}(\lVert\bar{X}^{i}\rVert+\lVert\bar{V}^{i}\rVert+\sqrt{T}\lVert\bar{\Psi}^{i}\rVert)\right)\left(\lVert\bar{X}^{1}-\bar{X}^{2}\rVert_{L^{2}(\Omega)}+\sqrt{T}\lVert\bar{\Psi}^{1}-\bar{\Psi}^{2}\rVert_{L^{2}(\Omega)}\right),$

for some constant $C>0$ independent of $T$ and the initial data.

###### Proof.

Properties (a), (b) and the right-continuity of $V_{t}$ are consequences of
[8, Theoreme 3.1], while (e) follows from [8, Theoreme 3.2].

By property (b), $V_{t}$ is the $L^{2}$-projection of the zero function onto
the closed, convex subset $U[X_{t}]-\partial I_{\mathscr{K}}(X_{t})$, and so
by (3.9) it follows that $\langle V,U-V-\xi\rangle\geq 0$ for all
$\xi\in\partial I_{\mathscr{K}}(X_{t})$. Furthermore, since $U[X_{t}]-V_{t}$
belongs to the cone $\partial I_{\mathscr{K}}(X_{t})$, it follows that both
$0$ and $2(U[X_{t}]-V_{t})$ belong to $\partial I_{\mathscr{K}}(X_{t})$; hence
by the previous inequality $U[X_{t}]-V_{t}\perp V_{t}$. On the other hand, by
property (a) and (3.14) it is clear that $V_{t}\in T_{X_{t}}\mathscr{K}$. Then
it follows from the characterization (3.9) that
$V_{t}=\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}U[X_{t}]$. Since
$\Phi\ast\rho_{t}(X_{t})\in\mathscr{H}_{X_{t}}$, it is clear from (3.17) that
$\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}(U[X_{t}])=\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}\bar{\Psi}-\Phi\ast\rho_{t}(X_{t})$.

The stability estimates (f) follow from instead considering (4.9) from the
point of view of the proper, convex and lower semi-continuous functional
$\bar{\mathcal{W}}_{\Phi}(X)=\mathcal{W}_{\Phi}(X)+I_{\mathscr{K}}(X)$, and
that (4.9) is equivalent to
$\dot{X}_{t}+\partial\bar{\mathcal{W}}_{\Phi}(X_{t})\ni\bar{\Psi}$. Then
(4.15) follows from [8, Lemme 3.1]. On the other hand, (4.16) is a consequence
of [31, Theorem 2]; in their notation, our setting translates to the Hilbert
triple $V=H=V^{\prime}=L^{2}(\Omega)$ and quantities
$w(u_{0},f)=\lVert\bar{X}\rVert_{L^{2}(\Omega)}+\lVert\bar{V}\rVert_{L^{2}(\Omega)}$
and $\lVert
f\rVert_{H^{1}(0,T;V^{\prime})}^{2}=T\lVert\bar{\Psi}\rVert^{2}_{L^{2}(\Omega)}$.
∎

######  Remark 4.8 (The initial velocity).

Corollary 4.7 allows us to have $\bar{V}\in
T_{\bar{X}}\mathscr{K}\setminus\mathscr{H}_{\bar{X}}$, and still have the
right-derivative $V_{t}\to\bar{V}$ as $t\to 0+$; this would then mean that the
initial concentrated mass has different initial velocities. However, if our
aim is to study (1.1) with initial data $\bar{\rho}$ and $\bar{v}\in
L^{2}(\mathbb{R},\bar{\rho})$, this does not make much sense, as the natural
choice of $\bar{V}$ would be $\bar{v}\circ\bar{X}\in\mathscr{H}_{\bar{X}}$.

######  Remark 4.9 (Energy dissipation).

We will not directly make use of the energy dissipation relation (4.14) in
this study, but merely point out that energy dissipation plays a role in a
recent study of a kinetic Cucker–Smale model [27, Section 3].

We can think of picking the minimal element in the set $U[X_{t}]-\partial
I_{\mathscr{K}}(X_{t})$, cf. (4.10), as finding the velocity $V_{t}$ closest
to the prescribed velocity $U[X_{t}]$ which still allows $X_{t}$ to remain in
$\mathscr{K}$. Note that in (4.11) we have, in analogy with the discrete
quantity $\boldsymbol{\psi}(t)$ from Section 2, introduced $\Psi_{t}$, which
satisfies the corresponding continuum version of the discrete relation (2.9).
That is, it is the projection of the natural velocity $\bar{\Psi}$ on the
tangent cone $T_{X_{t}}\mathscr{K}$.

If the relation $V_{t}\in\mathscr{H}_{X_{t}}$ in (4.13) were to hold for all
times, this is one of the hallmarks of the so-called sticky Lagrangian
solutions, and we recall their definition [6, Definition 3.7] next.

######  Definition 4.10 (Sticky Lagrangian solutions).

A Lagrangian solution is called sticky if

(4.17) $\text{for any}\ s\leq t\ \text{we have}\
\Omega_{X_{s}}\subset\Omega_{X_{t}}.$

Because of the implications (3.19) and (3.21), any sticky Lagrangian solution
satisfies the monotonicity condition

(4.18) $\partial I_{\mathscr{K}}(X_{s})\subset\partial
I_{\mathscr{K}}(X_{t}),\quad\mathscr{H}_{X_{t}}\subset\mathscr{H}_{X_{s}}\enspace\text{for
any }s\leq t.$

Another property of sticky solutions which will prove useful below is the
following,

(4.19) $\xi\in\partial
I_{\mathscr{K}}(X_{t})\implies\mathop{}\mathrm{P}_{\mathscr{H}_{X_{s}}}\xi\in\partial
I_{\mathscr{K}}(X_{t}).$

To prove this we will use the characterization from Lemma 3.1. Let $\Xi$ be
the corresponding primitive (3.18) of $\xi$, then we know that $\Xi(m)=0$ for
$m\in\Omega\setminus\Omega_{X_{t}}$, and $\Xi(m)\geq 0$ for
$m\in\Omega_{X_{t}}$. By (4.18), $\mathop{}\mathrm{P}_{\mathscr{H}_{X_{s}}}$
will only modify $\xi$ in $\Omega_{X_{s}}\subset\Omega_{X_{t}}$, so let us
consider a maximal interval $(\alpha_{s},\beta_{s})\subset\Omega_{X_{s}}$
which is necessarily contained in a maximal interval
$(\alpha_{t},\beta_{t})\subset\Omega_{X_{t}}$. On $(\alpha_{s},\beta_{s})$,
the projection will replace $\xi$ with its average value over this interval,
which means that the corresponding primitive of
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{s}}}\xi$ on this interval will be
$\Xi(\alpha_{s})+\frac{m-\alpha_{s}}{\beta_{s}-\alpha_{s}}(\Xi(\beta_{s})-\Xi(\alpha_{s}))\geq
0$, while on the rest of $(\alpha_{t},\beta_{t})$ its value is $\Xi(m)$.
Hence, the primitive of $\mathop{}\mathrm{P}_{\mathscr{H}_{X_{s}}}\xi$ also
belongs to $\mathscr{N}_{X_{t}}$, meaning (4.19) holds.

Sticky Lagrangian solutions satisfy additional properties, and analogous to
[6, Proposition 3.8] we showcase some of these below.

######  Proposition 4.11 (Projection formulas for sticky Lagrangian
solutions).

Let $X_{t}\colon[0,\infty)\to\mathscr{K}$ be a sticky Lagrangian solution of
(4.2) in the sense of Definition 4.10. Then

(4.20) $V_{t}\in\mathscr{H}_{X_{t}},\ \text{and so also}\
\Psi_{t}\in\mathscr{H}_{X_{t}},\ \text{for all}\ t\geq 0.$

Furthermore, for $0\leq s\leq t$, $X_{t}$ and $\Psi_{t}$ satisfy

(4.21) $\Psi_{s}-\Psi_{t}\in\partial
I_{\mathscr{K}}(X_{t})\implies\Psi_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\Psi_{s},$
(4.22)
$X_{t}=\mathop{}\mathrm{P}_{\mathscr{K}}\left(X_{s}+(t-s)\Psi_{s}-\int_{s}^{t}\Phi\ast\rho_{r}(X_{r})\mathop{}\\!\mathrm{d}r\right).$

In particular, we have the formulas

(4.23)
$V_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}U[X_{t}]=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}-\Phi\ast\rho_{t}(X_{t}),\
\text{and so also}\
\Psi_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi},$ (4.24)
$X_{t}=\mathop{}\mathrm{P}_{\mathscr{K}}\left(\bar{X}+\int_{0}^{t}U[X_{s}]\mathop{}\\!\mathrm{d}s\right)=\mathop{}\mathrm{P}_{\mathscr{K}}\left(\bar{X}+t\bar{\Psi}-\int_{0}^{t}\Phi\ast\rho_{s}(X_{s})\mathop{}\\!\mathrm{d}s\right).$

###### Proof.

The inclusion (4.20) follows from the right-continuity of $V_{t}$ and (4.13)
in Corollary 4.7 together with the monotonicity property (4.18). The identity
(4.23) follows from applying the projection to (4.2) with $\dot{X}_{t}$
replaced with $V_{t}$. As a consequence of (4.23) and (4.18) we then have
$\Psi_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\Psi_{s}$. On the other
hand, $\mathop{}\mathrm{P}_{\mathscr{H}_{X_{s}}}\Psi_{t}=\Psi_{t}$ since
$\Psi_{t}$ is already constant where $X_{s}$ is constant. Then, from (4.2) it
follows that $\bar{\Psi}-\Psi_{t}\in\partial I_{\mathscr{K}}(X_{t})$, and
applying $\mathop{}\mathrm{P}_{\mathscr{H}_{X_{s}}}$ to the left-hand side we
obtain $\Psi_{s}-\Psi_{t}$, which means that (4.21) follows from (4.19); this
also implies $\Psi_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\Psi_{s}$,
since $\partial I_{\mathscr{K}}(X_{t})\subset\mathscr{H}_{X_{t}}^{\perp}$.

We then note that for $0\leq s\leq r\leq t$ we have

$\Psi_{s}-\Phi\ast\rho_{r}(X_{r})-V_{r}=\Psi_{s}-\Psi_{r}\in\partial
I_{\mathscr{K}}(X_{r})\subset\partial I_{\mathscr{K}}(X_{t}),$

and integrating this expression we obtain, cf. Remark 3.4,

$(t-s)\Psi_{s}-\int_{s}^{t}\Phi\ast\rho_{r}(X_{r})\mathop{}\\!\mathrm{d}r+X_{s}-X_{t}\in\partial
I_{\mathscr{K}}(X_{t}).$

Then (4.22) is a consequence of (3.13), while (4.24) follows from $s=0$ and
$\Psi_{0}=\bar{\Psi}$. ∎

######  Remark 4.12 (Semigroup property).

From (4.21) and (4.22) we see that the sticky Lagrangian solution is a
semigroup $\hat{S}_{t}\colon(\bar{X},\bar{\Psi})\to(X_{t},\Psi_{t})$.
Furthermore, since $\bar{V}=\bar{\Psi}-\Phi\ast\bar{\rho}(\bar{X})$,
$\rho_{t}=X_{t\\#}\mathfrak{m}$, and $V_{t}=\Psi_{t}-\Phi\ast\rho_{t}(X_{t})$,
this shows that also $\check{S}_{t}\colon(\bar{X},\bar{V})\to(X_{t},V_{t})$ is
a semigroup.

######  Remark 4.13 (Reduction to pressureless Euler).

In the case $\phi\equiv 0$ we have $\Phi\equiv 0$ and $\bar{\Psi}=\bar{V}$,
thereby recovering the projection formulas in [24, Theorem 2.6] for the
pressureless Euler system.

Observe that $\frac{1}{h}(X_{t-h}-X_{t})\in T_{X_{t}}\mathscr{K}$ and is
uniformly bounded for any for any $0<h<t$, hence it has a subsequence
converging weakly to $-V^{\prime}\in T_{X_{t}}\mathscr{K}$ by the definition
(3.15) and the cone being weakly closed. Then, by the characterization (3.17),
any such weak limit must be nonincreasing on connected components of
$\Omega_{X_{t}}$, which is analogous to the monotonicity relation leading to
the barycentric lemma (2.13) in the particle case.

###  4.3. Sticky particle solutions

Since the dynamics derived in [21] is based on sticky particles, and both [24]
and [6] use limits of particle solutions to deduce sticky behavior, we will do
the same here. Let us recall the motivation of the particle solutions from
Section 2.1, in particular we had $\boldsymbol{m}\in\mathbb{R}^{N}_{+}$. For
convenience, following [6], we introduce the set $\mathbb{M}^{N}$ defined as

$\mathbb{M}^{N}\coloneqq\left\\{\boldsymbol{m}\in\mathbb{R}_{+}^{N}\colon\sum_{i=1}^{N}m_{i}=1\right\\}.$

For $\boldsymbol{m}\in\mathbb{M}^{N}$ we can define a partition of $[0,1)$
through the quantities

(4.25) $\theta_{0}\coloneqq
0,\qquad\theta_{i}\coloneqq\sum_{j=1}^{i}m_{j},\quad i\in\\{1,\dots,N\\}.$

Let us denote by $\chi_{A}$ the characteristic function of a subset
$A\subset[0,1]$. Then we can define the finite-dimensional Hilbert space

(4.26)
$\mathscr{H}_{\boldsymbol{m}}\coloneqq\left\\{X=\sum_{i=1}^{N}x_{i}\chi_{[\theta_{i-1},\theta_{i})}\colon(x_{1},\dots,x_{N})=\boldsymbol{x}\in\mathbb{R}^{N}\right\\}\subset
L^{2}(\Omega)$

and its convex cone

(4.27)
$\mathscr{K}_{\boldsymbol{m}}\coloneqq\left\\{X=\sum_{i=1}^{N}x_{i}\chi_{[\theta_{i-1},\theta_{i})}\colon(x_{1},\dots,x_{N})=\boldsymbol{x}\in\mathbb{K}^{N}\right\\}\subset\mathscr{K}\subset
L^{2}(\Omega).$

For $X^{N}\in\mathscr{K}_{\boldsymbol{m}}$ defined through
$\boldsymbol{x}\in\mathbb{K}^{N}$ and $\boldsymbol{m}\in\mathbb{M}^{N}$, we
can introduce the empirical measure $\rho^{N}_{\boldsymbol{x}}$ and its
cumulative distribution $M^{N}$,

$\rho^{N}_{\boldsymbol{x}}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}},\qquad
M^{N}(x)=\sum_{i=1}^{N}m_{i}H(x-x_{i})=\sum_{i=1}^{N}\theta_{i}\chi_{[x_{i},x_{i+1})}(x),$

where $H(x)$ is the right-continuous Heaviside function. Then we know from
before that $M^{N}$ is the generalized inverse of $X^{N}$. Moreover, we see
that convolution of $\Phi$ with $\rho^{N}_{\boldsymbol{x}}$ yields exactly the
“discrete convolution” $\Phi_{\boldsymbol{m}}^{\ast}$ from Section 2,

$(\Phi\ast\rho^{N})(x)=\sum_{j=1}^{N}m_{j}\Phi(x-x_{j})=\sum_{j=1}^{N}\int_{\theta_{j-1}}^{\theta_{j}}\Phi(x-X^{N}(\omega))\mathop{}\\!\mathrm{d}\omega=\int_{\Omega}\Phi(x-X^{N}(\omega))\mathop{}\\!\mathrm{d}\omega.$

######  Remark 4.14.

Let $\boldsymbol{x},\boldsymbol{y}\in\mathbb{K}^{N}$ and respectively define
$X,Y\in\mathscr{K}_{\boldsymbol{m}}$ for $\boldsymbol{m}\in\mathbb{M}^{N}$ and
the empirical measures
$\rho_{\boldsymbol{x}},\rho_{\boldsymbol{y}}\in\mathscr{P}(\mathbb{R})$. Then
we have the following identities

$W_{p}(\rho_{\boldsymbol{x}},\rho_{\boldsymbol{y}})=\lVert
X-Y\rVert_{L^{p}(\mathbb{R})}=\lVert\boldsymbol{x}-\boldsymbol{y}\rVert_{\boldsymbol{m},p},$

that is, we recover the $\boldsymbol{m}$-weighted Euclidean $p$-norm from
(2.4).

Next we want to formulate the particle dynamics from Section 2 using the above
framework. For a given $\boldsymbol{m}\in\mathbb{M}^{N}$, suppose
$\bar{X}^{N}\in\mathscr{K}_{\boldsymbol{m}}$ and
$\bar{V}^{N}\in\mathscr{H}_{\bar{X}^{N}}$. We may without loss of generality
assume the initial particles to be separate, i.e.,
$\bar{\boldsymbol{x}}\in\mathrm{int}(\mathbb{K}^{N})$; otherwise, we could
reduce and relabel the particles. Then we may take
$\bar{\boldsymbol{v}}\in\mathbb{R}^{N}$, since
$T_{\bar{\boldsymbol{x}}}\mathbb{K}^{N}=\mathbb{R}^{N}$, and we define

(4.28)
$X^{N}_{t}\coloneqq\sum_{i=1}^{N}x_{i}(t)\chi_{[\theta_{i-1},\theta_{i})},\qquad\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X_{t}^{N}=V^{N}_{t}\coloneqq\sum_{i=1}^{N}v_{i}(t)\chi_{[\theta_{i-1},\theta_{i})}.$

Recalling (2.1), we compute

$\bar{\Psi}^{N}=\Psi[\bar{X}^{N},\bar{V}^{N}]=\bar{V}^{N}+(\Phi\ast\bar{\rho}^{N})(\bar{X}^{N})=\sum_{i=1}^{N}\left[\bar{v}_{i}+\sum_{j=1}^{N}m_{j}\Phi(\bar{x}_{i}-\bar{x}_{j})\right]\chi_{[\theta_{i-1},\theta_{i})}=\sum_{i=1}^{N}\bar{\psi}_{i}\chi_{[\theta_{i-1},\theta_{i})}$

and

$U[X_{t}^{N}]=\bar{\Psi}^{N}-(\Phi\ast\rho_{t}^{N})(X_{t}^{N})=\sum_{i=1}^{N}\left[\bar{\psi}_{i}-\sum_{j=1}^{N}m_{j}\Phi(x_{i}(t)-x_{j}(t))\right]\chi_{[\theta_{i-1},\theta_{i})}$

If we now let $X_{t}^{N}$ satisfy

$\dot{X}^{N}_{t}=U[X_{t}^{N}]=\bar{\Psi}^{N}-(\Phi\ast\rho^{N}_{t})(X^{N}_{t}),\quad
X_{0}^{N}=\bar{X}^{N},$

this corresponds exactly to the particle evolution (2.2) for the coefficients
of the step functions (4.28), which by Lemma 2.1 is uniquely determined with
our inelastic collision rules. Here we have by construction
$V^{N}_{0}=\bar{V}^{N}\in\mathscr{H}_{X^{N}_{0}}$. Since there are $N$
discontinuities in the initial $\bar{X}$, i.e., $N$ particles, there can at
most be $N-1$ collisions. Hence there will be a set of strictly increasing
collision times $\\{t_{k}\\}_{k=0}^{N^{*}+1}$, where for convenience we have
included $t_{0}=0$ and $t_{N^{*}+1}=+\infty$, so that $N^{*}\leq N-1$ is the
number of actual collisions. If we now consider $t\in[t_{k},t_{k+1})$, by
construction we have the relations

$\mathscr{H}_{X^{N}_{t}}=\mathscr{H}_{X^{N}_{t_{k}}},\qquad
V_{t}+\Phi\ast\rho^{N}_{t}(X^{N}_{t})=V_{t_{k}}^{N}+\Phi\ast\rho^{N}_{t_{k}}(X^{N}_{t_{k}})\iff\Psi_{t}^{N}=\Psi_{t_{k}}^{N}.$

At the time of collisions $\\{t_{k}\\}_{k=1}^{N^{*}}$, by the continuous
trajectory of $\boldsymbol{x}$ and inelastic collision rule (2.7) for
$\boldsymbol{v}$, we have

$X^{N}(t_{k}+)=X^{N}(t_{k}-),\qquad
V^{N}(t_{k}+)=\mathop{}\mathrm{P}_{\mathscr{H}_{X^{N}(t_{k})}}V^{N}(t_{k}-)\iff\Psi^{N}(t_{k}+)=\mathop{}\mathrm{P}_{\mathscr{H}_{X^{N}(t_{k})}}\Psi^{N}(t_{k}-).$

Note that we will occasionally write $X^{N}(t_{k}\pm)$ rather than
$X^{N}_{t_{k}\pm}$, etc., to avoid cluttered subscripts. Now, since (4.17) and
(4.18) hold by construction, it follows that
$\Psi^{N}_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X^{N}_{t}}}\bar{\Psi}^{N}$. It
remains to show that this sticky evolution is in fact a Lagrangian solution of
the differential inclusion (4.2), and the proof is similar to those of [24,
Theorem 4.2] and [6, Theorem 5.2].

######  Proposition 4.15.

Let
$(\bar{\boldsymbol{x}},\bar{\boldsymbol{v}},\boldsymbol{m})\in\mathrm{int}(\mathbb{K}^{N})\times\mathbb{R}^{N}\times\mathbb{M}^{N}$,
and let
$(\bar{X}^{N},\bar{V}^{N})\in\mathscr{K}_{\boldsymbol{m}}\times\mathscr{H}_{\boldsymbol{m}}$
be the corresponding step functions defined through (4.25)–(4.27). Then the
corresponding sticky particle solution $X^{N}_{t}$ is a sticky Lagrangian
solution of (4.2). In particular, it satisfies the relations of Proposition
4.11.

###### Proof.

From construction it is clear that the monotonicity property (4.17) is
satisfied. We will prove by induction on the collision times that $X^{N}_{t}$
satisfies (4.2), which by right-continuity is equivalent to

(4.29) $U[X^{N}_{t}]-V_{t}^{N}=\bar{\Psi}^{N}-\Psi^{N}_{t}\in\partial
I_{\mathscr{K}}(X_{t}^{N}).$

Consider $t\in[0,t_{1})$, then by construction
$\mathscr{H}_{X^{N}_{t}}=\mathscr{H}_{\bar{X}^{N}}$. Therefore,
$V_{t}^{N}=U[X_{t}^{N}]$, or $\Psi_{t}^{N}=\bar{\Psi}^{N}$, leading to
$0\in\partial I_{\mathscr{K}}(X^{N}_{t})$, which is true.

Assume next that (4.2) is satisfied for $t\in[t_{k-1},t_{k})$, and let us
consider $t\in[t_{k},t_{k+1})$ instead. By construction we have
$\mathscr{H}_{X^{N}_{t}}=\mathscr{H}_{X^{N}_{t_{k}}}$,
$V^{N}(t_{k}+)=\mathop{}\mathrm{P}_{\mathscr{H}_{X^{N}_{t}}}V^{N}(t_{k}-)$,
$\Psi^{N}(t_{k}+)=\mathop{}\mathrm{P}_{\mathscr{H}_{X^{N}_{t}}}\Psi^{N}(t_{k}-)$
and by hypothesis

$U[X^{N}(t_{k})]-V^{N}(t_{k}-)=\bar{\Psi}^{N}-\Psi^{N}(t_{k}-)\in\partial
I_{\mathscr{K}}(X^{N}_{t_{k}}).$

On the other hand, $-V^{N}(t_{k}-)\in T_{X^{N}_{t_{k}}}\mathscr{K}$, and also
$-\Psi^{N}(t_{k}-)\in T_{X^{N}_{t_{k}}}\mathscr{K}$, which by (3.23) yields
$\Psi^{N}(t_{k}-)-\Psi^{N}_{t_{k}}\in\partial I_{\mathscr{K}}(X^{N}_{t_{k}})$.
Now, since $\partial I_{\mathscr{K}}(X_{t_{k}})$ is a cone, we add the
previous equations together, use $\Psi^{N}_{t}=\Psi^{N}_{t_{k}}$ and the
monotonicity (4.18) to deduce that (4.29) holds. ∎

####  4.3.1. The globally sticky evolution

We now return to the Lagrangian solution from Theorem 4.6, and as in [24,
Lemma 5.1], see also [6, Remarks 5.3, 5.4], we want to use the sticky particle
evolution above to show that it is globally sticky in the sense of Definition
4.10. To this end we will use the fact that functions of the form (4.26) are
dense in $L^{2}(\Omega)$, and so there are sequences $\\{N_{n}\\}$ and
$(\bar{\boldsymbol{x}}_{n},\bar{\boldsymbol{v}}_{n},\bar{\boldsymbol{m}}_{n})\in\mathbb{K}^{N_{n}}\times\mathbb{R}^{N_{n}}\times\mathbb{M}^{N_{n}}$
such that the corresponding $(\bar{X}^{N_{n}},\bar{V}^{N_{n}})$ converge
strongly to $(\bar{X},\bar{V})$ in $L^{2}(\Omega)$. For ease of notation we
will then use the superscript $n$ rather than $N_{n}$ in the corresponding
quantities. The strong convergence and Lemma 4.1 then shows that also
$\Psi[\bar{X}^{n},\bar{V}^{n}]$ converges strongly to $\Psi[\bar{X},\bar{V}]$.
On the other hand, assuming only weak convergence for the velocity, we have

$X^{n}\to X,\ V^{n}\rightharpoonup V\ \text{in
}L^{2}(\Omega)\implies\Psi[X^{n},V^{n}]\rightharpoonup\Psi[X,V]\ \text{in
}L^{2}(\Omega).$

Then we have the following result.

######  Theorem 4.16.

Let
$(X^{n}_{t},V^{n}_{t})\in\mathscr{K}_{\boldsymbol{m}}\times\mathscr{H}_{\boldsymbol{m}}$,
where
$V^{n}_{t}\coloneqq\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X^{n}_{t}$,
be sticky particle solutions of (4.2), as defined in Proposition 4.15, for
which $\bar{X}_{n}$ and $\bar{V}_{n}$ respectively converge to $\bar{X}$ and
$\bar{V}$ in $L^{2}(\Omega)$. Then:

  1. (a)

$X^{n}_{t}$ converges to $X_{t}$ in $L^{2}(\Omega)$ uniformly in each compact
interval, where $X_{t}$ is Lipschitz curve with values in $\mathscr{K}$.

  2. (b)

The Lipschitz curve $X_{t}$ is a sticky Lagrangian solution of (4.2), in
particular $X_{t}$ and
$V_{t}=\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X_{t}$
satisfy the properties of Corollary 4.7 and Proposition 4.11.

  3. (c)

$V^{n}_{t}$ converges strongly to $V_{t}$ in $L^{2}(0,T;L^{2}(\Omega))$ for
every $T>0$.

  4. (d)

For any weak accumulation point $V^{\prime}$ of $V^{n}_{t}$, we have
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}V^{\prime}=V_{t}$. Similarly, for
$\Psi^{\prime}\coloneqq V^{\prime}+\Phi\ast\rho(X_{t})$ we have
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\Psi^{\prime}=\Psi_{t}$.

  5. (e)

Let $\mathcal{T}\subset(0,\infty)$ be the countable set of discontinuities for
$t\mapsto\lVert V_{t}\rVert_{L^{2}(\Omega)}$. Then $V^{n}_{t}\to V_{t}$ and
$\Psi^{n}_{t}\to\Psi_{t}$ in $L^{2}(\Omega)$ for every
$t\in[0,\infty)\setminus\mathcal{T}$.

###### Proof.

Property (a) follows from the stability estimate (4.15); indeed, on every
compact time interval $X^{n}_{t}$ is a Cauchy sequence in the closed cone
$\mathscr{K}\subset L^{2}(\Omega)$. Furthermore, by property (d) of Corollary
4.7, $X^{n}_{t}$ has Lipschitz-constant bounded by
$\lVert\bar{V}^{n}\rVert_{L^{2}(\Omega)}$. From the uniform convergence on
each compact and the strong convergence $\bar{V}^{n}\to\bar{V}$ it follows
that the limit function $X_{t}$ has Lipschitz-constant bounded by
$\lVert\bar{V}\rVert_{L^{2}(\Omega)}$.

Property (b): That $X_{t}$ is a Lagrangian solution follows from stability
results for gradient flows in Hilbert spaces; indeed, by assumption it is a
weak solution in the sense of [8, Definition 3.1]. That it is a Lagrangian
solution according to Definition 4.5, i.e., a strong solution in the sense of
[8, Definition 3.1], follows from [8, Proposition 3.2] and $X_{t}$ being
absolutely continuous on each compact due to (a). That $X_{t}$ is sticky
follows from that $X^{n}_{t}$ is sticky and the stability estimate (4.15), cf.
[6, Remark 5.4]. Indeed, from the strong convergence $\bar{X}^{n}\to\bar{X}$
it follows that $\Omega_{\bar{X}}\subset\Omega_{X_{t}}$. By uniqueness, we
obtain the same solution $(X_{t},V_{t})$ by taking
$(\bar{X},\bar{V})=(X_{s},V_{s})$ for $0<s<t$. and evolve this to time $t-s$;
then by the same argument $\Omega_{X_{s}}\subset\Omega_{X_{t}}$.

Property (c) follows directly from the stability estimate (4.16).

Property (d): Let $n_{k}$ be an arbitrary subsequence such that
$V^{n_{k}}_{t}\rightharpoonup V^{\prime}$ in $L^{2}(\Omega)$. Since the
subdifferential of $\mathcal{V}(X)$ is maximally monotone, its graph is also
strongly-weakly closed, meaning we can pass to the limit in $V^{n}_{t}\in
U[X_{t}^{n}]-\partial I_{\mathscr{K}}(X^{n}_{t})$ to obtain $V^{\prime}\in
U[X_{t}]-\partial I_{\mathscr{K}}(X_{t})$. Recalling that $\partial
I_{\mathscr{K}}(X)\subset\mathscr{H}_{X}^{\perp}$, we can project the previous
relation to find
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}V^{\prime}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}U[X_{t}]$.
Since $X_{t}$ is a sticky solution it then follows from (4.23) that
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}V^{\prime}=V_{t}$.

Property (e): Let $t\in(0,\infty)\setminus\mathcal{T}$, and let $n_{k}$,
$V^{\prime}$ be as in the previous point. From the strong convergence in point
(c), there exists a dense set $\mathcal{S}\subset(0,\infty)$ we can extract a
further subsequence, not relabeled, such that $V^{n_{k}}(s)\to V(s)$ for every
$s\in\mathcal{S}$. In particular, for $s\in\mathcal{S}$ such that $s<t$ we
have

$\lVert V^{\prime}\rVert_{L^{2}(\Omega)}\leq\limsup_{k\to\infty}\lVert
V_{t}^{n_{k}}\rVert_{L^{2}(\Omega)}\leq\limsup_{k\to\infty}\lVert
V_{s}^{n_{k}}\rVert_{L^{2}(\Omega)}=\lVert V_{s}\rVert_{L^{2}(\Omega)}.$

Since $t$ is a point of continuity, we may approach it from below to obtain

$\lVert V^{\prime}\rVert_{L^{2}(\Omega)}\leq\lVert
V_{t}\rVert_{L^{2}(\Omega)}\leq\lVert
U[X_{t}]-\xi\rVert_{L^{2}(\Omega)},\quad\forall\>\xi\in\partial
I_{\mathscr{K}}(X_{t}).$

Then, as $V_{t}$ is the unique minimal element of $U[X_{t}]-\partial
I_{\mathscr{K}}(X_{t})$, it follows that $V^{\prime}=V_{t}$ and

$\limsup_{k\to\infty}\lVert V^{n_{k}}_{t}\rVert_{L^{2}(\Omega)}\leq\lVert
V_{t}\rVert_{L^{2}(\Omega)},$

which in turn gives the strong convergence of $V^{n_{k}}_{t}$ to $V_{t}$.
Furthermore, since the subsequence was arbitrary, it follows that the whole
sequence $V^{n}_{t}$ converges strongly to $V_{t}$. ∎

######  Corollary 4.17.

Let $\bar{X}\in\mathscr{K}$ and $\bar{V}\in\mathscr{H}_{\bar{X}}$. The
corresponding Lagrangian solution $X_{t}$ of Theorem 4.6 is globally sticky.
In particular, the results of Proposition 4.11 apply to $X_{t}$ and its
velocity $V_{t}$ from Corollary 4.7. Furthermore, the identity
$V_{t}=v_{t}\circ X_{t}$ from (4.13) holds for all $t\geq 0$.

###### Proof.

Since step functions are dense in $L^{2}(\Omega)$, we can approximate
$\bar{X}$, $\bar{V}$ with $\bar{X}^{n}$, $\bar{V}^{n}$ which converge strongly
in $L^{2}(\Omega)$, and apply Theorem 4.16. ∎

######  Remark 4.18 (Projection formulas).

The attractive Euler–Poisson system, obtained by replacing the velocity-
alignment forcing term in (1.1b) with the Poisson force
$-\tfrac{\alpha}{2}(\operatorname*{sgn}\ast\rho_{t})\rho_{t}$ for $\alpha>0$,
also has sticky Lagrangian solutions. Its corresponding projection formula,
cf. [6, Example 6.9], is

$X_{t}=\mathop{}\mathrm{P}_{\mathscr{K}}\left(\bar{X}+t\bar{V}-\frac{\alpha}{4}t^{2}(2m-1)\right),$

which for $\alpha=0$ reduces to the formula for the pressureless Euler system
from [24] mentioned in Remark 4.13. This projection is particularly simple in
that the expression being projected can be computed independently of the
intermediate values $X_{s}$ for $0<s<t$; indeed, it is uniquely determined by
the initial data. This differs from the projection formula (4.24), which is
more similar to the projection formula in [6, Proposition 3.8] for a more
general sticking force term $f[\rho_{t}]$ with Lagrangian representation
$F[X_{t}]$. This aligns with the observation in [22] that one cannot take the
(free-flow) particle trajectories of [15], which evolve according to the
natural velocities $\bar{\psi}_{i}$ and allow for crossings, and project these
to recover the sticky particle trajectories.

##  5\. From gradient flows to other solutions

In this section we will use the sticky Lagrangian solution established in the
previous section to obtain solutions for other equations. Naturally, in the
spirit of [24, 6], we will return to the original Euler-alignment system (1.1)
and show that the Lagrangian solution provides us with a distributional
solution according to Definition 1.2.

On the other hand, another main objective of this study is to show that the
sticky particle dynamics obtained in [21] by means of the scalar balance law
(1.11) can be realized from a gradient flow point-of-view. To this end,
following the ideas of [10], we will see how this balance law can be formally
derived from a scalar conservation law, where the flux function is the
primitive of the prescribed velocity $U[X_{t}]$ in (4.3). More importantly, we
show that the Lagrangian solution also provides us with an entropy solution of
(1.11).

###  5.1. A distributional solution of the Euler-alignment system

Now we return to the Euler-alignment system, recalling the (non-complete)
metric space $(\mathscr{T}_{2},D_{2})$ from (3.6), (3.8), and propose the
following result.

######  Theorem 5.1.

Suppose $(\bar{\rho},\bar{v})\in\mathscr{T}_{2}$. Define $\bar{X}$ according
to (3.2), i.e., $\bar{\rho}=\bar{X}_{\\#}\mathfrak{m}$, and
$\bar{V}=\bar{v}\circ\bar{X}$. Then the Lagrangian solution of Theorem 4.6
provides us with a distributional solution
$(\rho_{t},v_{t})\in\mathscr{T}_{2}$ of (1.1) in the sense of Definition 1.2,
with $\rho_{t}$ and $v_{t}$ given by Corollaries 4.7 and 4.17. In particular,
since the Lagrangian solution is globally sticky,
$S_{t}\colon(\bar{\rho},\bar{v})\to(\rho_{t},v_{t})$ is a semigroup in the
metric space $(\mathscr{T}_{2},D_{2})$.

###### Proof.

By Corollary 4.17 there is $v_{t}\in L^{2}(\mathbb{R},\rho_{t})$ such that
$V_{t}=v_{t}\circ X_{t}$ for all $t\geq 0$. In turn, we can define
$\psi_{t}\in L^{2}(\mathbb{R},\rho_{t})$ as in (1.9). We start by showing that
$(\rho_{t},\psi_{t})$ is a distributional solution of (1.10), and proceed as
in the proof of [6, Theorem 3.5]. Given $\varphi\in
C_{\mathrm{c}}^{\infty}([0,T]\times\mathbb{R})$ we use (3.3) to write

$\displaystyle\int_{0}^{\infty}$
$\displaystyle\int_{\mathbb{R}}\left[\partial_{t}\varphi(t,x)\psi_{t}(t,x)+\partial_{x}\varphi(t,x)v_{t}(t,x)\psi_{t}(t,x)\right]\mathop{}\\!\mathrm{d}\rho_{t}(x)\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{\infty}\int_{\Omega}\left[\partial_{t}\varphi(t,X_{t}(m))+\partial_{x}\varphi(t,X_{t}(m))V_{t}(m)\right]\Psi_{t}(m)\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{\infty}\int_{\Omega}\left[\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}\varphi(t,X_{t}(m))\right]\bar{\Psi}(m)\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t=-\int_{0}^{\infty}\int_{\Omega}\varphi(t,X_{t}(m))\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\bar{\Psi}(m)\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t=0,$

where we have used $\bar{\Psi}-\Psi_{t}\in\partial
I_{\mathscr{K}}(X_{t})\subset\mathscr{H}_{X_{t}}^{\perp}$. Therefore, the
“momentum” equation (1.10b) is satisfied in distributions. A similar, even
more straightforward, argument shows that the continuity equation (1.10a),
which coincides with (1.1a), is satisfied. Next, as in the proof of [21,
Theorem 6.3], we will make use of that (1.10) holds distributionally to show
that the momentum equation (1.1b) holds in distributions. Indeed, we have

$\displaystyle\partial_{t}(\rho_{t}v_{t})+\partial_{x}(\rho_{t}v_{t}^{2})$
$\displaystyle=\partial_{t}(\rho_{t}\psi_{t})+\partial_{x}(\rho_{t}v_{t}\psi_{t})-\partial_{t}(\rho_{t}(\Phi\ast\rho_{t}))-\partial_{x}(\rho_{t}v_{t}(\Phi\ast\rho_{t}))$
$\displaystyle=-(\Phi\ast\rho_{t})[\partial_{t}\rho_{t}+\partial_{x}(\rho_{t}v_{t})]-\rho_{t}[\partial_{t}(\Phi\ast\rho_{t})+v_{t}\partial_{x}(\Phi\ast\rho_{t})]$
$\displaystyle=\rho_{t}(\phi\ast(\rho_{t}v_{t}))-\rho_{t}v_{t}(\phi\ast\rho_{t}).$

It remains to verify the initial conditions (1.12). The first limit follows
from the relation $\rho_{t}=X_{t\\#}\mathfrak{m}$ and the strong limit
$X_{t}\to\bar{X}$ in $L^{2}(\Omega)$. For the second limit we need to show
that

$\lim\limits_{t\to
0+}\int_{\mathbb{R}}\varphi(x)v_{t}(x)\mathop{}\\!\mathrm{d}\rho_{t}(x)=\int_{\mathbb{R}}\varphi(x)\bar{v}(x)\mathop{}\\!\mathrm{d}\bar{\rho}(x)\quad\text{for
every}\ \varphi\in C_{\text{b}}(\mathbb{R}),$

where $C_{\text{b}}(\mathbb{R})$ is the space of continuous, bounded
functions. However, this follows from
$\bar{V}=\bar{v}\circ\bar{X}\in\mathscr{H}_{\bar{X}}\subset
T_{\bar{X}}\mathscr{K}$, $V_{t}=v_{t}\circ X_{t}$ and (4.12). The semigroup
property is a consequence of the semigroup property of the sticky Lagrangian
solution, cf. Remark 4.12. ∎

###  5.2. An entropy solution of the scalar balance law

Let us now see how our notion of gradient flow solutions relate to the entropy
solutions studied by Leslie and Tan [21].

####  5.2.1. Recovering the balance law

As noticed in [10], a Lagrangian solution $(X_{t},V_{t})$ satisfies a
conservation law of the form

(5.1) $\partial_{t}M_{t}+\partial_{x}\mathcal{U}(t,M_{t})=0,$

where the flux function $\mathcal{U}(t,m)$ is the primitive of the prescribed
velocity $U_{t}$ satisfying $U_{t}-V_{t}\in\partial I_{\mathscr{K}}(X_{t})$.
Define the primitive of $U[X_{t}]$ from (4.3),

$\int_{0}^{m}U[X_{t}](\omega)\mathop{}\\!\mathrm{d}\omega=\int_{0}^{m}(\bar{\psi}\circ\bar{X})(\omega)\mathop{}\\!\mathrm{d}\omega-\int_{0}^{m}\int_{\Omega}\Phi(X_{t}(\omega)-X_{t}(\tilde{m}))\mathop{}\\!\mathrm{d}\tilde{m}\mathop{}\\!\mathrm{d}\omega\eqqcolon
A(m)-S[X_{t}](m),$

where

(5.2)
$A(m)\coloneqq\int_{0}^{m}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega$

corresponds to the flux function $A$ in (1.11), as defined in [21]. The second
term evaluated at $m=M_{t}(x)$ can be written as

$S[X_{t}](M_{t}(x))=\int_{0}^{M_{t}(x)}\int_{\Omega}\Phi(X_{t}(\omega)-y)\mathop{}\\!\mathrm{d}\rho_{t}(y)\mathop{}\\!\mathrm{d}\omega$

Applying the Vol’pert $BV$-chain rule, see [10, Lemma 4.2], and the fact that
$X_{t}(M_{t}(x))=x$ for $\rho_{t}$-a.e. $x$, with
$\rho_{t}=\partial_{x}M_{t}$, we obtain

$\partial_{x}S[X_{t}](M_{t}(x))=\left(\int_{\mathbb{R}}\Phi(x-y)\mathop{}\\!\mathrm{d}\rho_{t}(y)\right)\partial_{x}M_{t}=(\Phi\ast\rho_{t})\partial_{x}M_{t}$

Combining the above with (5.1) and rearranging the source term
$\partial_{x}S[X_{t}](M_{t})$ we obtain exactly the balance law (1.11). Note
that in [21] they work with a shifted flux function
$A\colon[-\frac{1}{2},\frac{1}{2}]\to\mathbb{R}$, where the lower integral
limit in (5.2) is $-1/2$, since they instead work with the primitive
$\tilde{M}_{t}=M_{t}-\frac{1}{2}$. The reason for this, in a sense, more
symmetric choice of primitive is to justify the relation
$\Phi\ast\rho_{t}=\phi\ast M_{t}$ for their compactly supported $\rho_{t}$. We
will see below that under our assumptions, $\Phi\ast\rho_{t}$ is a continuous
and linearly bounded function, meaning $(\Phi\ast\rho_{t})\rho_{t}$ is a well-
defined measure. Hence we prefer to work with this quantity as it is.

####  5.2.2. Entropy solutions, the Rankine–Hugoniot and Oleĭnik E conditions

Let $\eta\colon[0,1]\to\mathbb{R}$ be a Lipschitz and convex function, and
suppose $q\colon[0,1]\to\mathbb{R}$ satisfies
$q^{\prime}=\eta^{\prime}A^{\prime}$. Then $(\eta,q)$ is called an entropy-
entropy flux pair, and the entropy inequality associated with the balance law
(1.11) is

(5.3)
$\partial_{t}\eta(M_{t})+\partial_{x}q(M_{t})\leq(\Phi\ast\rho_{t})\partial_{x}\eta(M_{t}).$

######  Definition 5.2.

A function $M_{t}\colon[0,T]\times\mathbb{R}\to[0,1]$ is an entropy solution
to (1.11) if it is nondecreasing, and satisfies (5.3) in the weak sense for
every entropy-entropy flux pair.

By an approximation argument, one can equivalently require that the inequality
holds with the Kružkov entropy-entropy flux pair

(5.4) $\eta_{k}(m)=\lvert m-k\rvert,\qquad
q_{k}(m)=\operatorname*{sgn}(m-k)(A(m)-A(k))=\int_{k}^{m}\operatorname*{sgn}(\omega-k)\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega$

for $k$ in a dense subset of $\mathbb{R}$, see [14, Section 3]. We can then,
as in [21], derive the Rankine–Hugoniot and Oleĭnik E conditions. Assume
$M_{t}$ takes the values $M_{t}^{-}$ and $M^{+}_{t}$ on respectively the left-
and right-hand sides of a shock curve $\\{(x,t)\colon x=\sigma(t)\\}$ with
shock velocity $\dot{\sigma}(t)$. Then (5.3) leads to the inequality

$(\dot{\sigma}(t)+\Phi\ast\rho_{t}(\sigma(t)))[[\eta(M_{t})]]\geq[[q(M_{t})]].$

For the choice $(\eta,q)$ = $(\mathrm{Id},A)$, (1.11) shows that the above
inequality holds as an identity, yielding the Rankine–Hugoniot condition along
the shock curve,

(5.5)
$\dot{\sigma}(t)+\Phi\ast\rho_{t}(\sigma(t))=\frac{A(M_{t}^{+})-A(M_{t}^{-})}{M_{t}^{+}-M_{t}^{-}}.$

On the other hand, the Kružkov entropy-entropy flux pair (5.4) yields the
Oleĭnik E condition, cf. [26], [13, Section 8.4],

(5.6)
$\frac{A(M_{t}^{+})-A(k)}{M_{t}^{+}-k}\leq\dot{\sigma}(t)+\Phi\ast\rho_{t}(\sigma(t))\leq\frac{A(k)-A(M_{t}^{-})}{k-M_{t}^{-}},\qquad
k\in(M_{t}^{-},M_{t}^{+}).$

Observe that such a shock would correspond to a maximal interval
$(M_{t}^{-},M_{t}^{+})\subset\Omega_{X_{t}}$ for the corresponding Lagrangian
solution $X_{t}$, such that $(X_{t}(m),V_{t}(m))=(\sigma(t),\dot{\sigma}(t))$
for $m\in(M_{t}^{-},M_{t}^{+})$. From this point of view, the Rankine–Hugoniot
condition (5.5) expresses exactly the identity
$\Psi_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}$ for these $m$.
Furthermore, by the characterization in Lemma 3.1, the Oleĭnik E condition
(5.6) is equivalent to $\bar{\Psi}-\Psi_{t}\in\partial I_{\mathscr{K}}(X_{t})$
for the same $m$. Then, in analogy with how
$\Psi_{t}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}$ can be deduced
from $\bar{\Psi}-\Psi_{t}\in\partial I_{\mathscr{K}}(X_{t})$, (5.5) can be
deduced from (5.6). We may also recall the inequality (2.14) coming from the
barycentric lemma in the particle setting; this is exactly (5.6) for the
piecewise linear interpolation of $A$ with interpolation points given by
(4.25), cf. [21, Section 4], [10, Section 6].

####  5.2.3. Verifying the entropy admissibility

As mentioned in the introduction, in [21] the initial data is assumed to be
$(\bar{\rho},\bar{v})\in\mathscr{P}_{\mathrm{c}}(\mathbb{R})\times
L^{\infty}(\mathbb{R},\bar{\rho})$. In particular, this ensures that the flux
function $A$ is Lipschitz. Moreover, since the solution $(\rho_{t},v_{t})$
retains these properties, $\Phi\ast\rho_{t}$ remains a bounded, continuous
function for all $t\in[0,T]$. In turn, the source term
$(\Phi\ast\rho_{t})\rho_{t}$ remains a well-defined measure for these times,
and the distributional formulation of the balance law (1.11) makes sense.

We now show that the balance law, in particular the source term, also makes
sense for our solution $(\rho_{t},v_{t})\in\mathscr{T}_{2}$. By the pointwise
linear bound on $\Phi$ in Lemma 1.1 we have

$\lvert\Phi\ast\rho(x)\rvert\leq\int_{\mathbb{R}}\lvert\Phi(x-y)\rvert\mathop{}\\!\mathrm{d}\rho(y)\leq\Phi(1)\left(1+\lvert
x\rvert+\int_{\mathbb{R}}\lvert y\rvert\mathop{}\\!\mathrm{d}\rho(y)\right),$

which shows that the map
$f[\rho]\colon\mathscr{P}_{2}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$ given by
$f[\rho]=(\Phi\ast\rho)\rho$ is pointwise linearly bounded, cf. [6, Definition
6.1]. Moreover, by Lemma 4.1, we even have that $f[\rho]$ is uniformly
continuous in the sense of [6, Definition 6.2]. In particular, the source term
in (1.11) is well-defined, and its distributional formulation makes sense.

######  Theorem 5.3.

Suppose $(\bar{\rho},\bar{v})\in\mathscr{T}_{2}$, so that
$\bar{\psi}=\bar{v}+\Phi\ast\bar{\rho}\in L^{2}(\mathbb{R},\bar{\rho})$.
Define $\bar{M}(x)=\bar{\rho}((-\infty,x])$ and the flux function $A$. Let
$M_{t}$ be the generalized inverse of the corresponding Lagrangian solution
$X_{t}$ and recall $\rho_{t}=X_{t\\#}\mathfrak{m}$. Then $M_{t}$ is an entropy
solution of the scalar balance law (1.11) with initial value $\bar{M}$.

###### Proof.

The initial data is clear from $\bar{M}$ being the generalized inverse of
$X_{0}=\bar{X}$. The fact that $M_{t}$ satisfies (1.11) in the weak sense
follows from that $(\rho_{t},v_{t})$ satisfies the continuity equation (1.1a)
and integration by parts, using once more

$\bar{\psi}\circ\bar{X}-\psi_{t}\circ X_{t}=\bar{\Psi}-\Psi_{t}\in\partial
I_{\mathscr{K}}(X_{t})\subset\mathscr{H}_{X_{t}}^{\perp}.$

In fact, for this part we only use that
$\bar{\Psi}-\Psi_{t}\in\mathscr{H}_{X_{t}}^{\perp}$, equivalent to the
Rankine–Hugoniot condition. On the other hand, to show that the entropy
inequality (5.3) holds, we need $\bar{\Psi}-\Psi_{t}\in\partial
I_{\mathscr{K}}(X_{t})$, equivalent to the Oleĭnik E condition.

Let $M(x)=\rho((-\infty,x])$ for some $\rho\in\mathscr{P}_{2}(\mathbb{R})$,
such that the distributional derivative $\partial_{x}M=\rho$. Then for $f\in
H^{1}(\mathbb{R})$, the distributional derivative of $f\circ M\in
BV(\mathbb{R})$ is $\partial_{x}f(M)=f^{\prime}_{M}\rho$ for an
$f^{\prime}_{M}\in L^{2}(\mathbb{R},\rho)$ given by the $BV$-chain rule, cf.
[10, Lemma 4.2]. Following [10, Section 5], for a.e. $m\in\Omega$ we have
$\eta_{k,M}^{\prime}(X_{t})=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}(\operatorname*{sgn}(m-k))$,
$A_{M}^{\prime}(X_{t})=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}=\Psi_{t}$,
and
$q_{k,M}^{\prime}(X_{t})=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}(\operatorname*{sgn}(m-k)\bar{\Psi})$.
Let $0\leq\varphi\in C^{\infty}_{\mathrm{c}}([0,T]\times\mathbb{R})$ be a
smooth test function with compact support; we use this to show that (5.3)
holds by integrating by parts and using the push-forward relation (3.3). That
is, using that $M_{t}$ is a weak solution, we compute

$\displaystyle\int_{0}^{T}\left(\int_{\mathbb{R}}\left[\eta_{k}(M_{t})\partial_{t}\varphi(t,x)+q_{k}(M_{t})\partial_{x}\varphi(t,x)\right]\mathop{}\\!\mathrm{d}x+\int_{\mathbb{R}}\varphi(t,x)\eta^{\prime}_{k,M}(M_{t})(\Phi\ast\rho_{t})\mathop{}\\!\mathrm{d}\rho_{t}\right)\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad=-\int_{0}^{T}\int_{\mathbb{R}}\varphi(t,x)\left[q_{k,M}^{\prime}(t,x)-\eta_{k,M}^{\prime}(t,x)A_{M}^{\prime}(t,x)\right]\mathop{}\\!\mathrm{d}\rho_{t}\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad=-\int_{0}^{T}\int_{\Omega}\varphi(t,X_{t})\left[q_{k,M}^{\prime}(X_{t})-\eta_{k,M}^{\prime}(X_{t})A^{\prime}_{M}(X_{t})\right]\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad=-\int_{0}^{T}\int_{\Omega}\varphi(t,X_{t})\operatorname*{sgn}(m-k)\left[\bar{\Psi}-\Psi_{t}\right]\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t\geq
0,$

where the final inequality follows from (3.15). Indeed, since $\varphi\geq 0$
it follows from (3.17) that $\varphi(t,X_{t})\operatorname*{sgn}(m-k)\in
T_{X_{t}}\mathscr{K}$ for any time $t\geq 0$ and $k\in\mathbb{R}$. ∎

#### Possible uniqueness of entropy solutions

In [21] the authors extend the Kružkov doubling-of-variables argument to
account for the additional source term, thereby proving that entropy solutions
for the balance law (1.11) with initial data
$(\bar{\rho},\bar{v})\in\mathscr{P}_{\mathrm{c}}(\mathbb{R})\times
L^{\infty}(\mathbb{R},\bar{\rho})$ are unique, relying on the fact that the
measure remains compactly supported, and that $\bar{v}$ is essentially
bounded, meaning the flux function is Lipschitz.

In our case, the solutions belong to the space $\mathscr{T}_{2}$, and the flux
function is continuous, but not Lipschitz. However, uniqueness of entropy
solutions has been proved in [14] for conservation laws with the type of flux
function we have here. This suggests that the same may be true for balance
laws of the form (1.11), most likely with some requirements for the source
term. However, we deem the investigation of this possibility to be outside the
scope of our current study.

##  6\. Clustering formation for sticky dynamics

In their follow-up paper [22], the authors of [21] derive results on the
finite- and infinite-time clustering of their sticky-particle solutions of
(1.1). Here clusters refer to the connected components, or maximal intervals,
of $\Omega_{X_{t}}$. Their analysis is based on the flux function $A$ for the
balance law (1.11) and its lower convex envelope $A^{**}$. It is perhaps not
surprising that the flux function $A$, which encodes the continuum natural
velocity $\bar{\Psi}$, determines the clustering behavior of the Euler-
alignment system; in [17] it is shown that for the Cucker–Smale particle
dynamics and initial positions $\bar{\boldsymbol{x}}\in\mathbb{K}^{N}$, there
can be finite-time collisions only if the natural velocities are not ordered,
i.e., $\bar{\boldsymbol{\psi}}\notin\mathbb{K}^{N}$, where we recall (2.1) and
(2.3). In our continuum case, recalling
$\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}=\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}m}A^{**}$
from Proposition 3.2, we observe that any deviation of $A$ from $A^{**}$ means
that $\bar{\Psi}$ is not nondecreasing in this region, hence
$\bar{\Psi}\notin\mathscr{K}$, the convex cone from (3.10). Indeed, what is
called the supercritical region $\Sigma_{-}$ in [22], which for our definition
(5.2) of $A$ becomes

$\Sigma_{-}\coloneqq\left\\{m\in\Omega\colon A(m)>A^{**}(m)\right\\},$

is exactly where finite-time clustering happens. Similarly, their subcritical
region $\Sigma_{+}$ is for us

$\Sigma_{+}\coloneqq\left\\{m\in[0,1)\colon A^{**}\text{ is not linear on any
interval }[m,m^{\prime})\right\\},$

while the critical region $\Sigma_{0}$ becomes

$\Sigma_{0}\coloneqq\left\\{\bigcup[m^{\prime},m^{\prime\prime})\colon
A^{**}\text{ is linear and equal to }A\text{ on
}[m^{\prime},m^{\prime\prime})\right\\}.$

Since $A^{**}$ is convex,
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}m}A^{**}$ exists for
every $m\in\Omega$ and is a nondecreasing, right-continuous function.
Therefore, its range
$\operatorname*{ran}(\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}m}A^{**})$
defines an ordering on $\Omega$, dividing it into sets of nonincreasing
averaged natural velocities. Suppose $m\in\Omega$, and write
$\psi=\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m)\in\operatorname*{ran}(\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}m}A^{**})$.
Then we write $L(m)=(\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi})^{-1}(\psi)$,
that is, $L(m)$ is the preimage of $\psi$. This set then turns out to be
analogous to the set $L(m)$ defined in [22], where it for $m\notin\Sigma_{+}$
is defined as the maximal half-open interval containing $m$ on which $A^{**}$
is linear, and $\\{m\\}$ otherwise. Figure 1 provides an illustration of these
subsets for a generic flux function $A$. As shown in [22], the sets $L(m)$
play a central role in the asymptotic behavior.

Figure 1. A flux function $A$ and its lower convex envelope $A^{**}$. The non-
singleton subgroups $L(m)$ are shown together with the regions $\Sigma_{+}$,
$\Sigma_{0}$ and $\Sigma_{-}$.

###  6.1. Auxiliary estimates

From the characterization in Lemma 3.1 we know that an element $\xi\in\partial
I_{\mathscr{K}}(X)$ can only be nonzero in $\Omega_{X}$. Then for a.e.
$m\in\Omega\setminus\Omega_{X_{t}}$, the differential inclusion (4.2) will be
a differential equation since $\dot{X}_{t}(m)=U[X_{t}](m)$. In fact, by
(4.11), the Lagrangian solution satisfies the following differential equation
for all $t\geq 0$,

(6.1)
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X_{t}=\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}\bar{\Psi}-\int_{\Omega}\Phi(X_{t}-X_{t}(\omega))\mathop{}\\!\mathrm{d}\omega.$

Since our solutions are globally sticky, we have
$\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}\bar{\Psi}=\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}$,
and by (3.17) and (4.17)

$\mathscr{K}\subset T_{X_{t}}\mathscr{K}\subset
T_{X_{s}}\mathscr{K}\quad\text{for}\ 0\leq s\leq t.$

In particular, since $X_{0}=\bar{X}$ and
$V_{0}=\bar{V}\in\mathscr{H}_{\bar{X}}$, $X_{t}$ initially evolves according
to the continuum version of (2.2). Consider
$[\alpha_{i},\beta_{i})\subset\Omega$ for $i=1,2$, where
$\beta_{1}\leq\alpha_{2}$ such that the intervals do not overlap. Then we
introduce the averaged quantities

(6.2)
$Y^{i}_{t}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{i}}^{\beta_{i}}X_{t}(\omega)\mathop{}\\!\mathrm{d}\omega,\qquad\Gamma^{i}_{t}\coloneqq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{i}}^{\beta_{i}}\mathop{}\mathrm{P}_{T_{X_{t}}\mathscr{K}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega,$

where $Y_{t}^{1}\leq Y_{t}^{2}$ follows from $X_{t}\in\mathscr{K}$. Then we
can derive the following inequalities from (6.1),

(6.3a)
$\displaystyle\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}\left(Y^{2}_{t}-Y^{1}_{t}\right)$
$\displaystyle\geq\Gamma^{2}_{t}-\Gamma^{1}_{t}-2\Phi\left(\frac{1}{2}\left(Y^{2}_{t}-Y^{1}_{t}\right)\right),$
(6.3b)
$\displaystyle\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}\left(Y^{2}_{t}-Y^{1}_{t}\right)$
$\displaystyle\leq\Gamma^{2}_{t}-\Gamma^{1}_{t}.$

The inequality (6.3b) follows easily from $X_{t}\in\mathscr{K}$ and $\Phi$
being a nondecreasing function. Indeed, we have $J_{t}\geq 0$, where

$J_{t}\coloneqq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{2}}^{\beta_{2}}\int_{\Omega}\Phi(X_{t}(m)-X_{t}(\omega))\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}m-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{1}}^{\beta_{1}}\int_{\Omega}\Phi(X_{t}(m)-X_{t}(\omega))\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}m.$

In order to derive (6.3a), we estimate $J_{t}$ from above as in (4.4),

$\displaystyle J_{t}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{2}}^{\beta_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{1}}^{\beta_{1}}\int_{\Omega}\int_{X_{t}(\tilde{m})}^{X_{t}(m)}\phi(y-X_{t}(\omega))\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\tilde{m}\mathop{}\\!\mathrm{d}m$
$\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{2}}^{\beta_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{1}}^{\beta_{1}}\int_{X_{t}(\tilde{m})}^{X_{t}(m)}\phi\left(y-\frac{X_{t}(m)+X_{t}(\tilde{m})}{2}\right)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\tilde{m}\mathop{}\\!\mathrm{d}m$
$\displaystyle=2\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{2}}^{\beta_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\alpha_{1}}^{\beta_{1}}\Phi\left(\frac{X_{t}(m)-X_{t}(\tilde{m})}{2}\right)\mathop{}\\!\mathrm{d}\tilde{m}\mathop{}\\!\mathrm{d}m\leq
2\Phi\left(\frac{Y_{t}^{2}-Y_{t}^{1}}{2}\right),$

where the first inequality follows from $\phi(x)=\phi(\lvert x\rvert)$ being
radially nondecreasing, and the second inequality is a consequence of
$\Phi(x)$ being concave for $x\geq 0$ and Jensen’s inequality. Observe that
the inequalities (6.3) still holds if we replace one or both of the averaged
quantities with $X_{t}$ in a point.

Although $\Gamma^{2}_{t}-\Gamma_{t}^{1}$ in (6.3) depends on time, we will use
their relation with $A$ and $A^{**}$ to appropriately bound them with time-
independent quantities, so that the following result, comparable with [22,
Lemma 3.4], is applicable.

######  Lemma 6.1.

Suppose $X_{t}\colon[0,\infty)\to\mathscr{K}$, with $Y_{t}^{i}$ and
$\Gamma^{i}_{t}$ for $i\in\\{1,2\\}$ as in (6.2). We consider the following
cases:

  1. I.

If $\Gamma^{2}_{t}-\Gamma^{1}_{t}\geq 2\sigma>0$, then there is $\eta>0$
depending on $\Phi$ and $\sigma$ such that

$Y_{t}^{2}-Y_{t}^{1}\geq\min\left\\{Y_{0}^{2}-Y_{0}^{1}+\sigma
t,\eta\right\\}\geq\min\left\\{Y_{0}^{2}-Y_{0}^{1},\sigma t,\eta\right\\}.$

  2. II.

If $\Gamma^{2}_{t}-\Gamma^{1}_{t}\leq-\sigma<0$, then there exists some time
$\tau$, $\tau\leq(Y_{0}^{2}-Y_{0}^{1})/\sigma$, such that
$Y_{\tau}^{2}=Y_{\tau}^{1}$.

###### Proof.

Case I: Let $\eta>0$ be such that $2\Phi(\frac{1}{2}\lvert x\rvert)\leq\sigma$
whenever $\lvert x\rvert\leq\eta$, which is always possible, cf. Lemma 1.1. In
particular, if $\Phi$ is invertible, we can choose
$\eta=2\Phi^{-1}(\frac{1}{2}\sigma)$. Now, either
$Y_{t}^{2}-Y_{t}^{1}\geq\eta$, or $Y_{t}^{2}-Y_{t}^{1}<\eta$. In the latter
case, the right-hand side of (6.3a) is greater than or equal to $\sigma$.
Therefore, we may integrate the leftmost inequality of (6.3a) to obtain the
result.

Case II: This follows from integration of (6.3b). ∎

###  6.2. Clustering from the Lagrangian solutions

Note that $\Sigma_{-}$, like $\Omega_{X_{t}}$, is an open set in $\Omega$, and
so can be written as a countable union of disjoint open intervals. Let us
consider one such maximal interval $(m_{-},m_{+})\subset\Sigma_{-}$. Note that
if $\Omega_{\bar{X}}\cap\Sigma_{-}\neq\emptyset$, then any maximal interval
$(\alpha,\beta)\subset\Omega_{\bar{X}}$ intersecting $(m_{-},m_{+})$ must
necessarily be contained in $(m_{-},m_{+})$. If not, $\alpha<m_{+}<\beta$,
say, we have

$\frac{A(m_{+})-A(m_{-})}{m_{+}-m_{-}}>\frac{A(m_{+})-A(m)}{m_{+}-m},\quad
m_{-}<m<m_{+},$

by definition of $\Sigma_{-}$, and in particular this holds for
$m\in(\alpha,m_{+})$. However, since $\bar{\Psi}\in\mathscr{H}_{\bar{X}}$, its
slope must be constant on $(\alpha,\beta)$, and so the previous inequality
implies $A^{**}(m_{+})=A(m_{+})>A(\beta)\geq A^{**}(\beta)$, such that

$\frac{A^{**}(m_{+})-A^{**}(m_{-})}{m_{+}-m_{-}}>\frac{A^{**}(\beta)-A^{**}(m_{-})}{\beta-
m_{-}}.$

This contradicts $A^{**}$ being convex, cf. [22, Lemma 2.2]. A similar
argument holds for the case $\alpha<m_{-}<\beta$. This means that if
$m^{\prime}<m^{\prime\prime}$ and $L(m^{\prime})\neq L(m^{\prime\prime})$, we
must have $\bar{X}(m^{\prime})<\bar{X}(m^{\prime\prime})$, which is also the
content of [22, Lemma 3.5].

Based on our Lagrangian solutions, we now present a result corresponding to
[22, Theorem 1.7], where steps of our proof are very much inspired by theirs.

######  Theorem 6.2.

Suppose $m^{\prime},m^{\prime\prime}\in\Omega$. If
$\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m^{\prime})<\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m^{\prime\prime})$,
then there is a time-independent constant $c>0$ such that
$X_{t}(m^{\prime\prime})-X_{t}(m^{\prime})\geq c>0$ for all $t\geq 0$.
Furthermore, we have the following cases for $m\in\Omega$.

  1. I.

If $m\in\Sigma_{+}$, there is no finite- or infinite-time clustering at $m$.

  2. II.

If $m\in\Sigma_{-}$, there is a finite-time cluster at $m$ for sufficiently
large $t\geq 0$.

  3. III.

    1. (i)

Suppose $\int_{0}^{1}\frac{1}{\Phi(x)}\mathop{}\\!\mathrm{d}x=\infty$,
including the case $\phi\equiv 0$. If $[m^{\prime},m^{\prime\prime})\ni m$ is
a finite-time cluster, then either $m\in\Sigma_{0}$ and
$[m^{\prime},m^{\prime\prime})$ is an initial cluster, or $m\in\Sigma_{-}$ and
$[m^{\prime},m^{\prime\prime})\subset[m_{-},m_{+})$, where $(m_{-},m_{+})$ is
a connected component of $\Sigma_{-}$. No other finite-time clusters are
possible.

    2. (ii)

Suppose $\phi$ is not identically zero. If $m\notin\Sigma_{+}$ and
$\left\\{\bar{X}(\omega)\colon\omega\in L(m)\right\\}$ is bounded, there is an
infinite-time cluster at $m$, given by $L(m)$.

    3. (iii)

Suppose in the previous case (ii) that
$\int_{0}^{1}\frac{1}{\Phi(x)}\mathop{}\\!\mathrm{d}x<\infty$. Then $L(m)$ is
a finite-time cluster.

###### Proof.

Case I is a direct consequence of the first part of the theorem, which we
prove next. Consider $m^{\prime\prime}\in\Omega$ and write
$\psi^{\prime\prime}=\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m^{\prime\prime})$.
Then $L(m^{\prime\prime})$ may be a singleton $\\{m^{\prime\prime}\\}$ or an
interval $[m_{-}^{\prime\prime},m_{+}^{\prime\prime})$, and we write
$m_{-}^{\prime\prime}=\min L(m^{\prime\prime})$. For any $m^{\prime}\in\Omega$
with
$\psi^{\prime}=\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m^{\prime})<\psi^{\prime\prime}$
we must then have $L(m^{\prime})\neq L(m^{\prime\prime})$ and
$\bar{X}(m^{\prime})<\bar{X}(m^{\prime\prime})$. Assume for the case of
contradiction that $X_{t}(m_{-}^{\prime\prime})=X_{t}(m^{\prime})$ for some
$m^{\prime}<m_{-}^{\prime\prime}$, and let $t=\tau$ be the first time this
happens. For $t\in[0,\tau)$, $L(m^{\prime\prime})$ can then only collide with
mass to its right, and since
$A(m_{-}^{\prime\prime})=A^{**}(m_{-}^{\prime\prime})$, $A^{**}$ is increasing
and $A\geq A^{**}$, it follows that
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(m_{-}^{\prime\prime})\geq\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m_{-}^{\prime\prime})=\psi^{\prime\prime}$
in this time interval.

Now consider the interval $[m^{\prime},m_{-}^{\prime\prime})$ and introduce
the averaged quantities

$Y_{t}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m^{\prime}}^{m_{-}^{\prime\prime}}X_{t}(\omega)\mathop{}\\!\mathrm{d}\omega,\qquad\Gamma_{t}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m^{\prime}}^{m_{-}^{\prime\prime}}\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega,$

such that $Y_{\tau}=X_{\tau}(m_{-}^{\prime\prime})$. Let us write
$m_{+}^{\prime}=\sup L(m^{\prime})$, where if $L(m^{\prime})=\\{m^{\prime}\\}$
we have $m_{+}^{\prime}=m^{\prime}$, otherwise $m^{\prime}<m_{+}^{\prime}$.
Initially, $\mathop{}\mathrm{P}_{\mathscr{H}_{X_{0}}}\bar{\Psi}=\bar{\Psi}$
and we compute

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m^{\prime}}^{m_{-}^{\prime\prime}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega$
$\displaystyle=\frac{A(m_{+}^{\prime})-A(m^{\prime})+A(m_{-}^{\prime\prime})-A(m_{+}^{\prime})}{m_{-}^{\prime\prime}-m^{\prime}}\leq\frac{A^{**}(m_{+}^{\prime})-A^{**}(m^{\prime})+A^{**}(m_{-}^{\prime\prime})-A^{**}(m_{+}^{\prime})}{m_{-}^{\prime\prime}-m^{\prime}},$

where $A^{**}(m_{+}^{\prime})-A^{**}(m^{\prime})$ and
$A^{**}(m_{-}^{\prime\prime})-A^{**}(m_{+}^{\prime})$ may be zero, but not
simultaneously. When nonzero, we have
$A^{**}(m_{+}^{\prime})-A^{**}(m^{\prime})=(m_{+}^{\prime}-m^{\prime})\psi^{\prime}$
and
$A^{**}(m_{-}^{\prime\prime})-A^{**}(m_{+}^{\prime})=(m_{-}^{\prime\prime}-m_{+}^{\prime})\tilde{\psi}$,
where $\tilde{\psi}\in(\psi^{\prime},\psi^{\prime\prime})$ is the average
slope of $A^{**}$ on $(m_{+}^{\prime},m_{-}^{\prime\prime})$. In any case, we
find that $\Gamma_{0}\leq\psi^{\prime}$, and this remains an upper bound for
$\Gamma_{t}$ whenever mass within $[m_{+}^{\prime},m_{-}^{\prime\prime})$
collides, since the projection $\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}$ can
only make the extremal values of $\bar{\Psi}$ less extreme. Furthermore, for
$t\in[0,\tau)$, $[m_{+}^{\prime},m_{-}^{\prime\prime})$ can only interact with
mass to its left; since $A^{**}$ is increasing,
$A(m_{-}^{\prime})=A^{**}(m_{-}^{\prime})$ and $A\geq A^{**}$, collision with
mass to the left of $L(m^{\prime})$ can only decrease
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}$ on
$[m^{\prime},m_{-}^{\prime\prime})$. Then, since
$\psi^{\prime}<\psi^{\prime\prime}$ it follows from case II of Lemma 6.1 that
$X_{\tau}(m_{-}^{\prime\prime})>Y_{\tau}^{\prime}$, which contradicts our
assumption. In particular, mass labels belonging to distinct $L(m)$ can never
collide. If $L(m)=\\{m\\}$, then
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(m)=\bar{\Psi}(m)$.
Otherwise, if $L(m)=[m_{-},m_{+})$, then for any $m_{1},m_{2}\in(m_{-},m_{+})$

$\int_{m_{2}}^{m_{+}}\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega\leq\int_{m_{-}}^{m_{+}}\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega=\int_{m_{-}}^{m_{+}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega=\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m)\leq\int_{m_{-}}^{m_{1}}\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega.$

Following the proof of case I in Lemma 6.1, we find that the time-independent
constant $c$ can be chosen as
$c=\min\\{\bar{X}(m^{\prime\prime})-\bar{X}(m^{\prime}),\eta\\}$ for some
$\eta>0$ depending on $\psi^{\prime\prime}-\psi^{\prime}>0$.

Case II: Consider $m\in(m_{-},m_{+})$, where $(m_{-},m_{+})\subset\Sigma_{-}$
is a maximal interval. Now, consider any $m_{1},m_{2}\in(m_{-},m_{+})$ where
$\bar{X}(m_{1})<\bar{X}(m_{2})$. We will prove that the mass centers

$Y_{t}^{1}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m_{-}}^{m_{1}}X_{t}(\omega)\mathop{}\\!\mathrm{d}\omega,\qquad
Y_{t}^{2}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m_{2}}^{m_{+}}X_{t}(\omega)\mathop{}\\!\mathrm{d}\omega$

must coincide in finite time, which necessarily means that $X_{t}(m_{1})$ and
$X_{t}(m_{2})$ must as well. As long as $X_{t}(m_{1})<X_{t}(m_{2})$, by
definition of $\Sigma_{-}$, we must have

$\Gamma_{t}^{1}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m_{-}}^{m_{1}}\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega\geq\min_{m_{1}\leq
m\leq
m_{2}}\frac{A(m)-A(m_{-})}{m-m_{-}}\eqqcolon\Gamma^{1}>\frac{A(m_{+})-A(m_{-})}{m_{+}-m_{-}}$

and

$\Gamma_{t}^{2}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m_{2}}^{m_{+}}\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega\leq\max_{m_{1}\leq
m\leq
m_{2}}\frac{A(m_{+})-A(m)}{m_{+}-m}\eqqcolon\Gamma^{2}<\frac{A(m_{+})-A(m_{-})}{m_{+}-m_{-}}.$

The result then follows from case II of Lemma 6.1.

Case III: (i): From before we know that mass labels belonging to different
$L(m)$ will never cluster, so without loss of generality we consider
$L(m)\neq\\{m\\}$ with the associated value
$\psi=\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m)$. We note that $L(m)$ may
contain segments belonging to both $\Sigma_{0}$ and $\Sigma_{-}$. If
$m\in\Sigma_{-}$, we write $C(m)=[m_{+},m_{-})$, where
$(m_{-},m_{+})\subset\Sigma_{-}$ is the maximal interval containing $m$.
Alternatively, if $m\in\Sigma_{0}$ and there is an initial cluster at $m$, we
denote this cluster by $C(m)$. Otherwise, $C(m)=\\{m\\}$. We consider
$m^{\prime},m^{\prime\prime}\in L(m)$ with $m^{\prime}<m^{\prime\prime}$ and
$C(m^{\prime})\neq C(m^{\prime\prime})$, and aim to show that these sets
cannot cluster in finite time. We write $m_{-}^{\prime\prime}=\min
C(m^{\prime\prime})$ and $m_{+}^{\prime}=\sup C(m^{\prime})$. Following the
same arguments as in the first part of the proof, we deduce that
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(m_{-}^{\prime\prime})\geq\psi$;
similarly, if $C(m^{\prime})\neq\\{m^{\prime}\\}$ then
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m^{\prime}}^{m_{+}^{\prime\prime}}\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(\omega)\mathop{}\\!\mathrm{d}\omega\leq\psi$,
otherwise
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(m^{\prime})=\psi$.
Writing
$Y_{t}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{m^{\prime}}^{m_{+}^{\prime}}X_{t}(\omega)\mathop{}\\!\mathrm{d}\omega$
if $C(m^{\prime})\neq\\{m^{\prime}\\}$ and $Y_{t}=X_{t}(m^{\prime})$
otherwise, we define $l(t)=X_{t}(m_{-}^{\prime\prime})-Y_{t}$. Observe that
$X_{t}(m_{-}^{\prime\prime})-Y_{t}\leq
X_{t}(m^{\prime\prime})-X_{t}(m^{\prime})$, so $l(t)$ gives a lower bound on
their distance, and we have $l(0)>0$. From (6.3a) it follows that
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}l(t)\geq-2\Phi(l(t)/2)$,
which we integrate to obtain

$\int_{l(t)}^{l(0)}\frac{\mathop{}\\!\mathrm{d}x}{\Phi(\frac{x}{2})}\leq 2t.$

From our assumption, $l(t)$ can only become zero in infinite time.

(ii): Consider $m\in\Omega$ with $L(m)=[m_{-},m_{+})$ and
$\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m)=\psi$, where we necessarily
have $A(m_{\pm})=A^{**}(m_{\pm})$. Since $m\mapsto X_{t}(m)$ is nondecreasing
and $\bar{X}(\omega)$ is bounded for $\omega\in L(m)$, both
$\lim\limits_{\omega\to m_{-}+}\bar{X}(m)=\bar{X}(m_{-})$ and
$\lim\limits_{\omega\to m_{+}-}\bar{X}(m)\eqqcolon\bar{X}(m_{+}-)$ are finite.
From before we know that it is only possible for $X_{t}(m_{-})$ to collide
with mass to its right, leading to the inequality
$\mathop{}\mathrm{P}_{\mathscr{H}_{X_{t}}}\bar{\Psi}(m_{-})\geq\mathop{}\mathrm{P}_{\mathscr{K}}\bar{\Psi}(m_{-})=\psi$.

On the other hand, for any $m^{\prime}\in L(m)$, the set $[m^{\prime},m_{+})$
can only collide with mass to its left, leading to