# Equivalence of entropy solutions and gradient flows for pressureless 1D
Euler systems

José Antonio Carrillo  Mathematical Institute, University of Oxford, Oxford
OX2 6GG, United Kingdom<EMAIL_ADDRESS>and  Sondre Tesdal Galtung
Department of Mathematical Sciences, NTNU – Norwegian University of Science
and Technology, 7491 Trondheim, Norway<EMAIL_ADDRESS>
###### Abstract.

We study distributional solutions of pressureless Euler systems on the line.
In particular we show that Lagrangian solutions [7], introduced by Brenier,
Gangbo, Savaré and Westdickenberg, and entropy solutions [37], studied by
Nguyen and Tudorascu for the Euler–Poisson system, are equivalent. For the
Euler–Poisson system this can be seen as a generalization to second-order
systems of the equivalence between $L^{2}$-gradient flows and entropy
solutions for a first-order aggregation equation proved by Bonaschi, Carrillo,
Di Francesco and Peletier [4]. The key observation is an equivalence between
Oleĭnik’s E-condition for conservation laws and a characterization due to
Natile and Savaré of the normal cone for $L^{2}$-gradient flows. This new
equivalence allows us to define unique solutions after blow-up for classical
solutions of the Euler–Poisson system with quadratic confinement due to
Carrillo, Choi and Zatorska [14], as well as to describe their asymptotic
behavior.

###### Key words and phrases:

Pressureless Euler system, Euler–Poisson system, entropy solutions, gradient
flows, sticky particles, front tracking

###### 1991 Mathematics Subject Classification:

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

##  1\. Introduction

We will study distributional solutions of the Cauchy problem for the 1D
pressureless Euler system

(1.1a) $\displaystyle\partial_{t}\rho+\partial_{x}(\rho v)$ $\displaystyle=0,$
(1.1b) $\displaystyle\partial_{t}(\rho v)+\partial_{x}(\rho v^{2})$
$\displaystyle=f[\rho],$

with initial density $\rho(0,x)=\rho_{0}(x)$ and velocity $v(0,x)=v_{0}(x)$,
where $f[\rho]$ is a force distribution generated by the fluid itself.
Furthermore, we assume initial data $\rho_{0}\in\mathscr{P}_{2}(\mathbb{R})$,
the subspace of probability measures $\mathscr{P}(\mathbb{R})$ with bounded
second moments, and $v_{0}\in L^{2}(\mathbb{R},\rho_{0})$, such that the
initial kinetic energy is finite. Moreover, we assume the map
$f[\rho]\colon\mathscr{P}_{2}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$, where
$\mathscr{M}(\mathbb{R})$ are the signed Borel measures with finite total
variation, is absolutely continuous with respect to $\rho$, that is,

(1.2) $f[\rho]=f_{\rho}\rho,\qquad f_{\rho}\in L^{2}(\mathbb{R},\rho),$

where $f_{\rho}$ is the Radon–Nikodym derivative of $f[\rho]$ with respect to
$\rho$. The system (1.1) was studied under these assumptions in [7] with
forcing and no forcing in [8, 35]. A special case of (1.1) which is of
interest to us is the pressureless Euler–Poisson system, studied in, e.g.,
[36, 37] in the form

(1.3a) $\displaystyle\partial_{t}\rho+\partial_{x}(\rho v)$ $\displaystyle=0,$
(1.3b) $\displaystyle\partial_{t}(\rho v)+\partial_{x}(\rho v^{2})$
$\displaystyle=-\left(\alpha\Phi_{x}+\beta\right)\rho,$

where $\alpha,\beta\in\mathbb{R}$, while
$\Phi_{x}(t,x)=\phi^{\prime}\ast\rho(t,x)$ with $\phi(x)=\frac{1}{2}\lvert
x\rvert$ such that $\partial^{2}_{x}\Phi=\rho$ in the distributional sense.
Note that in [36, 37] one only specifies $\partial^{2}_{x}\Phi=\rho$, which
determines $\Phi_{x}$ up to a constant which can easily be absorbed in
$\beta$, and this explains why they work with a different $\phi$ corresponding
to the Heaviside function. We prefer to work with the symmetric $\phi$ given
above, which then can be thought of as a 1D Poisson potential. A positive or
negative sign for $\alpha$ then yields respectively an attractive or repulsive
force.

We are particularly interested in the two, seemingly distinct, notions of
solution introduced in [7] and [36, 37]. Each of which yields a unique
solution to their respective systems under appropriate assumptions on the
forcing terms, even faced with mass concentration. Our aim is to show that
these notions in fact are equivalent. More precisely, we prove the equivalence
of Lagrangian solutions [7] and entropy solutions [36, 37] for the Euler
system (1.1), which contains the attractive and repulsive Euler–Poisson
systems (1.3) as special cases. A very recent paper [43] describes this
equivalence between sticky particles flow described in [28] and entropy
solutions of (1.1) for $f[\rho]\equiv 0$ .

An analogous equivalence has been shown in [4] for the nonlocal interaction
equation where a probability measure $\mu\in\mathscr{P}_{2}(\mathbb{R})$
satisfies

(1.4)
$\partial_{t}\mu+\partial_{x}\left(\mu\;\partial_{x}\phi\ast\mu\right)=0,\qquad
t>0,\quad x\in\mathbb{R}$

for an interaction potential $\Phi(x)=\alpha\lvert x\rvert$ with
$\alpha\in\\{-1,1\\}$, and the Burgers-type conservation law

(1.5) $\partial_{t}M+\partial_{x}\mathcal{A}(M)=0,\qquad t>0,\quad
x\in\mathbb{R}$

with flux function $\mathcal{A}(m)=\alpha(m-m^{2})$ and $M$ being the
cumulative distribution function of $\mu$, $M(t,x)=\mu(t,(-\infty,x])$. Here
$\alpha=1$ results in an attractive potential, while $\alpha=-1$ gives a
repulsive potential. The dynamics of (1.4) is relatively simple in the sense
that an attractive potential promotes accumulation of mass $\mu$, while a
repulsive potential spreads the mass. This is particularly clear for a sum of
Dirac masses $\mu=\sum_{i=1}^{n}m_{i}\delta_{x_{i}}$, i.e., a set of $n$
“particles”, as an attractive potential will aggregate the particles into one
particle of unit mass, while a repulsive potential will instantly diffuse the
Dirac measures into a measure which is absolutely continuous with respect to
the Lebesgue measure. The aggregation and diffusion turn out to have a one-to-
one correspondence with respectively shock and rarefaction waves of the
conservation law (1.5), see [4, Section 2]. In fact, there the equivalence of
(1.4) and (1.5) is first established at the particle level before passing to a
limit to cover the general case.

Note that (1.4) is a first-order system, unlike the second-order Euler–Poisson
system (1.3), and so it does not feature the issues of sub- and supercritical
velocities in the repulsive case. Indeed, if particles in the repulsive (1.4)
are initially separate, they will never meet, as the inter-particle repulsive
forces will only drive them further apart. On the other hand, for particles in
the repulsive (1.3), for certain, so-called supercritical, initial velocity
configurations, particles might collide despite the mutually repulsive forces
acting on them. This also raises the question of what is the “natural” or
physical way of extending the solutions beyond collision-time, or singularity
formation.

In this connection to particle dynamics, we also confirm the final remark of
Nguyen and Tudorascu [37] that their entropy solution of the Euler–Poisson
system coincides with the corresponding Lagrangian solution of Brenier et. al
[7] in the attractive Poisson case on the basis that they both can be realized
as limits of the same sticky discrete particle systems; indeed, we find these
notions of solution are equivalent directly from their definitions regardless
of the attractive/repulsive character of the force. Furthermore, we deal with
the following generalization of (1.3),

(1.6) $\displaystyle\partial_{t}\rho+\partial_{x}(\rho v)$ $\displaystyle=0,$
$\displaystyle\partial_{t}(\rho v)+\partial_{x}(\rho v^{2})$
$\displaystyle=-\gamma\rho v-\lambda^{2}(\tilde{\phi}^{\prime}\ast\rho)\rho.$

In the above, $\gamma\geq 0$ is a linear damping parameter, while
$\tilde{\phi}=\frac{1}{2}x^{2}-\lvert x\rvert$ represents a potential with
Coulomb repulsion and quadratic confinement. This system has been studied in
detail for in [14] for classical solutions up until blow-up, or mass
concentration, for which precise critical thresholds are given. We want to use
the aforementioned solution concept(s) to study (1.6) beyond the time of blow-
up and its asymptotic behavior.

There is a rich literature on pressureless Euler systems, and so we will only
present a few works relevant to us, mostly in the one-dimensional case. The
system (1.1) with $f[\rho]\equiv 0$, often named simply the pressureless Euler
system, has been thoroughly studied, and has found applications for instance
in modeling the formation of the early Universe, in terms of “sticky dust”,
cf. [49, 42]. Indeed, in [8] the authors show that sticky, or adhesive,
particle evolution can be formulated in terms of a scalar conservation law and
give rise to weak solutions of this system satisfying an entropy condition; in
particular, they construct weak solutions for compactly supported
$\rho_{0}\in\mathscr{P}(\mathbb{R})$, denoted
$\mathscr{P}_{\mathrm{c}}(\mathbb{R})$, and continuous and bounded velocity
$v_{0}\in C_{\mathrm{b}}(\mathbb{R})$. This system has been studied from a
slightly different point of view in [22], which also covers the self-
gravitating case (1.3) with $\beta=0$ and $\alpha=1$, where one employs a
continuation of characteristics-type argument. In [5] the authors introduce
the notion of duality solutions for the pressureless Euler system, which are
also related to solutions of an associated scalar conservation law, and prove
existence and uniqueness of these. Another existence and uniqueness result is
found in [27] for $v_{0}$ bounded and measurable with respect to the
nonnegative Radon measure $\rho_{0}$, where one relies on an entropy
inequality, see (1.7) below, and weak continuity of the initial energy.

In [35], the authors study the pressureless system using adhesive dynamics and
gradient flows with Wasserstein metrics while recovering the results of [8].
The authors in [17] give a more direct proof of construction of sticky
solutions of (1.3) in the attractive case using projections. Inspired by [8],
the authors of [36] use particle approximations and scalar conservation laws
to introduce what they call entropy solutions of the Euler–Poisson system
(1.3). Later they refined their results in [37] to allow for the same
assumptions on initial data as in [7], as well as treating additional viscous
terms in the right-hand side.

Several works for the Euler–Poisson system deal with classical solutions up
until blow-up [23, 45, 46, 13, 14, 2]. Their strategy is based on explicit
representations of the solutions along characteristics, leading to critical
thresholds for the existence and uniqueness of classical solutions, see [12]
for numerical illustrations. An existence and uniqueness result of entropy
solutions related to ours using the framework of conservation laws [32] covers
(1.3) in the attractive case, see Remark 5.9 for more details. Furthermore, in
the recent work [26] uniqueness of entropy solutions for the attractive
Euler–Poisson system (1.3) with $\alpha=1$, $\beta=0$, and initial data
$\rho_{0}\in\mathscr{P}_{\mathrm{c}}(\mathbb{R})$ and $v_{0}\in
L^{\infty}(\mathbb{R},\rho_{0})$ is established. Note however that in this
work, like in [27], entropy solutions are defined directly for the system
(1.3) through a one-sided Lipschitz condition, see (1.7) below, and weak
continuity of $\rho v^{2}$ to $\rho_{0}v_{0}^{2}$ as $t\to 0+$, see Remark
5.10 for more details.

Other pressureless Euler-type systems [44] have recently been treated using
the connection between sticky particle solutions and entropy solutions of
scalar balance laws. For instance, solutions of the Euler-alignment system
have been obtained for compactly supported
$\rho_{0}\in\mathscr{P}(\mathbb{R})$ and $v_{0}\in
L^{\infty}(\mathbb{R},\rho_{0})$ in [33], where $f[\rho]$ is replaced by an
alignment force of Cucker–Smale-type involving the velocity $v$.

Connections between scalar conservation laws, optimal transport, monotone
nondecreasing rearrangements, and gradient flows have been explored in several
papers [3, 15, 16, 6, 4, 21].

Finally, we emphasize that a technical difficulty one needs to overcome is the
lack of Lipschitz bounds for the flux of the associated conservation law to
(1.1). As in [37], we then build upon the uniqueness results of [24], which is
based on earlier ideas of [11, 30] rather than classical existence and
uniqueness results for scalar conservation laws, cf. [31, 9, 25].

Since we only consider systems without pressure terms in the flux, we will
from here on omit the word pressureless when we speak of the systems (1.1) and
(1.3).

###  1.1. Main results

Here we present a formal overview of our main results, namely the equivalence
of the Lagrangian and entropy solution concepts for the Euler system (1.1).

A first clue to the connection between these solution concepts is the fact
that they both apply a “trick” to reduce the second-order system for the
variables $\rho$ and $v$ to a first order equation. For Lagrangian solutions,
the new unknown is the optimal transport map, or monotone rearrangement,
$X\colon[0,T]\times(0,1)\to\mathbb{R}$, which pushes the Lebesgue measure
$\mathfrak{m}$ on $(0,1)$ into $\rho(t,\cdot)$, that is,
$X(t,\cdot)_{\\#}\mathfrak{m}=\rho(t,\cdot)$. For a given time $t\in[0,T]$,
the map $X(t,\cdot)$ belongs to the cone $\mathscr{K}$ of nondecreasing
functions in $L^{2}(0,1)$. Note that concentration of mass in $\rho(t,\cdot)$
corresponds to “flat” sections of $X(t,\cdot)$, and in this setting another
useful subspace of $L^{2}(0,1)$ is the set $\mathscr{H}_{X}$ of functions
which are constant wherever $X$ is constant. The time-evolution of the
transport map is subject to a differential inclusion, rather than an equation,
involving a velocity $U$, which may very well depend on $X$, prescribed by the
forcing term in the Euler system. This differential inclusion reads
$U-\dot{X}\in N_{X}\mathscr{K}$, where $N_{X}\mathscr{K}$ is the normal cone
of $\mathscr{K}$ at $X$. Sometimes, however, when there is mass concentration,
the prescribed velocity $U$ may be incompatible with $X$ remaining in the cone
$\mathscr{K}$, that is, $U$ does not belong to $T_{X}\mathscr{K}$, the tangent
cone of $\mathscr{K}$ at $X$. This is analogous to when in the method of
characteristics, the characteristics are prescribed velocities which lead them
to cross. In such situations, one way of ensuring that $X$ remains
nondecreasing is to let its velocity $\dot{X}$ be given by the
$L^{2}$-projection of $U$ onto $\mathscr{H}_{X}\subset T_{X}\mathscr{K}$,
i.e., $\dot{X}=\operatorname{P}_{\mathscr{H}_{X}}U$. This then ensures a flat
section remains flat, moving with a common velocity. Of course, we then
require $U-\operatorname{P}_{\mathscr{H}_{X}}U\in N_{X}\mathscr{K}$ for $X$ to
still satisfy the differential inclusion. It then turns out that for all
times, the velocity of $X$ coincides with $L^{2}$-projection
$\operatorname{P}_{T_{X}\mathscr{K}}U$ of $U$ onto the tangent cone
$T_{X}\mathscr{K}$, i.e., the minimal element of the set $U-N_{X}\mathscr{K}$;
hence, this acts as a selection principle. We refer to Section 3 for details
on this solution framework.

On the other hand, the new unknown in the entropy solution-setting is the
distribution function of $\rho$, that is
$M\colon[0,T]\times\mathbb{R}\to[0,1]$. In order to combine the two equations
of the Euler system into a single conservation law, one then needs to devise
an appropriate time-dependent flux function $\mathcal{U}(t,M)$, which in fact
turns out to be the primitive of the prescribed velocity $U$ of the Lagrangian
solutions. Here, concentration of mass in $\rho(t,\cdot)$ corresponds to
“jumps” in $M(t,\cdot)$, i.e., a shock for the conservation law. If $M$ is
sufficiently regular, say, for simplicity, a piecewise smooth solution of the
conservation law, we know from the theory that for $M$ to be an admissible
weak solution, such jumps must satisfy the Rankine–Hugoniot condition which
determines the shock velocity as the ratio of the jump in $\mathcal{U}(t,M)$
to the jump in $M$. Moreover, uniqueness can be established if $M$ is an
entropy solution, which in this case amounts to the jumps satisfying an
inequality known as Oleĭnik’s E-condition, see [39]. This condition determines
whether a jump should be sustained as a shock, or if it should be smoothed
out, turning it into a rarefaction wave. This framework is detailed in Section
4.

We emphasize that the condition above should not be confused with another
inequality [38], see also [19, Section 11.2], frequently called the Oleĭnik
entropy condition and used to single out the desired solution. For (1.1) with
$f[\rho]\equiv 0$ this inequality takes the following form, cf. [27, 36, 35];
for a.e. $t>0$ we have

(1.7)
$\frac{v(t,x_{2})-v(t,x_{1})}{x_{2}-x_{1}}\leq\frac{1}{t}\enspace\text{for
$\rho$-a.e.}\ x_{1}\leq x_{2}.$

Note that for a given probability measure $\rho$, if its corresponding optimal
transport map $X$ and distribution function $M$ both are taken to be right-
continuous, $X$ and $M$ are each other’s generalized inverses, see, e.g.,
[20]. Then at least for the initial data, there is a correspondence between
the two notions of solution. A more interesting issue is whether this
reciprocity remains valid as these functions evolve in time, as it is not
obvious that these procedures “pick” the same solutions. The key to answering
this question lies in a more interesting correspondence between elements of
the two notions presented above. For instance, as a consequence of being each
others generalized inverses, a horizontal segment of an optimal transport map
corresponds to a vertical segment of the distribution function. Then it would
perhaps not be so surprising if the mechanisms preserving these segments, the
projection $\operatorname{P}_{\mathscr{H}_{X}}U$ for the former and the
Rankine–Hugoniot condition for the latter, were equivalent. A closer look at
the definition of the projection $\operatorname{P}_{\mathscr{H}_{X}}$ and the
relation between the prescribed velocity $U$ and flux function $\mathcal{U}$
reveals that this is indeed the case.

The next connection is perhaps less obvious, but thanks to a characterization
of the normal cone $N_{X}\mathscr{K}$ due to Natile and Savaré [35], one can
show that the requirement that the projection satisfies
$U-\operatorname{P}_{\mathscr{H}_{X}}U\in N_{X}\mathscr{K}$ is equivalent to
the Oleĭnik E-condition for $M$ and $\mathcal{U}(t,M)$.

Concept $\backslash$ Framework | Lagrangian solution |  | Entropy solution
---|---|---|---
Solution variable | Optimal transport map $X$ |  | Distribution function $M$
Shock admissibility | $\dot{X}=\operatorname{P}_{\mathscr{H}_{X}}U$ | $\iff$ | Rankine–Hugoniot condition
Selection principle | $U-\operatorname{P}_{\mathscr{H}_{X}}U\in N_{X}\mathscr{K}$ |  | Oleĭnik E-condition
Table 1. Correspondence table for the notions of solution.

These correspondences are summarized in Table 1 and lie at the heart of the
argument for our main result, which can be informally stated as follows.

######  Main result.

Lagrangian $\iff$ Entropy solutions for 1D pressureless Euler systems. The
equivalence is based on showing the correspondence of the concepts in Table 1.

Section 5 contains precise statements and proofs of the above results.
Finally, we emphasize that the one-dimensional setting is essential for both
frameworks: For the Lagrangian solutions, it gives a well-defined ordering to
define the cone $\mathscr{K}$. On the other hand, for the entropy solutions it
allows us to work with the distribution function $M$ rather than the measure
$\rho$, which on the line uniquely determine one another. The one-dimensional
setting is also needed for the reciprocal relationship between the optimal
transport map and distribution function of a probability measure, i.e., that
they are each other’s generalized inverses. Since the 1D setting is essential
for both solution concepts, it seems unlikely that these results may be
generalized to higher dimensions, at least in any naive way. However, on the
line one might allow the inclusion of more general forcing terms, e.g., the
alignment force studied in the recent work [33].

Following [7], Section 2 is devoted to motivate the Lagrangian solutions using
particle dynamics, as this concept might be more intuitive in this simpler
setting. The particle dynamics will be studied further in Section 6 in both
repulsive and attractive Poisson forces showcasing the right continuation
after collisions between particles dictated by the Lagrangian and entropy
solution concepts. We finally discuss in Section 7 the repulsive Poisson force
case with/without confinement and/or damping. These novel cases show the full
power of this equivalence to characterize the unique continuations after blow-
up in these additional examples pressureless Euler systems of the type (1.6).

##  2\. Motivation from particle dynamics

In this section we will motivate the need for well-defined notions of solution
from a particle-point of view, as the sticky particle dynamics lays the
foundations for the works [7] and [36, 37]. In particular we want to, as in
[7], set the stage for the Lagrangian solutions. This allows us to introduce a
few concepts from convex analysis in a somewhat simple setting, which we can
then recall as specific cases when we present the Lagrangian solutions in
generality in Section 3. We will also return to the particle dynamics in
Section 6 where we will study the particle dynamics for (1.3) in detail.

When we talk about particle solutions of (1.3), we mean linear combinations of
Dirac measures in the following form, which is typically called an empirical
measure in the case of $\rho$,

(2.1) $\rho(t,x)=\sum_{i=1}^{n}m_{i}\delta_{x_{i}(t)},\qquad\rho
v(t,x)=\sum_{i=1}^{n}m_{i}v_{i}(t)\delta_{x_{i}(t)},$

where $x_{i}(t)$ for $i\in\\{1,\dots,n\\}$ is the trajectory of a particle
with velocity $v_{i}(t)$ and mass $m_{i}>0$. Since we are considering
$\rho\in\mathscr{P}(\mathbb{R})$, the masses should always sum to one, i.e.,
$\sum_{i=1}^{n}m_{i}=1$. For the rest of this section we assume that the $i$th
particle has initial position $x_{i}^{0}$ and velocity $v_{i}^{0}$, where the
initial positions are distinct and ordered:

$x_{1}^{0}<x_{2}^{0}<\cdots<x_{n-1}^{0}<x_{n}^{0}.$

At some later time, particles may meet, and so we will have to impose a rule
for resolving the dynamics in those cases; for now, the only requirement we
impose is that particle trajectories may not cross, that is, the initial
ordering of particles is retained.

###  2.1. First considerations and simplifications

Initially, the $n$ distinct particles will be governed by the particle system
corresponding to (1.3) given by

(2.2) $\ddot{x}_{i}=-\frac{\alpha}{2}\sum_{j=1,j\neq
i}^{n}m_{j}\operatorname*{sgn}(x_{i}-x_{j})-\beta,\qquad i\in\\{1,\dots,n\\},$

which can seen as Newton’s second law for a set of particles moving in a
conservative field governed by the scalar potential
$E_{\mathrm{p}}(\\{x_{i}\\})$ where

$E_{\mathrm{p}}(\\{x_{i}\\})=\frac{\alpha}{2}\sum_{i=1}^{n}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{n}m_{i}m_{j}\phi(x_{i}-x_{j})+\beta\sum_{i=1}^{n}m_{i}x_{i}=\frac{\alpha}{4}\sum_{i=1}^{n}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{n}m_{i}m_{j}\lvert
x_{i}-x_{j}\rvert+\beta\sum_{i=1}^{n}m_{i}x_{i}.$

The first term on the right-hand side of (2.2), scaled by $\alpha$, is the
force per unit mass coming from the Poisson interaction forces between
particles, and is therefore an internal force for the system. On the other
hand, the second term scaled by $\beta$ is the force per unit mass coming from
an external force, which could be an external gravitational or electrical
field. Introducing the kinetic energy
$E_{\mathrm{k}}(\\{x_{i}\\})=\frac{1}{2}\sum_{i=1}^{n}m_{i}\dot{x}_{i}^{2}$,
then the Euler–Lagrange equations for the Lagrangian
$L=E_{\mathrm{k}}-E_{\mathrm{p}}$ yields (2.2). Moreover, the energy
$E_{\mathrm{k}}+E_{\mathrm{p}}$, which corresponds to the Hamiltonian after
introducing the momentum $p_{i}=m_{i}\dot{x}_{i}$, is conserved. The total
momentum $\sum_{i=1}^{n}m_{i}\dot{x}_{i}$ however, is not conserved because of
the external force. If we introduce the center of mass
$\bar{x}\coloneqq\sum_{i=1}^{n}m_{i}x_{i}$ and average velocity, which here
coincides with the total momentum, $\bar{v}=\sum_{i=1}^{n}m_{i}\dot{x}_{i}$,
we find that $\dot{\bar{x}}=\bar{v}$. Furthermore, $\dot{\bar{v}}=-\beta$
since the sum of internal forces equals zero by Newton’s third law, meaning
that the total momentum changes at a linear rate, while the center of mass
will follow a parabolic trajectory. If we instead of the absolute positions
$x_{i}$ consider the positions $x_{i}^{\prime}$ relative to the center of
mass, i.e., $x_{i}^{\prime}=x_{i}-\bar{x}$, we can consider the system (2.2)
without external forces. As we have seen, the external force only affect the
center of mass $\bar{x}$, while the internal forces only affect the relative
positions $x_{i}^{\prime}$ governed by (2.2) with $x_{i}$ replaced by
$x_{i}^{\prime}$ and $\beta=0$. The relative center of mass
$\sum_{i=1}^{n}m_{i}x_{i}^{\prime}=0$ and momentum
$\sum_{i=1}^{n}m_{i}\dot{x}_{i}^{\prime}=0$ are conserved, and the kinetic
energy $E_{\mathrm{k}}=\frac{1}{2}\sum_{i=1}^{n}m_{i}\dot{x}_{i}^{2}$ can be
decomposed as

$\frac{1}{2}\sum_{i=1}^{n}m_{i}\dot{x}_{i}^{2}=\frac{1}{2}\dot{\bar{x}}^{2}+\frac{1}{2}\sum_{i=1}^{n}m_{i}(\dot{x}_{i}^{\prime})^{2}.$

Therefore, we could without loss of generality consider the dynamics of the
relative positions, not relabeled with primes, governed by (2.2) with
$\beta=0$, and add the motion of the center of mass afterwards to obtain the
full dynamics. However, we choose to keep $\beta$ in our approach, for ease of
comparison to the works [36, 37].

###  2.2. Particle dynamics as weighted $\ell^{2}$-gradient flow

In this section, we follow [7] in motivating the differential inclusion
formulation of the particle dynamics. For ease of notation, let us write
$\mathbf{m}=(m_{1},\dots,m_{n})$, $\mathbf{x}=(x_{1},\dots,x_{n})$ and
$\mathbf{v}=(v_{1},\dots,v_{n})\in\mathbb{R}^{n}$, such that
$\mathbf{m}\in\mathbb{R}^{n}_{+}$ while
$\mathbf{x},\mathbf{v}\in\mathbb{R}^{n}$. Let us for the moment consider a
slightly more general setting than in the previous section, where the dynamics
initially are given by the accelerated motion

(2.3)
$\dot{\mathbf{x}}=\mathbf{v},\qquad\dot{\mathbf{v}}=\mathbf{a}_{\mathbf{m}}(\mathbf{x}),$

where
$\mathbf{a}_{\mathbf{m}}=(a_{\mathbf{m},1}(\mathbf{x}),\dots,a_{\mathbf{m},n}(\mathbf{x}))$
is defined for $\mathbf{x}\in\mathbb{R}^{n}$. For the set of particles
$\mathbf{x}\in\mathbb{R}^{n}$, our rule of non-crossing trajectories, or well-
ordering, is equivalent to requiring $\mathbf{x}$ to lie in the convex cone

(2.4) $\mathbb{K}^{n}\coloneqq\\{\mathbf{x}\in\mathbb{R}^{n}\colon x_{j}\leq
x_{j+1},\,j\in\\{1,\dots,n-1\\}\\}.$

When a non-empty subset of particles collide, we have
$\mathbf{x}\in\partial\mathbb{K}^{n}$, the boundary of the convex set. To be
precise, let us introduce the set of indices for collided particles
$\Omega_{\mathbf{x}}\coloneqq\\{j\colon
x_{j}=x_{j+1},\,j\in\\{1,\dots,n-1\\}\\}$, then
$\partial\mathbb{K}^{n}=\\{\mathbf{x}\in\mathbb{K}^{n}\colon\Omega_{\mathbf{x}}\neq\emptyset\\}$.
Now, since $\mathbf{x}$ is required to lie within $\mathbb{K}^{n}$, not all
velocities are admissible for a given $\mathbf{x}$. The set of admissible
velocities at position $\mathbf{x}$ is given by the tangent cone
$T_{\mathbf{x}}\mathbb{K}^{n}$, defined as

(2.5)
$T_{\mathbf{x}}\mathbb{K}^{n}\coloneqq\mathrm{cl}\\{\vartheta(\mathbf{y}-\mathbf{x})\colon\mathbf{y}\in\mathbb{K}^{n},\vartheta\geq
0\\},$

the closure of line segments in the direction $\mathbf{y}-\mathbf{x}$ for
which $\mathbf{y}$ belongs to $\mathbb{K}^{n}$. One can show that (2.5) is
equivalent to

(2.6) $T_{\mathbf{x}}\mathbb{K}^{n}=\\{\mathbf{v}\in\mathbb{R}^{n}\colon
v_{j}\leq v_{j+1}\>\forall{j,j+1}\in\Omega_{\mathbf{x}}\\}.$

We see that if $\Omega_{\mathbf{x}}=\emptyset$ then all velocities are
admissible, i.e., $T_{\mathbf{x}}\mathbb{K}^{n}=\mathbb{R}^{n}$. However, when
particles collide at time $t$, i.e., $\mathbf{x}(t)\in\partial\mathbb{K}^{n}$,
we have to change the dynamics (2.3) in order to stay within the cone. One
could imagine several ways of doing this, but assuming inelastic collisions,
one is led to a projection

$\mathbf{v}(t+)=\operatorname{P}_{T_{\mathbf{x}(t)}\mathbb{K}^{n}}\mathbf{v}(t-),$

where $\operatorname{P}_{T_{\mathbf{x}(t)}\mathbb{K}^{n}}$ is the weighted
$\ell^{2}$-projection onto $T_{\mathbf{x}(t)}\mathbb{K}^{n}$ with respect to
the induced norm

$\lVert\mathbf{v}\rVert_{\mathbf{m}}=\left(\sum_{i=1}^{n}m_{i}v_{i}^{2}\right)^{1/2}.$

It is then also true that the new velocity is minimizing in the sense
$(\mathbf{v}(t-)-\mathbf{v}(t+))\cdot\mathbf{u}\leq 0$ for all $\mathbf{u}\in
T_{\mathbf{x}(t)}\mathbb{K}^{n}$. Thinking of this change in velocity as the
result of an instantaneous force, this force must be an element of the normal
cone $N_{\mathbf{x}(t)}\mathbb{K}^{n}$, defined by

$N_{\mathbf{x}}=\\{\mathbf{n}\in\mathbb{R}^{n}\colon\mathbf{n}\cdot(\mathbf{y}-\mathbf{x})\leq
0\;\forall\mathbf{y}\in\mathbb{K}^{n}\\}.$

which coincides with the subdifferential $\partial
I_{\mathbb{K}^{n}}(\mathbf{x})$ of the indicator function $I_{\mathbb{K}^{n}}$
of $\mathbb{K}^{n}$ at the point $\mathbf{x}$. This then suggests the
following generalization of (2.3) to describe the dynamics also during
collisions, namely the second order differential inclusion

(2.7)
$\dot{\mathbf{x}}=\mathbf{v},\qquad\dot{\mathbf{v}}+N_{\mathbf{x}}\mathbb{K}^{n}\ni\mathbf{a}_{\mathbf{m}}(\mathbf{x}).$

It was shown in [35] that along a curve $t\mapsto\mathbf{x}(t)$ satisfying a
global stickiness condition, there is a monotonicity property for the normal
cones; that is, $N_{\mathbf{x}(s)}\mathbb{K}^{n}\subset
N_{\mathbf{x}(t)}\mathbb{K}^{n}$ for $s<t$. This property can then be used to
reduce the second order differential inclusion (2.7) to a first order
inclusion. Under this monotonicity, if there is an
$\xi\colon[0,\infty)\to\mathbb{R}^{n}$ with $\xi(t)\in
N_{\mathbf{x}(t)}\mathbb{K}^{n}$, e.g.,
$\mathbf{a}_{\mathbf{m}}-\dot{\mathbf{v}}$ in (2.7), then

$\int_{s}^{t}\xi(r)\mathop{}\\!\mathrm{d}r\in N_{\mathbf{x}(t)}\mathbb{K}^{n}$

for all $t>s$. Formally, we then have

$\mathbf{v}(t)+N_{\mathbf{x}(t)}\mathbb{K}^{n}\ni\mathbf{v}(s)+\int_{s}^{t}\mathbf{a}_{\mathbf{m}}(\mathbf{x}(r))\mathop{}\\!\mathrm{d}r,$

and so we can rewrite (2.7) as the first-order differential inclusion

(2.8)
$\dot{\mathbf{x}}+N_{\mathbf{x}}\mathbb{K}^{n}\ni\mathbf{u},\qquad\dot{\mathbf{u}}=\mathbf{a}_{\mathbf{m}}(\mathbf{x}).$

Now, the idea presented in [7] is that one can introduce a well-defined
solution of the differential inclusion (2.8) by choosing
$\dot{\mathbf{x}}=\mathbf{v}$ where the velocity $\mathbf{v}$ is the minimal
element in $\mathbf{u}-N_{\mathbf{x}}\mathbb{K}^{n}$, i.e.,
$\mathbf{v}=\operatorname{P}_{T_{\mathbf{x}}\mathbb{K}^{n}}\mathbf{u}$. Then,
since we define the particle solution to be the trajectory for which the
velocity is the unique element in the tangent cone
$T_{\mathbf{x}}\mathbb{K}^{n}$ closest to the free velocity $\mathbf{u}$ with
respect to the weighted $\ell^{2}$-norm $\lVert\cdot\rVert_{\mathbf{m}}$, this
can be regarded as a discrete version of $L^{2}$-gradient flow. In the next
section we will see how this principle can be adapted to the continuous
setting and an acceleration induced by the forcing term $f[\rho]$ in (1.1).

##  3\. $L^{2}$-gradient flow

We will here present the concept of Lagrangian solutions for the Euler system
(1.1) as introduced by Brenier, Gangbo, Savaré and Westdickenberg [7]. Note
that for the first order interaction equation (1.4) in [4], this corresponds
to their $L^{2}$-gradient flow solutions. This notion of solution relies on
concepts from convex analysis and theory on differential inclusions found in
the monograph of Brézis [10], in addition to results of Natile and Savaré [35]
from their study of the Euler system without forcing, i.e., $f[\rho]\equiv 0$.
For the reader’s convenience we will restate several of their definitions and
results which we make use of. We refer to Section 3, see also Section 6, of
[7] for the complete picture.

###  3.1. Elements of optimal transport

Consider the space $\mathscr{P}(\mathbb{R}^{m})$ of Borel probability measures
on $\mathbb{R}^{m}$. The push-forward $\nu\coloneqq Y_{\\#}\mu$ of a measure
$\mu\in\mathscr{P}(\mathbb{R}^{m})$ under the Borel map
$Y\colon\mathbb{R}^{m}\to\mathbb{R}^{n}$ is the measure $\nu$ defined through
$\nu(A)=\mu(Y^{-1}(A))$ for all Borel sets $A\subset\mathbb{R}^{n}$. For any
Borel map $\varphi\colon\mathbb{R}^{n}\to[0,\infty]$ we then have the change-
of-variable formula

(3.1)
$\int_{\mathbb{R}^{n}}\varphi(y)\mathop{}\\!\mathrm{d}Y_{\\#}\mu(y)=\int_{\mathbb{R}^{m}}\varphi(Y(x))\mathop{}\\!\mathrm{d}\mu(x).$

For $p\in[1,\infty)$ we let $\mathscr{P}_{p}(\mathbb{R}^{n})$ denote the space
of all $\rho\in\mathscr{P}(\mathbb{R}^{n})$ with bounded $p$th moment, i.e.,
$\int_{\mathbb{R}^{n}}\lvert x\rvert^{p}\mathop{}\\!\mathrm{d}\rho(x)<\infty$.
Then the $p$-Wasserstein distance between two measures
$\rho_{1},\rho_{2}\in\mathscr{P}_{p}(\mathbb{R}^{n})$ can be defined as

(3.2)
$d_{\mathscr{W}_{p}}(\rho_{1},\rho_{2})^{2}\coloneqq\min\left\\{\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\lvert
x-y\rvert^{p}\mathop{}\\!\mathrm{d}\varrho(x,y)\colon\varrho\in\mathscr{P}(\mathbb{R}^{n}\times\mathbb{R}^{n}),\>\varpi^{i}_{\\#}\varrho=\rho_{i}\right\\},$

where $\varpi^{i}(x_{1},x_{2})=x_{i}$ is the projection onto the $i$th
coordinate. It can be shown that there always exists an optimal transport plan
$\varrho$ which achieves the infimum of the integral in (3.2). We will be
concerned with $p=2$ and measures on the line, i.e., $n=1$. In this setting
there is a unique optimal plan, which can be characterized explicitly.

For any $\rho\in\mathscr{P}(\mathbb{R})$ we define its right-continuous
cumulative distribution function

$M_{\rho}(x)\coloneqq\rho((-\infty,x])\quad\text{for all }x\in\mathbb{R},$

noting that $\partial_{x}M_{\rho}=\rho$ in $\mathscr{D}^{\prime}(\mathbb{R})$,
i.e., in the distributional sense. Then we can define its right-continuous
generalized inverse as

$X_{\rho}(m)\coloneqq\inf\\{x\colon M_{\rho}(x)>m\\}\quad\text{for all
}m\in\Omega,$

where $\Omega\coloneqq(0,1)$, which is clearly nondecreasing. This is in fact
the optimal transport map pushing the one-dimensional Lebesgue measure
$\mathfrak{m}\coloneqq\mathcal{L}^{1}|_{\Omega}$ to $\rho$ on $\mathbb{R}$.
the Lebesgue measure on $\Omega$, meaning

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

for any Borel map $\varphi\colon\mathbb{R}\to[0,\infty]$ by (3.1). For
instance, $\rho\in\mathscr{P}_{2}(\mathbb{R})$ if and only if $X_{\rho}\in
L^{2}(\Omega)$. The Hoeffding–Frechét theorem, cf. [47, Theorem 2.18], then
characterizes the optimal transport plan $\varrho$ between $\rho_{1}$ and
$\rho_{2}$ as

(3.3) $\varrho=(X_{\rho_{1},\rho_{2}})_{\\#}\mathfrak{m},\quad
X_{\rho_{1},\rho_{2}}=(X_{\rho_{1}},X_{\rho_{2}})\quad\text{for all
}m\in\Omega.$

Therefore we have that their 2-Wasserstein distance is given by

$d_{\mathscr{W}_{2}}(\rho_{1},\rho_{2})^{2}=\int_{\Omega}\lvert
X_{\rho_{1}}(m)-X_{\rho_{2}}(m)\rvert^{2}\mathop{}\\!\mathrm{d}m=\lVert
X_{\rho_{1}}-X_{\rho_{2}}\rVert^{2}_{L^{2}(\Omega)}.$

The map $\rho\mapsto X_{\rho}$ is an isometry between
$\mathscr{P}_{2}(\mathbb{R})$ and $\mathscr{K}$, where

(3.4) $\mathscr{K}\coloneqq\\{X\in L^{2}(\Omega)\colon X\text{ is
nondecreasing}\\},$

and without loss of generality we may consider right-continuous
representatives in $\mathscr{K}$, which are defined everywhere.

####  3.1.1. Comparison to the particle case

The idea is now to replace the set of particles labeled by
$i\in\\{1,\dots,n\\}$ with “particles” labeled by $m\in\Omega=(0,1)$ and which
at time $t$ has position $X(t,m)\in\mathbb{R}$. Of course, these are no longer
discrete particles, but a continuum of mass distributed over the real line.
For a fixed $t$, we know from the previous considerations that $X$ can be
uniquely characterized by a measure $\rho$, i.e., it is the nondecreasing and
right-continuous map $X\colon\Omega\to\mathbb{R}$ such that

$X(m)\leq x\iff m\leq\rho((-\infty,x])\qquad\text{for all }x\in\mathbb{R}.$

Going the other way, to any solution $(\rho,v)$ of (1.1) we can associate a
map $X\colon[0,\infty)\times\Omega\to\mathbb{R}$ with $X(t,\cdot)$
nondecreasing, and a velocity $V\colon[0,\infty)\times\Omega\to\mathbb{R}$
such that for $t\geq 0$ we have

(3.5) $X(t,\cdot)_{\\#}\mathfrak{m}=\rho(t,\cdot),\qquad
V(t,\cdot)=v(t,X(t,\cdot))=\partial_{t}X(t,\cdot).$

The idea is now, analogous to the discrete particle case, to associate (1.1)
to a differential inclusion for the pair $(X,V)$. Here the cone
$\mathbb{K}^{n}$ is replaced with the set of optimal transport maps
$\mathscr{K}$ defined in (3.4), which can be shown to be a closed, convex cone
in $L^{2}(\Omega)$. To handle mass concentration, corresponding to particle
collisions in the discrete case, we will recall some results on convex
analysis.

###  3.2. Elements of convex analysis

The normal cone $N_{X}\mathscr{K}$ of $\mathscr{K}$ at $X\in\mathscr{K}$ is

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

where, as in the discrete case, $N_{X}\mathscr{K}=\partial
I_{\mathscr{K}}(X)$, the subdifferential at $X$ of the indicator function
$I_{\mathscr{K}}\colon L^{2}(\Omega)\to[0,+\infty]$,

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

Moreover, the tangent cone $T_{X}\mathscr{K}$ at $X$ can be defined as the
polar cone of $N_{X}\mathscr{K}$, that is,

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

We recall from the particle dynamics in Section 2.2 that there were two
equivalent characterizations (2.5) and (2.6) of the tangent cone
$T_{\mathbf{x}}\mathbb{K}^{n}$. This is true also in our current setting, for
the tangent cone $T_{X}\mathscr{K}$ as well as the normal cone
$N_{X}\mathscr{K}$, and it will be convenient to work with both
characterizations. For the alternative descriptions, we need, analogously to
the set of collided particles $\Omega_{\mathbf{x}}$, the set of concentrated
mass $\Omega_{X}$, namely

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

The following characterization can be found in [7, Lemma 2.3] and builds upon
[35, 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

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

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

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

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

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

Moreover, we have the tangent cone characterization below, found in [7, Lemma
2.4].

######  Lemma 3.2 (Characterization of the tangent cone $T_{X}\mathscr{K}$).

Given $X\in\mathscr{K}$, then

(3.9) $T_{X}\mathscr{K}=\left\\{U\in L^{2}(\Omega)\colon U\text{ is
nondecreasing on each interval }(m_{l},m_{r})\subset\Omega_{X}\right\\}.$

Based on the set $\Omega_{X}$ in (3.7) we can for $X\in\mathscr{K}$ also
introduce the closed subspace $\mathscr{H}_{X}\subset L^{2}(\Omega)$ given by

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

For $U\in L^{2}(\Omega)$, the $L^{2}$-projection onto this subspace is then
given by

(3.11) $\operatorname{P}_{\mathscr{H}_{X}}U=\begin{cases}U,&\text{for a.e.\
}m\in\Omega\setminus\Omega_{X}\\\
\frac{1}{m_{r}-m_{l}}\int_{m_{l}}^{m_{r}}U(\omega)\mathop{}\\!\mathrm{d}\omega,&\text{for
a.e.\ }m\in(m_{l},m_{r}),\text{ a maximal interval of }\Omega_{X}.\end{cases}$

As can be expected from the particle dynamics, such a projection can be
applied to the velocity to ensure that $X$ stays in the cone $\mathscr{K}$. A
direct consequence of Lemma 3.2 is the equivalence relation

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

Moreover, if $X_{1},X_{2}\in\mathscr{K}$, we have the implication

(3.12)
$\Omega_{X_{1}}\subset\Omega_{X_{2}}\implies\mathscr{H}_{X_{2}}\subset\mathscr{H}_{X_{1}}$

Observe that if we take $U\in\mathscr{H}_{X}$ then the integral in (3.6)
vanishes, meaning $N_{X}\mathscr{K}\subset\mathscr{H}_{X}^{\perp}$ for all
$X\in\mathscr{K}$, where $\mathscr{H}_{X}^{\perp}$ is the orthogonal
complement of $\mathscr{H}_{X}$. As a consequence,

(3.13) $Y=\operatorname{P}_{\mathscr{K}}X\implies
Y=\operatorname{P}_{\mathscr{H}_{Y}}X,\quad\mathscr{H}_{Y}\subset\mathscr{H}_{X}.$

In fact, [7, Lemma 2.5] provides a more precise characterization of
$\mathscr{H}_{X}$ in terms of $N_{X}\mathscr{K}$. Indeed, we have

$\mathscr{H}_{X}^{\perp}=\left\\{W\in L^{2}(\Omega)\colon W=0\text{ a.e.\ in
}\Omega\setminus\Omega_{X},\int_{m_{l}}^{m_{r}}W(m)\mathop{}\\!\mathrm{d}m=0\>\forall\text{
maximal interval }(m_{l},m_{r})\subset\Omega_{X}\right\\},$

and it is the closed, linear subspace of $L^{2}(\Omega)$ generated by
$N_{X}\mathscr{K}$. Moreover it satisfies

(3.14) $\mathscr{H}_{X}^{\perp}=\left\\{W\in
L^{2}(\Omega)\colon\int_{\Omega}W(m)\varphi(X(m))\mathop{}\\!\mathrm{d}m=0\>\forall\varphi\in
C_{\mathrm{b}}(\mathbb{R})\right\\}.$

###  3.3. Lagrangian solutions

Following the same arguments as for the particle dynamics in Section 2.2, we
are led to study first-order differential inclusions of the form

(3.15) $\dot{X}(t)+\partial I_{\mathscr{K}}(X(t))\ni
V^{0}+\int_{0}^{t}F[X(s)]\mathop{}\\!\mathrm{d}s\text{ for a.e. }t\geq
0,\qquad X(0)=\lim\limits_{t\to 0+}X(t)=X^{0}.$

Here $\dot{X}$ is the differential map of
$X\colon[0,\infty)\to\mathscr{K}\subset L^{2}(\Omega)$.

####  3.3.1. The Lagrangian force representation

The link between (3.15) and the Euler system (1.1) lies in the Lagrangian
representation $F\colon\mathscr{K}\to L^{2}(\Omega)$ of the force distribution
$f\colon\mathscr{P}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$, which must satisfy
the distributional identity

(3.16)
$\int_{\mathbb{R}}\varphi(x)f[\rho](\mathop{}\\!\mathrm{d}x)=\int_{\Omega}\varphi(X_{\rho}(m))F[X](m)\mathop{}\\!\mathrm{d}m\text{
for all }\varphi\in\mathscr{D}(\mathbb{R})\text{ and
}\rho\in\mathscr{P}(\mathbb{R}),$

where $X_{\rho}\in\mathscr{K}$ and $(X_{\rho})_{\\#}\mathfrak{m}=\rho$. From
(3.14) we see that $F[X]$ is not uniquely defined unless
$\mathscr{H}_{X}^{\perp}=\\{0\\}$, meaning $\mathscr{H}_{X}=L^{2}(\Omega)$,
that is, $\Omega_{X}=\emptyset$; this is exactly when $X$ is strictly
increasing. In order to facilitate the study of solutions for (3.15), in
addition to (3.16) we will require $F$ to be everywhere defined in
$\mathscr{K}$ and satisfy additional boundedness and continuity properties.

######  Definition 3.3 (Boundedness).

The operator $F\colon\mathscr{K}\to L^{2}(\Omega)$ is bounded if there is a
constant $C\geq 0$ such that

(3.17) $\lVert F[X]\rVert_{L^{2}(\Omega)}\leq C\left(1+\lVert
X\rVert_{L^{2}(\mathbb{R})}\right)\text{ for all }X\in\mathscr{K}.$

We say it is pointwise linearly bounded if there is a constant
$C_{\mathrm{p}}\geq 0$ such that

$\lvert F[X](m)\rvert\leq C_{\mathrm{p}}\left(1+\lvert X(m)\rvert+\lVert
X\rVert_{L^{1}(\Omega)}\right).$

Observe that if $F$ is pointwise linearly bounded, then it is bounded with
$C=2C_{\mathrm{p}}$ in (3.17). Recall that a modulus of continuity $\omega$ is
a continuous and concave function $\omega\colon[0,\infty)\to[0,\infty)$
satisfying $0=\omega(0)<\omega(r)$ for all $r>0$.

######  Definition 3.4 (Uniform continuity).

We say that $F\colon\mathscr{K}\to L^{2}(\Omega)$ is uniformly continuous if
there is a modulus of continuity $\omega$ such that

(3.18) $\lVert F[X_{1}]-F[X_{2}]\rVert_{L^{2}(\Omega)}\leq\omega\left(\lVert
X_{1}-X_{2}\rVert_{L^{2}(\Omega)}\right)\quad\text{for all
}X_{1},X_{2}\in\mathscr{K}.$

Similarly, $F$ is Lipschitz continuous if (3.18) holds with $\omega(r)=Lr$ for
some $L>0$ and all $r\geq 0$.

If a uniformly continuous $F$ is defined by (3.16) on the convex subset
$\mathscr{K}_{\mathrm{si}}\subset\mathscr{K}$ of strictly increasing maps and
satisfies (3.18) on $\mathscr{K}_{\mathrm{si}}$, it admits a unique extension
to $\mathscr{K}$ which preserves the compatibility (3.16) and continuity
(3.18) conditions.

A final important property of $F$ concerns its action on the subset
$\Omega_{X}$ where $X$ is constant, i.e., where mass concentrates. For a
sticky evolution of $X$, this set would be nondecreasing in time, and since
the force map determines the change in velocity, it would be natural to assume
$F[X]\in\mathscr{H}_{X}$ for $X\in\mathscr{K}$ in this setting. However, it
turns out that the following weaker condition is sufficient to preserve the
sticky property.

######  Definition 3.5 (Sticking).

The map $F\colon\mathscr{K}\to L^{2}(\Omega)$ is called sticking if

(3.19) $F[X]-\operatorname{P}_{\mathscr{H}_{X}}F[X]\in\partial
I_{\mathscr{K}}(X)\enspace\text{for all }X\in\mathscr{K}.$

####  3.3.2. Definition, existence and uniqueness

The definition of a Lagrangian solution of (3.15) is given in [7, Definition
3.4], which we recall below.

######  Definition 3.6 (Lagrangian solutions of (3.15)).

Let $F\colon\mathscr{K}\to L^{2}(\Omega)$ be a uniformly continuous operator
and let $X^{0}\in\mathscr{K}$, $V^{0}\in L^{2}(\Omega)$ be given. A Lagrangian
solution to (3.15) with initial data $(X^{0},V^{0})$ is a curve
$X\in\mathrm{Lip}_{\mathrm{loc}}([0,\infty);\mathscr{K})$ satisfying
$X(0)=X^{0}$ and (3.15).

If we introduce the velocity $U$ prescribed by the initial data $V^{0}$ and
the force $F$, i.e.,

(3.20) $U(t)\coloneqq V^{0}+\int_{0}^{t}F(X(s))\mathop{}\\!\mathrm{d}s,$

we see that (3.15) is equivalent to

(3.21a) $\displaystyle\dot{X}(t)$ $\displaystyle+\partial
I_{\mathscr{K}}(X(t))\ni U(t)\text{ for a.e.\ }t\geq 0,$ $\displaystyle\qquad
X(0)$ $\displaystyle=X^{0},$ (3.21b) $\displaystyle\dot{U}(t)$
$\displaystyle=F(X(t)),$ $\displaystyle U(0)$ $\displaystyle=V^{0},$

where we note that the continuity of $F$ yields $U\in
C^{1}([0,\infty);L^{2}(\Omega))$. Here and for the rest of the paper we use
the term prescribed velocity to distinguish $U$ from the actual velocity of
$X$, since the two do not coincide in general; indeed, this happens when
$U\notin T_{X}\mathscr{K}$.

Lagrangian solutions exhibit several useful properties, contained in the
following [7, Theorem 3.5].

######  Lemma 3.7 (Properties of Lagrangian solutions).

Let $F\colon\mathscr{K}\to L^{2}(\Omega)$ be a uniformly continuous operator
and let $(X,U)$ be a corresponding Lagrangian solution of (3.21). Then the
following properties hold:

  * •

The right-derivative of $X(t)$ exists for all $t\geq 0$ and equals

$V\coloneqq\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X.$

  * •

The velocity $V(t)$ for $t\geq 0$ is the minimal element of the closed and
convex set $U(t)-\partial_{\mathscr{K}}I(X(t))$. Note that if we replace
$\dot{X}$ with $V$ then (3.15) and (3.21) hold for all $t\geq 0$.

  * •

For all $t\geq 0$, the minimal element $V$ is the projection on the tangent
cone given by

$V(t)=\operatorname{P}_{T_{X}(t)\mathscr{K}}U(t)$

  * •

$V(t)$ is right-continuous for all $t\geq 0$, and in particular

$\lim\limits_{t\to 0+}V(t)=V^{0}\iff V^{0}\in T_{X^{0}}\mathscr{K}.$

Moreover, if $\mathcal{T}\subset(0,\infty)$ is the subset of times at which
$t\mapsto\lVert V(t)\rVert_{L^{2}(\Omega)}$ is continuous, then
$(0,\infty)\setminus\mathcal{T}$ is negligible, and $V$ is continuous, $X$ is
differentiable in $L^{2}(\Omega)$ at every point of $\mathcal{T}$. Setting
$\rho(t)=X(t)_{\\#}\mathfrak{m}$, there exists a unique map $v(t,\cdot)\in
L^{2}(\Omega,\rho)$ such that

$\dot{X}(t)=V(t)=\operatorname{P}_{\mathscr{H}_{X(t)}}U(t)=v(t,\cdot)\circ
X(t)\in\mathscr{H}_{X(t)},\quad\text{for every }t\in\mathcal{T}.$

  * •

If moreover $F$ is linked to $f$ by (3.16),
$\rho_{0}=X^{0}_{\\#}\mathfrak{m}$, $V^{0}=v_{0}\circ X^{0}$, then $(\rho,v)$
defined above is a distributional solution to (1.1) such that

$\lim\limits_{t\to 0+}\rho(t,\cdot)=\rho_{0}\text{ in
}\mathscr{P}_{2}(\mathbb{R}),\qquad\lim\limits_{t\to
0+}\rho(t,\cdot)v(t,\cdot)=\rho_{0}v_{0}\text{ in }\mathscr{M}(\mathbb{R})$

Assuming $F$ to be merely pointwise linearly bounded and uniformly continuous,
one can show existence of Lagrangian solutions to (3.21). We will however be
interested in the cases where we also have uniqueness, and to this end we have
the following result from [7, Theorem 3.6].

######  Theorem 3.8 (Existence and uniqueness of Lagrangian solutions).

Assume $F\colon\mathscr{K}\to L^{2}(\Omega)$ is Lipschitz. Then for every
$T>0$ and $(X^{0},V^{0})\in\mathscr{K}\times L^{2}(\Omega)$ there exists a
unique Lagrangian solution $X$ to (3.15) for $t\in[0,T]$.

Let us return to the matter of sticky evolution: it turns out that we may
define a subclass of the Lagrangian solutions with this behavior.

######  Definition 3.9 (Sticky Lagrangian solutions).

A Lagrangian solution is called sticky if for $t_{1}\leq t_{2}$ we have
$\Omega_{X(t_{1})}\subset\Omega_{X(t_{2})}$.

Because of the implications (3.8) and (3.12), any sticky Lagrangian solution
satisfies the monotonicity condition

$\partial I_{\mathscr{K}}(X(t_{1}))\subset\partial
I_{\mathscr{K}}(X(t_{2})),\quad\mathscr{H}_{X(t_{2})}\subset\mathscr{H}_{X(t_{1})}\enspace\text{for
any }t_{1}\leq t_{2}.$

From this property one may derive the following representation formulas for
sticky solutions, [7, Proposition 3.8].

######  Proposition 3.10 (Projection formula).

If $X$ is a sticky Lagrangian solution, then

$V(t)\in\mathscr{H}_{X(t)}\enspace\text{for all times }t\geq 0$

and $(X,V)$ satisfy

(3.22) $\displaystyle X(t)$
$\displaystyle=\operatorname{P}_{\mathscr{K}}\left(X^{0}+\int_{0}^{t}U(s)\mathop{}\\!\mathrm{d}s\right)=\operatorname{P}_{\mathscr{K}}\left(X^{0}+tV^{0}+\int_{0}^{t}(t-s)F[X(s)]\mathop{}\\!\mathrm{d}s\right),$
(3.23) $\displaystyle V(t)$
$\displaystyle=\operatorname{P}_{\mathscr{H}_{X(t)}}U(t)=\operatorname{P}_{\mathscr{H}_{X(t)}}\left(V^{0}+\int_{0}^{t}F[X(s)]\mathop{}\\!\mathrm{d}s\right).$

The above $L^{2}$-projection onto the cone $\mathscr{K}$ in (3.4) can be
computed through the following useful characterization from [35, Theorem 3.1].

######  Lemma 3.11.

Let $f\in L^{2}(0,1)$ with primitive
$F(m)=\int_{0}^{m}f(\omega)\mathop{}\\!\mathrm{d}\omega$. Then the projection
of $f$ onto the cone $\mathscr{K}$ is given by the right-derivative of the
double Legendre–Fenchel transform $F^{**}$ of $F$:

$\operatorname{P}_{\mathscr{K}}f=\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}m}F^{**}(m).$

For a continuous function $F\colon[0,1]\to\mathbb{R}$, $F^{**}$ coincides with
its lower convex envelope on $[0,1]$, denoted by $F_{\smallsmile}$, which is
the case in the above lemma.

As mentioned before, there is a sufficient criterion for a Lagrangian solution
to be sticky. Indeed, by [7, Theorem 3.11], if the force $F$ is Lipschitz
continuous and sticking in the sense of Definition 3.5, then every Lagrangian
solution of (3.15) with $(X^{0},V^{0})\in\mathscr{K}\times\mathscr{H}_{X^{0}}$
is (globally) sticky, and given by the projection formula of Proposition 3.10.

######  Remark 3.12 (Generalized Lagrangian solutions).

In [7, Section 4] the authors introduce the notion of generalized Lagrangian
solutions. We will not go into detail on these, as they lack uniqueness with
respect to initial data. However, we mention that forcing the evolution of $X$
to be sticky, despite $F$ not satisfying the sticking condition (3.19), which
would correspond to replacing $F[X]$ with
$\operatorname{P}_{\mathscr{H}_{X}}F[X]$ in (3.15) (which would still satisfy
(3.16)), yields such a generalized Lagrangian solution. We will compare the
Lagrangian solution to this type of generalized solution in Example 6.1 and in
Section 7.1.

###  3.4. Lagrangian solutions for the Euler system

From the final bullet point of Lemma 3.7 we have that a Lagrangian solution of
the differential inclusion (3.15) provides us with a solution of the original
Euler system (1.1). In this section we will see how suitable assumptions on
the force distribution $f[\rho]$ allow us to apply Theorem 3.8 for the
differential inclusion involving its Lagrangian representation $F[X]$, and
thereby infer that this solution also is a stable selection with respect to
initial data in an appropriate metric space, that is

(3.24)
$\mathscr{T}_{2}\coloneqq\left\\{(\rho,v)\colon\rho\in\mathscr{P}_{2}(\mathbb{R}),v\in
L^{2}(\mathbb{R},\rho)\right\\}$

endowed with the metric

(3.25) $d_{\mathscr{T}_{2}}((\rho_{1},v_{1}),(\rho_{2},v_{2}))^{2}\coloneqq
d_{\mathscr{W}_{2}}(\rho_{1},\rho_{2})^{2}+d_{\mathscr{V}_{2}}((\rho_{1},v_{1}),(\rho_{2},v_{2}))^{2},$

involving the semi-distance

$d_{\mathscr{V}_{2}}((\rho_{1},v_{1}),(\rho_{2},v_{2}))^{2}\coloneqq\int_{\mathbb{R}\times\mathbb{R}}\lvert
v_{1}(x)-v_{2}(y)\rvert^{2}\mathop{}\\!\mathrm{d}\varrho(x,y)=\int_{\Omega}\lvert
v_{1}(X_{\rho_{1}}(m))-v_{2}(X_{\rho_{2}}(m))\rvert^{2}\mathop{}\\!\mathrm{d}m,$

where, recalling (3.3), $\varrho$ is the unique optimal transport map between
the measures $\rho_{1}$ and $\rho_{2}$. The monotone rearrangements of the
measures are $X_{\rho_{1}}$ and $X_{\rho_{2}}$, and defining
$V_{\rho_{i}}=v_{i}\circ X_{i}$ for $i=1,2$ we observe by (3.3) that we in
fact have the relation

(3.26) $d_{\mathscr{T}_{2}}((\rho_{1},v_{1}),(\rho_{2},v_{2}))^{2}=\lVert
X_{\rho_{1}}-X_{\rho_{2}}\rVert_{L^{2}(\Omega)}^{2}+\lVert
V_{\rho_{1}}-V_{\rho_{2}}\rVert_{L^{2}(\Omega)}^{2}.$

Next, in a similar vein to the Lagrangian setting, we define the notion of
boundedness and continuity in this metric space.

######  Definition 3.13 (Boundedness).

A map $f\colon\mathscr{P}_{2}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$ is
bounded if there is a constant $C\geq 0$ such that

$\lVert f_{\rho}\rVert^{2}_{L^{2}(\mathbb{R},\rho)}\leq
C\left(1+\int_{\mathbb{R}}\lvert
x\rvert^{2}\mathop{}\\!\mathrm{d}\rho\right)\text{ for all
}\rho\in\mathscr{P}_{2}(\mathbb{R}).$

The map is called pointwise linearly bounded if there is $C_{\mathrm{p}}\geq
0$ such that

(3.27) $\lvert f_{\rho}(x)\rvert\leq C_{\mathrm{p}}\left(1+\lvert
x\rvert+\int_{\mathbb{R}}\lvert
y\rvert\mathop{}\\!\mathrm{d}\rho(y)\right)\text{ for a.e.\ $x\in\mathbb{R}$
and all $\rho\in\mathscr{P}_{2}(\mathbb{R})$. }$

######  Definition 3.14 (Uniform continuity I).

A map $f\colon\mathscr{P}_{2}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$ as in
(1.2) is uniformly continuous if there exists a modulus of continuity $\omega$
such that

(3.28)
$d_{\mathscr{T}_{2}}((\rho_{1},f_{\rho_{1}}),(\rho_{2},f_{\rho_{2}}))\leq
d_{\mathscr{W}_{2}}(\rho_{1},\rho_{2})\text{ for all
}\rho_{1},\rho_{2}\in\mathscr{P}_{2}(\mathbb{R}).$

As before, if $\omega(r)=Lr$ for some $L\geq 0$ and all $r\geq 0$, then $f$ is
Lipschitz continuous.

Recall from Section 3.2 the correspondence between measures
$\rho\in\mathscr{P}_{2}(\mathbb{R})$ and optimal transport maps $X\in
L^{2}(\Omega)$, namely $X\in\mathscr{K}$ and $X_{\\#}\mathfrak{m}=\rho$. For
such correspondences we want a map $F\colon\mathscr{K}\to L^{2}(\Omega)$
satisfying (3.16), and a possible choice is simply $F[X]=f_{\rho}\circ X$ for
$f_{\rho}$ given by (1.2). Then the boundedness and continuity properties for
the force distribution $f$ in Definitions 3.13 and 3.14 translates into the
corresponding properties for $F$ in Definitions 3.3 and 3.4. However,
different choices for $F$ can be convenient, and for this one can make use of
another definition.

######  Definition 3.15 (Uniform continuity II).

The map $f\colon\mathscr{P}_{2}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$ is
densely uniformly continuous if (3.28) holds for measures that are uniformly
continuous with respect to the Lebesgue measure with bounded densities. Dense
Lipschitz continuity is defined analogously.

By [7, Lemma 6.4], if
$f\colon\mathscr{P}_{2}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$ is densely
uniformly continuous, there exists a unique uniformly continuous map
$F\colon\mathscr{K}\to L^{2}(\Omega)$ such that (3.16) holds for all
$(X,\rho)$ satisfying $X\in\mathscr{K}$ and $X_{\\#}\mathfrak{m}=\rho$.
Furthermore, we can then define a densely uniformly continuous force
distribution $f$ to be sticking if its corresponding map $F$ described above
satisfies Definition 3.5.

Having established these definitions, [7, Theorem 6.6] tells us that if
$f\colon\mathscr{P}_{2}(\mathbb{R})\to\mathscr{M}(\mathbb{R})$ is densely
Lipschitz continuous, there exists a stable selection of a solution $(\rho,v)$
of (1.1) with respect to initial data $(\rho_{0},v_{0})$ in
$(\mathscr{T}_{2},d_{\mathscr{T}_{2}})$.

At the end of Section 6 in [7] the authors give several examples of force
functionals covered by their theory. Below we see how two of these connect our
examples in Section 7.

####  3.4.1. Euler–Poisson system

Theorem 3.8 can be applied to (1.3) where according to [7, Example 6.9],
compare also with [4, Proposition 2.10], we have
$F[X](m)=-\alpha(m-\frac{1}{2})-\beta$. That is, the force operator $F[X]$
does not even depend on $X$, meaning it is Lipschitz continuous in the sense
of Definition 3.4. Furthermore, this means we can compute the prescribed
velocity $U$ in (3.20) for all times directly:

(3.29) $U(t,m)=V^{0}(m)+tA(m),\qquad
A(m)\coloneqq-\alpha\left(m-\frac{1}{2}\right)-\beta.$

In this case, the force operator $f[\rho]$ turns out to be pointwise linearly
bounded and densely uniformly continuous. Moreover, for $\alpha\geq 0$ it is
also sticking, meaning the unique Lagrangian solution is given by the
projection formula of Proposition 3.10. We study the non-sticking case in
Section 7.1.

####  3.4.2. Euler–Poisson system with quadratic confinement

A more general force distribution compared to the above is given in [7,
Example 6.10]. Given $\sigma\in L^{\infty}(\mathbb{R})$ they define
$f[\rho]=-\rho\partial_{x}q_{\sigma}$ for all
$\rho\in\mathscr{P}_{2}(\mathbb{R})$, where $q_{\sigma}$ satisfies
$-\partial_{xx}q_{\sigma}=\lambda(\rho-\sigma)$. Of course, this only defines
$\partial_{x}q_{\sigma}$ up to a constant, and making the same choice for the
$\rho$-part as the previous example and defining
$S_{\sigma}(x)=\int_{0}^{x}\sigma(y)\mathop{}\\!\mathrm{d}y$ one arrives at
the following Lagrangian representation

$F[X](m)=-\lambda\left(m-\frac{1}{2}-S_{\sigma}(X(m))\right)\quad\text{for all
}m\in\Omega.$

Then $F\colon\mathscr{K}\to L^{2}(\Omega)$ is pointwise linearly bounded and
Lipschitz-continuous. Choosing $\sigma\equiv 1$ such that $S_{\sigma}(x)=x$
and $\lambda>0$ this corresponds to a repulsive Poisson force combined with a
confining force driving mass towards the origin. In Section 7.2 we will
consider a variant of this studied in [14] where the confining force comes
from a quadratic confinement potential which drives mass toward the center of
mass $\bar{x}(t)$. In the above situation this behavior can be achieved by
choosing $S_{\sigma}(t,x)=x-\bar{x}(t)$.

##  4\. Entropy solutions

In this section, following the terminology used for the first order
interaction equation (1.4) in [4], we will introduce the concept of entropy
solutions for the Euler system (1.1). This was already done for the specific
case of the Euler–Poisson system (1.3) by Nguyen and Tudorascu [36, 37], which
in turn is based on the work of Brenier and Grenier [8] on the Euler system
without forcing. The main idea is to approximate the density $\rho$ with an
empirical measure, and noting that its piecewise constant distribution
function satisfies a conservation law with a piecewise linear flux function,
much in the spirit of front tracking [18]. Then, applying the calculus for
functions of bounded variation ($BV$) developed by Vol’pert [48], it turns out
that the corresponding solution of the conservation law gives rise to
solutions of the Euler system, and a solution of the original problem is
obtained from a limiting argument. This idea can be regarded as using the flow
along characteristics to combine (1.1a) and (1.1b) into a single scalar
conservation law for an appropriate flux function, analogous to what was done
in [33], and this is how we motivate the flux function below. Under our
assumptions, the flux function will in general be time-dependent and not
Lipschitz-continuous in the solution variable. Therefore, as in [37], we will
rely on results developed by Golovaty and Nguyen [24], rather than the
standard existence and uniqueness theory for conservation laws which typically
requires the flux function to be Lipschitz.

###  4.1. Derivation of the scalar conservation law

Here we will derive a scalar conservation law with the aim of finding
solutions of (1.1). To this end we introduce the the right-continuous
distribution functions

(4.1) $M(t,x)=\int_{(-\infty,x]}\mathop{}\\!\mathrm{d}\rho(t,y),\qquad
Q(t,x)=\int_{(-\infty,x]}v(t,y)\mathop{}\\!\mathrm{d}\rho(t,y),$

where we write $M(0,x)=M^{0}(x)$ and $Q(0,x)=Q^{0}(x)$. We know from our
assumptions on $\rho_{0}$ and $v_{0}$ that these functions are initially well-
defined functions of bounded variation. Then, integrating (1.1a) and (1.1b)
together with (1.2) yields two advection-type equations

(4.2a) $\displaystyle\partial_{t}M+v\partial_{x}M$ $\displaystyle=0,$ (4.2b)
$\displaystyle\partial_{t}Q+v\partial_{x}Q$
$\displaystyle=\int_{(-\infty,x]}f_{\rho}\mathop{}\\!\mathrm{d}\rho(t,y).$

Now the goal is to rewrite (4.2a) as a conservation law for $M$ by using the
relation $\partial_{x}Q=v\partial_{x}M$ coming from (4.1), and expressing $Q$
as a function of $M$ using the evolution equation (4.2b). From (4.2a) we see
that $M$ is constant along characteristics traveling with velocity $v$, which
we want to take advantage of in the second equation. One way of deriving the
conservation laws corresponding to the Euler ($f[\rho]\equiv 0$) and the
Euler–Poisson system in [8, 36] is then to introduce characteristics
$\eta(t,x)$ satisfying

$\partial_{t}\eta(t,x)=v(t,\eta(t,x)),\qquad\eta(0,x)=x$

as in [33], cf. Example 4.3 below. However, in order to highlight the
connection to the Lagrangian solutions of Section 3, we will instead use the
optimal transport map $X$ defined through the push-forward relation
$\rho=X_{\\#}\mathfrak{m}$. We recall that $X(t,m)$, with $X(0,m)=X^{0}(m)$,
is the right-continuous generalized inverse $M^{-1}(t,m)$ of $M(t,x)$. Then,
if $\rho(t,\cdot)$ contains no atoms, we have $M(t,X(t,m))=m$, which together
with (4.2a) implies that $\dot{X}(t,m)=v(t,X(t,m))$. Assuming smooth, i.e.,
atomless, solutions, we can recast (4.2b) as

$\partial_{t}Q(t,X(t,m))+v(t,X(t,m))\partial_{x}Q(t,X(t,m))=\int_{\mathbb{R}}\chi_{(-\infty,0]}(y-X(t,m))f_{\rho}\mathop{}\\!\mathrm{d}\rho(t,y).$

Using the push-forward relation, this is equivalent to

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}Q(t,X(t,m))=\int_{0}^{m}F[X](t,\omega)\mathop{}\\!\mathrm{d}\omega,$

where $F$ corresponds to the Lagrangian representation (3.16) of the forcing
term from Section 3, and we recall that one possible choice for this is
$F[X]=f_{\rho}\circ X$. Integrating this equation in time and interchanging
the order of the integrals leads to

$Q(t,X(t,m))=Q(0,X(0,m))+\int_{0}^{m}\int_{0}^{t}F[X](s,\omega)\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}\omega.$

Plugging in for $m=M(t,x)$ we arrive at

(4.3) $\displaystyle Q(t,x)$
$\displaystyle=Q^{0}(X^{0}(M(t,x)))+\int_{0}^{M(t,x)}\int_{0}^{t}F[X](s,m)\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}m$
$\displaystyle=\int_{0}^{M(t,x)}\left(v_{0}(X(0,m))+\int_{0}^{t}F[X](s,m)\mathop{}\\!\mathrm{d}s\right)\mathop{}\\!\mathrm{d}m\eqqcolon\mathcal{U}(t,M(t,x)),$

and we have, rather formally, expressed $Q(t,x)$ as a function of $M(t,x)$.
Using (4.3) to define the flux function $\mathcal{U}(t,m)$, we express (4.2a)
as the scalar conservation law

(4.4) $\partial_{t}M+\partial_{x}\mathcal{U}(t,M)=0.$

Note that the flux function is exactly the primitive of the prescribed
velocity $U$ in (3.20) from Section 3. Differentiating $\mathcal{U}(t,M)$ with
respect to $x$ we get a Radon measure which is absolutely continuous with
respect to $\rho(t,x)=\partial_{x}M(t,x)$, and the Radon–Nikodym derivative of
$\partial_{x}Q$ with respect to $\partial_{x}M$ corresponds to the velocity of
the characteristics for the conservation law (4.4), which shows a formal
connection between the two solution concepts. If $\rho$ contains atoms, then
$M$ contains jumps, and for any sufficiently regular flux function we can
apply a $BV$-chain rule as developed in [48]. Note that in general,
compositions of $BV$ functions are not necessarily of bounded variation, cf.
[29], and in order to have a chain rule for general $BV$ functions we would
need the outer function to be at least Lipschitz. However, in our one-
dimensional setting where $M$ is the distribution function of a probability
measure, we can require less regularity. The following result, found in [36,
Lemma 2.1], will be of use.

######  Lemma 4.1.

Let $\mu$ be a Borel probability measure on $\mathbb{R}$ and let $M$ be its
right-continuous distribution function. Write

$\mu=\sum_{j\in\mathcal{J}}m_{j}\delta_{x_{j}}+\rho,$

where $\\{x_{j}\\}_{j\in\mathcal{J}}$ is the set of (at most countable)
discontinuities of $M$, with $m_{j}\coloneqq\mu({x_{j}})$. If $\rho$ is
nonzero, then we have

(4.5) $M_{\\#}\rho=\chi_{J^{c}}\enspace\text{for}\enspace
J\coloneqq\bigcup_{j\in\mathcal{J}}\left(M(x_{j}-),M(x_{j})\right).$

Now we can state the generalized $BV$-chain rule, comparable to the results
[36, Theorem 2.2, Corollary 2.3] which were proved with approximation
arguments.

######  Lemma 4.2.

Consider $p\in[1,\infty]$ and a function $f\in L^{p}(0,1)$, for which we
define $F(m)\coloneqq\int_{0}^{m}f(\omega)\mathop{}\\!\mathrm{d}\omega$. If
$M$ is the right-continuous cumulative distribution function of some Borel
probability measure $\mu$ on $\mathbb{R}$, then $F\circ M\in BV(\mathbb{R})$
with distributional derivative $g\mu$, where

(4.6) $g(x)\coloneqq\begin{cases}f\circ M(x),&\mu({x})=0,\\\ \frac{F\circ
M(x)-F\circ M(x-)}{\mu({x})},&\mu({x})\neq 0\end{cases}$

in the $\mu$-a.e. sense. Furthermore $g\in L^{p}(\mathbb{R},\mu)$ with $\lVert
g\rVert_{L^{p}(\mathbb{R},\mu)}\leq\lVert f\rVert_{L^{p}(0,1)}$, where the
inequality is replaced by equality if $\mu$ is diffuse, i.e., there are no
$\mu$-atoms.

Note that the function $g$ in (4.6), which in fact is the Radon–Nikodym
derivative of the (signed) Radon measure $\partial_{x}F(M)$ with respect to
the Radon measure $\partial_{x}M$, can be equivalently written as

(4.7) $g(x)=\int_{0}^{1}f((1-s)M(x-)+sM(x))\mathop{}\\!\mathrm{d}s,$

a representation due to Vol’pert [48] for compositions of general Lipschitz
functions and $BV$ functions.

###### Proof of Lemma 4.2.

Note that $f\colon\mathbb{R}\to\mathbb{R}$ is a function of bounded variation
if and only if we for any partition $\\{x_{i}\\}$ of $\mathbb{R}$ have

$\operatorname*{T.\\!V.}(f)\coloneqq\sup_{x_{i}}\sum_{i}\lvert
f(x_{i})-f(x_{i-1})\rvert<\infty.$

Since $M$ is a monotone increasing function, any partition $\\{x_{i}\\}_{i}$
of $\mathbb{R}$ defines a partition $\\{M(x_{i})\\}_{i}$ of $[0,1]$, and the
set of such partitions of $[0,1]$ generated by partitions of $\mathbb{R}$ is
strictly smaller than the set of all possible partitions of $[0,1]$ since
there is no way to pick $x_{i}$ such that $M(x_{i})\in J$, the ‘jump set’ of
$M$. Indeed, the range of $F\circ M$ is strictly included in the range of $F$.
Hence

$\sup_{x_{i}}\sum_{i}\lvert
F(M(x_{i}))-F(M(x_{i-1}))\rvert\leq\sup_{m_{i}}\sum_{i}\lvert
F(m_{i})-F(m_{i-1})\rvert,$

from which we observe it to be sufficient if $F\in BV([0,1])$. Now, since
$F\in W^{1,p}(0,1)$, it is absolutely continuous and consequently of bounded
variation, and so $F\circ M\in BV(\mathbb{R})$.

A (signed) Radon measure on $\mathbb{R}$ is uniquely determined by its
distribution function, and we note that $M$ can be expressed as

$M(x)=\int_{(-\infty,x]}\mathop{}\\!\mathrm{d}\rho+\sum_{j\in\mathcal{J}}m_{j}\chi_{(-\infty,x]}(x_{j}).$

To verify (4.6) we shall similarly rewrite $F\circ M$ in a form from which it
is clear that it is the distribution function of the signed measure
$g\mu\ll\mu$ with $g$ as in (4.6). Indeed, by Lemma 4.1 we have

$\displaystyle\int_{0}^{M(x)}f(m)\mathop{}\\!\mathrm{d}m$
$\displaystyle=\int_{0}^{1}\chi_{(0,M(x)]}\chi_{J^{c}}f(m)\mathop{}\\!\mathrm{d}m+\int_{0}^{1}\chi_{(0,M(x)]}\chi_{J}f(m)\mathop{}\\!\mathrm{d}m$
$\displaystyle=\int_{\mathbb{R}}\chi_{(0,M(x)]}(M(y))f(M(y))\mathop{}\\!\mathrm{d}\rho(y)+\sum_{j\in\mathcal{J}}\chi_{(-\infty,x]}(x_{j})\int_{M(x_{j}-)}^{M(x_{j})}f(m)\mathop{}\\!\mathrm{d}m$
$\displaystyle=\int_{(-\infty,x]}f\circ
M\mathop{}\\!\mathrm{d}\rho+\sum_{j\in\mathcal{J}}\frac{F(M(x_{j}))-F(M(x_{j}-))}{m_{j}}m_{j}\chi_{(-\infty,x]}(x_{j}),$

from which the result follows. The last statement for $p\in[1,\infty)$ can be
proved using Lemma 4.1 and Jensen’s inequality, namely

$\displaystyle\lVert g\rVert_{L^{p}(\mathbb{R},\mu)}^{p}$
$\displaystyle=\int_{\mathbb{R}}\lvert
g(x)\rvert^{p}\mathop{}\\!\mathrm{d}\mu=\int_{\mathbb{R}}\lvert f\circ
M(x)\rvert^{p}\mathop{}\\!\mathrm{d}\rho+\sum_{j\in\mathcal{J}}m_{j}\left|\frac{1}{m_{j}}\int_{M(x_{j}-)}^{M(x_{j})}f(m)\mathop{}\\!\mathrm{d}m\right|^{p}$
$\displaystyle\leq\int_{J^{c}}\lvert
f(m)\rvert^{p}\mathop{}\\!\mathrm{d}m+\int_{J}\lvert
f(m)\rvert^{p}\mathop{}\\!\mathrm{d}m=\lVert f\rVert_{L^{p}(0,1)}^{p}.$

The case of $p=\infty$, i.e., $F$ Lipschitz, follows from a similar
decomposition. ∎

We see from (4.3) that the integral term in the velocity $U$ at time $t$
involves the generalized inverse $X(s,m)$ for $s\in[0,t]$. As it turns out, in
some cases one can use a priori knowledge about the solution to express
$\mathcal{U}(t,M)$ in a more explicit form. Below we give some examples of
such cases, and the first of these corresponds to the Euler–Poisson system
(1.3).

######  Example 4.3 (Poisson force).

From the assumption $\rho_{0}\in\mathscr{P}(\mathbb{R})$ and the mass
conservation due to (1.3a) we have that a solution of (1.3) must satisfy
$\rho\in\mathscr{P}(\mathbb{R})$. This allows us to rewrite (1.3b) as

$\displaystyle(\rho v)_{t}+(\rho v^{2})_{x}$
$\displaystyle=-\frac{\alpha}{2}\left(\int_{(-\infty,x)}\rho(t,y)\mathop{}\\!\mathrm{d}y+\int_{(-\infty,x]}\rho(t,y)\mathop{}\\!\mathrm{d}y-1\right)\rho-\beta\rho$
$\displaystyle=-\frac{\alpha}{2}(M(t,x-)+M(t,x)-1)\rho-\beta\rho=-\partial_{x}\left(\frac{\alpha}{2}(M(t,x)^{2}-M(t,x))+\beta
M(t,x)\right),$

where in the last identity we have used (4.6). Integrating (1.3) we arrive at
the nonconservative, transport-like system

(4.8)
$\partial_{t}M+v\partial_{x}M=0,\qquad\partial_{t}Q+v\partial_{x}Q=\frac{\alpha}{2}\left(M^{2}-M\right)+\beta
M.$

Since $\partial_{x}Q=v\partial_{x}M$ and $M$ is constant along
characteristics, we see that if we can find a flux function
$\mathcal{V}^{0}\colon[0,1]\to\mathbb{R}$ satisfying the compatibility
condition $\mathcal{V}^{0}(M^{0})=Q^{0}$, then the transport equations above
can be combined into a single conservation law, as shown in [36]. The
appropriate function $\mathcal{V}^{0}$, depending only on the initial data of
(1.3), is given by

$\mathcal{V}^{0}(m)=\int_{0}^{m}v_{0}((M^{0})^{-1}(\omega))\mathop{}\\!\mathrm{d}\omega,$

where $(M^{0})^{-1}(m)$ is the right-continuous generalized inverse of
$M^{0}(x)$. Note that the integrand is exactly $V^{0}(m)\coloneqq v_{0}\circ
X^{0}$, that is, $\mathcal{V}^{0}$ is the primitive of $V^{0}$. In the end we
obtain the conservation law (4.4) with flux function

(4.9)
$\mathcal{U}(t,m)=\mathcal{V}^{0}(m)+t\mathcal{A}(m),\qquad\mathcal{A}(m)=-\frac{\alpha}{2}(m^{2}-m)-\beta
m.$

Observe that the flux function $\mathcal{U}(t,m)$ is the primitive of the
prescribed velocity $U(t,m)$ from (3.29).

######  Example 4.4 (Poisson force with linear damping).

We can consider a slightly more general system than (1.3) by including a
linear damping term for the momentum, that is, we look at

(4.10) $\displaystyle\partial_{t}\rho+\partial_{x}(\rho v)$ $\displaystyle=0,$
$\displaystyle\partial_{t}(\rho v)+\partial_{x}(\rho v^{2})$
$\displaystyle=-\left(\tfrac{\alpha}{2}(\operatorname*{sgn}\ast\rho)+\beta\right)\rho-\gamma\rho
v.$

Here $\gamma\in\mathbb{R}$ should satisfy $\gamma>0$ to be considered a
damping term. As before, the idea is to integrate (4.10) and consider the
cumulative distributions (4.1) of $\rho$ and $\rho v$. Analogous to (4.8) we
then arrive at the transport-like equations

$\partial_{t}M+v\partial_{x}M=0,\qquad\partial_{t}Q+v\partial_{x}Q=-\left(\frac{\alpha}{2}(M^{2}-M)+\beta
M+\gamma Q\right).$

The second equation can be rewritten, using an integrating factor, as

$\partial_{t}(\mathrm{e}^{\gamma t}Q)+v\partial_{x}(\mathrm{e}^{\gamma
t}Q)=-\mathrm{e}^{\gamma t}\left(\frac{\alpha}{2}(M^{2}-M)+\beta M\right),$

which we can integrate along characteristics, where $M$ is constant. As
before, choosing $\mathcal{V}^{0}$ such that $Q(0,x)=\mathcal{V}^{0}(M(0,x))$
we find that $M$ satisfies the conservation law (4.4), this time with a time-
dependent flux function $\mathcal{U}(t,m)$ given by

(4.11) $\mathcal{U}(t,m)=\mathrm{e}^{-\gamma
t}\mathcal{V}^{0}(m)+\frac{1-\mathrm{e}^{-\gamma
t}}{\gamma}\mathcal{A}(m)=\mathcal{V}^{0}(m)-\frac{1-\mathrm{e}^{-\gamma
t}}{\gamma}(\gamma\mathcal{V}^{0}(m)-\mathcal{A}(m)),$

with $\mathcal{A}$ as defined in (4.9). Note that for $\gamma=0$ we have to
interpret the fraction above as a limit $\gamma\to 0$, for which we recover
(4.9). We observe that the damping reduces the time it takes for the forcing
term to dominate the flux function.

Note that (4.10) is not directly covered by (1.1), as the forcing term now
depends on the velocity in addition to the density. However, this can still be
seen as a Lipschitz perturbation of a gradient flow, as argued in [7], see
Section 7.2.

###  4.2. Entropy solutions for the scalar conservation law

We now consider existence and uniqueness of weak solutions for (4.4).
Introducing a convex, Lipschitz entropy function
$\eta\colon[0,1]\to\mathbb{R}$ with corresponding entropy-flux function
$q\colon\mathbb{R}\times[0,1]\to\mathbb{R}$ satisfying
$\partial_{m}q(t,m)=\eta^{\prime}(m)\partial_{m}\mathcal{U}(t,m)$, one
typically requires an entropy solution to satisfy

$\partial_{t}\eta(M)+\partial_{x}q(t,M)\leq 0$

in the distributional sense, cf. [19, Section 4.5.1]. That is, considering the
associated Cauchy problem for (4.4), for any $\varphi\in
C_{\mathrm{c}}^{\infty}([0,T)\times\mathbb{R})$ with $\varphi\geq 0$ we have

(4.12)
$\int_{0}^{T}\int_{\mathbb{R}}\left[\eta(M)\varphi_{t}+q(t,M)\varphi_{x}\right]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\int_{\mathbb{R}}\varphi(0,x)\eta(M^{0}(x))\mathop{}\\!\mathrm{d}x\geq
0.$

Since such convex functions can be seen as limits of piecewise affine
functions, this is equivalent to (4.12) being true for the Krǔzkov entropy-
entropy flux pair

(4.13) $\eta_{k}(m)=\lvert m-k\rvert,\qquad
q_{k}(t,m)=\operatorname*{sgn}(m-k)\left(\mathcal{U}(t,m)-\mathcal{U}(t,k)\right)=\int_{k}^{m}\operatorname*{sgn}(\omega-k)U(t,\omega)\mathop{}\\!\mathrm{d}\omega,$

for all $k\in\mathbb{R}$, see, e.g., [25]. In fact, it is sufficient to
consider $k$ in a dense set of $\mathbb{R}$, cf. [40, Proposition 2] or [24,
Section 3].

In Example 4.3 we saw that the flux function $\mathcal{U}(t,M)$ corresponding
to the Euler–Poisson system takes the form

$\mathcal{U}(t,m)=\int_{0}^{m}V^{0}(\omega)\mathop{}\\!\mathrm{d}\omega+t\int_{0}^{m}A(\omega)\mathop{}\\!\mathrm{d}\omega,$

where $A(m)=-\alpha(m-\frac{1}{2})-\beta$ and $V^{0}=v_{0}\circ X^{0}\in
L^{2}(0,1)$ due to $v_{0}\in L^{2}(\mathbb{R},\rho_{0})$. In particular, we
see that $m\mapsto\mathcal{U}(t,m)\in W^{1,2}(0,1)$, which means that the flux
function is in general not Lipschitz-continuous. This contrasts the case
studied in [8], where they assume $v_{0}\in L^{\infty}(\mathbb{R},\rho_{0})$.
Then their flux function $\mathcal{U}(t,M)\equiv\mathcal{V}^{0}(M)$ is
Lipschitz and one can employ the standard assumptions used in the classical
version of Krǔzkov’s doubling of variables-argument, cf. [9], to prove
uniqueness for (4.4). Another issue with the flux function (4.9) may be
degenerate in the sense that $m\mapsto\mathcal{U}(t,m)$ is affine on non-
degenerate intervals, and this can cause issues in existence proofs for (4.4).

The above issues for the Euler–Poisson flux (4.9) motivated the
generalizations of existence and uniqueness proofs found in [24], where they
consider scalar conservation laws on the line where the flux function is
merely continuous in $m$ and may in general depend on both time $t$ and
position $x$. These results were then used in [37] to relax the assumptions
used to study the Euler–Poisson system (1.3) in [36]. We follow [37,
Definition 3.1] in defining entropy solutions of (4.4).

######  Definition 4.5 (Entropy solution of (4.4)).

Let $M^{0}\in L^{1}_{\mathrm{loc}}(\mathbb{R})$. A function $M\in
L^{\infty}\left((0,T)\times\mathbb{R}\right)\cap
C\left([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R})\right)$ is an entropy solution
of (4.4) if

(4.14) $\int_{0}^{T}\int_{\mathbb{R}}\left(\lvert
M-k\rvert\varphi_{t}+\operatorname*{sgn}(M-k)(\mathcal{U}(t,M)-\mathcal{U}(t,k))\varphi_{x}\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\int_{\mathbb{R}}\lvert
M^{0}(x)-k\rvert\varphi(0,x)\mathop{}\\!\mathrm{d}x\geq 0$

for all $k$ in a dense set of $\mathbb{R}$ and nonnegative test functions
$\varphi\in C^{\infty}_{\mathrm{c}}([0,T)\times\mathbb{R})$.

As usual, replacing the entropy-entropy flux pair with the identity
$\mathrm{Id}$ and $\mathcal{U}(t,M)$ and removing the nonnegativity
requirement on $\varphi$ in (4.14), we recover the definition of a weak
solution, i.e.,

(4.15)
$\int_{0}^{T}\int_{\mathbb{R}}\left(M\varphi_{t}+\mathcal{U}(t,M)\varphi_{x}\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\int_{\mathbb{R}}M^{0}(x)\varphi(0,x)\mathop{}\\!\mathrm{d}x=0.$

####  4.2.1. The Rankine–Hugoniot- and Oleĭnik E-condition

For piecewise smooth solutions of (4.4), the requirements for weak and entropy
solutions can be reduced to conditions on the discontinuities, cf. [25,
Chapter 2]. To this end, assume $M(t,x)$ takes the values $M_{l}(t)$ and
$M_{r}(t)$ to the left and right of some shock curve $\Gamma=\\{(t,x)\colon
x=s(t)\\}$. Denoting by $[[M]]$ the jump in $M$ across the curve $\Gamma$, and
correspondingly $[[\eta(M)]]$, $[[q(t,M)]]$ for the other quantities, this
leads to the inequality

$\int_{\Gamma}\varphi(t,x)\left([[\eta(M)]]\dot{s}(t)-[[q(t,M)]]\right)\geq
0,$

which gives us $\dot{s}(t)[[\eta(M)]]\geq[[q(t,M)]]$. For $\eta(M)=M$ and
$q(t,M)=\mathcal{U}(t,M)$ the above is an equality and we get the
Rankine–Hugoniot condition

(4.16) $\dot{s}(t)=\frac{[[\mathcal{U}(t,M)]]}{[[M]]}.$

Using the Krǔzkov entropy-flux pair, the inequality (4.14) is equivalent to

(4.17)
$\frac{\mathcal{U}(t,M_{r})-\mathcal{U}(t,k)}{M_{r}-k}\leq\dot{s}(t)\leq\frac{\mathcal{U}(t,k)-\mathcal{U}(t,M_{l})}{k-M_{l}}$

for any $M_{l}<k<M_{r}$, which is known as the Oleĭnik E-condition, cf. [39,
19]. In fact, for the distributional entropy inequality to reduce to the
Oleĭnik E-condition, it is enough for the weak solution to be of class
$BV_{\mathrm{loc}}$, see [19, Section 4.5], which is exactly what we are
interested in.

####  4.2.2. Existence and uniqueness results

To establish existence and uniqueness of entropy solutions of (4.4) we have to
make some assumptions on its flux function. However, as it stands, the formal
expression for $\mathcal{U}(t,m)$ in (4.3) has an apparent intricate
dependence on the generalized inverse of the solution $M$ up until time $t$;
that is, for $F[X]=f_{\rho}\circ M^{-1}$ we have

$\mathcal{U}(t,m)=\int_{0}^{m}\left((v_{0}\circ
M_{0}^{-1})(\omega)+\int_{0}^{t}f_{\rho}(M^{-1}(s,\omega))\mathop{}\\!\mathrm{d}s\right)\mathop{}\\!\mathrm{d}\omega.$

On the other hand, in cases such as Examples 4.3 and 4.4 the part involving
the forcing term $f_{\rho}$ simplifies, and we we are led to consider flux
functions of the form

(4.18)
$\mathcal{U}(t,m)=\mathcal{V}^{0}(m)+\sum_{i=1}^{n}\int_{0}^{t}b(s)\mathop{}\\!\mathrm{d}s\int_{0}^{m}F_{i}(\omega)\mathop{}\\!\mathrm{d}\omega\eqqcolon\sum_{i=0}^{n}B_{i}(t)\mathcal{F}_{i}(m)$

for $\mathcal{V}^{0}=v_{0}\circ M^{-1}_{0}\in L^{2}(0,1)$, $F_{i}\in
L^{p}(0,1)$ and $b_{i}\in L^{1}(0,T)$. Indeed, since $\mathcal{U}(t,m)$ in
(4.3) can be thought of as the primitive of the prescribed velocity $U$
defined in (3.20), it is natural to expect $U$, and then also $\mathcal{U}$,
to be a function of $t$ for fixed $m\in\Omega$. The functions
$\mathcal{F}_{i}$ in (4.18) are continuous, while $B_{i}\in
L^{\infty}(0,T)\subset L^{2}(0,T)$, which means that our problem is covered by
the theory in [24]. In particular we have the following crucial
$L^{1}$-contraction principle [24, Theorem 2.2], the proof of which relies on
establishing a Kato-type inequality. Here, for $c\in\mathbb{R}$, its positive
part is written $c^{+}=\max\\{c,0\\}$.

######  Theorem 4.6 ($L^{1}$-contraction).

Assume $M$ and $\tilde{M}$ are entropy solutions of (4.4) with flux function
of the form (4.18) and initial data $M^{0},\tilde{M}^{0}\in
L^{\infty}(\mathbb{R})$. Then

(4.19)
$\int_{\mathbb{R}}(M(t,x)-\tilde{M}(t,x))^{+}\mathop{}\\!\mathrm{d}x\leq\int_{\mathbb{R}}(M^{0}(x)-\tilde{M}^{0}(x))^{+}\mathop{}\\!\mathrm{d}x.$

Several important results can be deduced from Theorem 4.6, cf. Corollaries 2.8
and 2.9 in [24]. For instance, since the constant functions 0 and 1 are
entropy solutions, we see that if $M$ is an entropy solution with initial data
$M^{0}(x)\in[0,1]$, then we must have $M(t,x)\in[0,1]$ for a.e. $x$. Moreover,
we would like to use it to show that the distributional derivatives of entropy
solutions are probability measures, and to this end we introduce some useful
function spaces as in [37, Definition 3.2].

######  Definition 4.7.

A function $M\colon\mathbb{R}\to\mathbb{R}$ belongs to the set $\mathscr{R}$
if it is nondecreasing, right-continuous and has limits $0$ and $1$ at
respectively $-\infty$ and $+\infty$. Moreover, $M\in\mathscr{R}$ belongs to
$\mathscr{R}_{p}$ for $p\geq 1$ if its distributional derivative
$\partial_{x}M$ has finite $p$-moment, that is, $\int_{\mathbb{R}}\lvert
x\rvert^{p}\partial_{x}M$ is finite.

We note that the distributional derivative $\partial_{x}M$ of any function
$M\in\mathscr{R}$ is a Borel probability measure on $\mathbb{R}$ with
$M(x)=\partial_{x}M((-\infty,x])$. The map $M\mapsto\partial_{x}M$ is
injective from $\mathscr{R}$ ($\mathscr{R}_{p}$) to $\mathscr{P}(\mathbb{R})$
($\mathscr{P}_{p}(\mathbb{R})$). The space
$(\mathscr{P}_{p}(\mathbb{R}),d_{\mathscr{W}_{p}})$ is a complete metric space
equipped with the $p$-Wasserstein metric $d_{\mathscr{W}_{p}}$. In view of the
above discussion, $\mathscr{R}_{p}$ from Definition 4.7 is a metric space with
the induced metric

(4.20) $d_{\mathscr{R}_{p}}(M,\tilde{M})\coloneqq
d_{\mathscr{W}_{p}}(\partial_{x}M,\partial_{x}\tilde{M})=\left(\int_{0}^{1}\left\lvert
X(m)-\tilde{X}(m)\right\rvert^{p}\mathop{}\\!\mathrm{d}m\right)^{1/p},$

where $X$ and $\tilde{X}$ are the respective right-continuous generalized
inverses of $M$ and $\tilde{M}$. Moreover, as a consequence of Fubini’s
theorem we have

(4.21) $d_{\mathscr{W}_{1}}(\partial_{x}M,\partial_{x}\tilde{M})=\lVert
X-\tilde{X}\rVert_{L^{1}(0,1)}=\lVert M-\tilde{M}\rVert_{L^{1}(\mathbb{R})},$

cf. [41, Proposition 2.17], [47, Remarks 2.19]. The following important
corollary of Theorem 4.6 guarantees that entropy solutions of (4.4) have a
representative which remains in the set $\mathscr{R}$.

######  Corollary 4.8.

Let $M$ be a entropy solution of (4.4) with flux function (4.18) and initial
data $M^{0}\in\mathscr{R}$. Then there exists a unique solution $\tilde{M}\in
C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}))$ such that
$M(t,\cdot)\in\mathscr{R}$ and $\tilde{M}(t,\cdot)=M(t,\cdot)$ as functions in
$L^{1}_{\mathrm{loc}}(\mathbb{R})$ for $t\in[0,T]$.

The proof of this result is identical to that of [37, Proposition 3.16]. As a
consequence of Corollary 4.8, we will from now on, when there is no confusion,
always identify the entropy solution with its unique representative in
$\mathscr{R}$. Furthermore, for an entropy solution $M\in\mathscr{R}$ of (4.4)
with initial data $M^{0}\in\mathscr{R}$, we will, when there is no confusion,
identify the respective distributional derivatives with
$\rho,\rho_{0}\in\mathscr{P}(\mathbb{R})$ for ease of notation.

So far we have been considering uniqueness, but we would of course also like
to have existence of entropy solutions. This is established using vanishing
viscosity methods in [24, Theorem 3.7], from which the next result follows
directly.

######  Theorem 4.9 (Existence and uniqueness).

Assume a flux function $\mathcal{U}(t,m)$ of the form (4.18). For any
$M^{0}\in L^{\infty}(\mathbb{R})$, (4.4) has a unique entropy solution $M\in
C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}))$ satisfying

$\lVert M\rVert_{L^{\infty}((0,T)\times\mathbb{R})}\leq\lVert
M^{0}\rVert_{L^{\infty}(\mathbb{R})}.$

Moreover, if $M^{0}\in L^{1}(\mathbb{R})$, then $M\in
C([0,T];L^{1}(\mathbb{R}))$.

At this stage, we have verified that entropy solutions for (4.4) in the sense
of Definition 4.5 exist, are unique, and belong to the set $\mathscr{R}$ such
that their distributional derivatives belong to $\mathscr{P}(\mathbb{R})$. The
next result, apart from being a useful property of the entropy solutions, will
help us establish that entropy solutions with initial data in
$\mathscr{R}_{p}$ remain in this space.

######  Lemma 4.10 (Lipschitz-continuity for $L^{1}$-norm).

Let $M(t)\in\mathscr{L}^{\infty}((0,T)\times\mathbb{R})\cap
C([0,T];\mathscr{L}^{1}_{\mathrm{loc}}(\mathbb{R}))$ be the entropy solution
of (4.4) with flux function (4.18) and $M(t,\cdot)\in\mathscr{R}$. Then, for
$0\leq s<t\leq T$ we have

(4.22) $\lVert M(t)-M(s)\rVert_{L^{1}(\mathbb{R})}\leq\lvert
t-s\rvert\sum_{i=0}^{n}\lVert F_{i}\rVert_{L^{p}(0,1)}\lVert
B_{i}\rVert_{L^{\infty}(s,t)}.$

Moreover, $M\in BV_{\mathrm{loc}}([0,T]\times\mathbb{R})$.

###### Proof.

The argument follows the lines of the proof of [25, Theorem 7.10] or [19,
Theorem 4.3.1], with some adjustments to cater for our non-Lipschitzian flux
function. Let $\chi^{\varepsilon}(t)$ for $t\in[0,T]$ be a smooth
approximation of the characteristic function for the interval
$[s,t]\subset[0,T]$ such that $\lim\limits_{\varepsilon\to
0+}\chi^{\varepsilon}=\chi_{[s,t]}$. Furthermore we choose
$\varphi_{\varepsilon}(t,x)=\chi^{\varepsilon}(t)\phi(x)$ for an arbitrary
$\phi\in C_{\mathrm{c}}^{\infty}(\mathbb{R})$. For this test function, the
weak formulation of (4.4) reads

$\int_{0}^{T}\int_{\mathbb{R}}\left(M\partial_{t}\varphi_{\varepsilon}+\mathcal{U}(t,M)\partial_{x}\varphi_{\varepsilon}\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\int_{\mathbb{R}}M^{0}(x)\varphi_{\varepsilon}(0,x)\mathop{}\\!\mathrm{d}x=0,$

and letting $\varepsilon\to 0+$ we obtain

$-\int_{\mathbb{R}}\phi(x)(M(t,x)-M(s,x))\mathop{}\\!\mathrm{d}x+\int_{s}^{t}\int_{\mathbb{R}}\phi^{\prime}(x)\mathcal{U}(r,M(r,x))\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}r=0.$

Since $\phi$ was arbitrary we have

$\displaystyle\lVert M(t,\cdot)-M(s,\cdot)\rVert_{L^{1}(\mathbb{R})}$
$\displaystyle=\sup_{\lvert\phi\rvert\leq
1}\int_{\mathbb{R}}\phi(x)(M(t,x)-M(s,x))\mathop{}\\!\mathrm{d}x$
$\displaystyle=\int_{s}^{t}\sup_{\lvert\phi\rvert\leq
1}\int_{\mathbb{R}}\phi^{\prime}(x)\mathcal{U}(r,M(r,x))\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}r$
$\displaystyle=\int_{s}^{t}\operatorname*{T.\\!V.}(\mathcal{U}(r,M(r,x)))\mathop{}\\!\mathrm{d}r.$

If $m\mapsto\mathcal{U}(t,m)$ were Lipschitz, i.e., $p=\infty$ in (4.18), we
could have finished the argument by using $\operatorname*{T.\\!V.}(M(t,x))=1$.
We will instead use Lemma 4.2 to estimate
$\operatorname*{T.\\!V.}(\mathcal{U}(t,M))$ as follows,

$\displaystyle\left\lvert\int_{\mathbb{R}}\phi^{\prime}(x)\mathcal{U}(t,M(t,x))\mathop{}\\!\mathrm{d}x\right\rvert$
$\displaystyle=\left\lvert-\int_{\mathbb{R}}\phi(x)\partial_{x}\mathcal{U}(t,M(t,x))\right\rvert=\left\lvert-\sum_{i=0}^{n}B_{i}(t)\int_{\mathbb{R}}\phi(x)\mathcal{F}_{i,M}^{\prime}(t,x)\mathop{}\\!\mathrm{d}\rho(t,x)\right\rvert$
$\displaystyle\leq\sum_{i=0}^{n}\lvert
B_{i}(t)\rvert\left(\int_{\mathbb{R}}\lvert\phi(x)\rvert^{p^{\prime}}\mathop{}\\!\mathrm{d}\rho(t,x)\right)^{\frac{1}{p^{\prime}}}\lVert\mathcal{F}_{i,M}^{\prime}\rVert_{L^{p}(\mathbb{R},\rho)}\leq\sum_{i=0}^{n}\lvert
B_{i}(t)\rvert\lVert F_{i}\rVert_{L^{p}(0,1)}.$

Taking supremum over $\lvert\phi\rvert\leq 1$ and combining with the previous
estimate we obtain (4.22). Note that we could also have arrived at the same
bound by estimating the total variation of $\mathcal{U}(t,M(t,\cdot))$ as in
the proof of Lemma 4.2. Since $\mathscr{R}\subset BV(\mathbb{R})$, then as in
the proof of [19, Theorem 4.3.1] it follows from the characterization of
$BV_{\mathrm{loc}}$-functions in [19, Theorem 1.7.2] and (4.22) that $M\in
BV_{\mathrm{loc}}([0,T]\times\mathbb{R})$. ∎

Moreover, the estimate on the total variation of the flux function
$\mathcal{U}(t,M)$ from (4.18) suggests the following generalization of Lemma
4.2.

######  Corollary 4.11.

Assume $p\in[1,\infty]$, take $\mathcal{U}(t,m)$ of the form (4.18), and let
$M$ be the right-continuous distribution function of a measure
$\mu\in\mathscr{P}(\mathbb{R})$. Then $\mathcal{U}(t,M)\in BV(\mathbb{R})$ and
$\partial_{x}\mathcal{U}(t,M)=g(t,\cdot)\mu$ for some $g(t,\cdot)\in
L^{p}(\mathbb{R},\mu)$ where

$g(t,x)=\begin{cases}\sum_{i=0}^{n}B_{i}(t)F_{i}\circ M(x),&\mu(\\{x\\})=0,\\\
\sum_{i=0}^{n}B_{i}(t)\frac{\mathcal{F}_{i}\circ M(x)-\mathcal{F}_{i}\circ
M(x-)}{\mu(\\{x\\})},&\mu(\\{x\\})\neq 0,\end{cases}$

and

$\lVert g(t,\cdot)\rVert_{L^{p}(\mathbb{R},\mu)}\leq\sum_{i=0}^{n}\lvert
B_{i}(t)\rvert\lVert F_{i}\rVert_{L^{p}(0,1)}.$

This also slightly generalizes the corresponding result for the Euler–Poisson-
type flux $\mathcal{U}(t,\cdot)=\mathcal{F}+t\Psi$ in [37, Proposition 2.1],
where $\mathcal{F}\in W^{1,p}(0,1)$ and $\Psi\in C^{1}([0,1])$. The proof of
Corollary 4.11 is a straightforward adaption of the proof of Lemma 4.2; that
is, the structure of the flux function allows us to apply the same arguments
termwise.

Let us from now on denote by $\mathcal{U}_{M}^{\prime}(t,x)$ the Radon–Nikodym
derivative of $\partial_{x}\mathcal{U}(t,M)$ with respect to $\partial_{x}M$,
and according to Corollary 4.11 this takes the form

$\mathcal{U}_{M}^{\prime}(t,x)=\sum_{i=0}^{n}B_{i}(t)\mathcal{F}_{i,M}^{\prime}(t,x),$

where $\mathcal{F}_{i,M}^{\prime}\in L^{p}(\mathbb{R},\partial_{x}M(t,\cdot))$
satisfies
$\partial_{x}\mathcal{F}_{i}(M(t,\cdot))=\mathcal{F}_{i,M}^{\prime}(t,x)\partial_{x}M(t,\cdot)$
by Lemma 4.2. From the properties of entropy solutions and Corollary 4.11 it
follows that

$\int_{0}^{T}\lVert\mathcal{U}^{\prime}_{M}(t,\cdot)\rVert_{L^{p}(\mathbb{R},\rho(t))}\mathop{}\\!\mathrm{d}t\leq
T\sum_{i=0}^{n}\lVert B_{i}\rVert_{L^{\infty}(0,T)}\lVert
F_{i}\rVert_{L^{p}(0,1)}<\infty.$

######  Lemma 4.12.

Let $M(t,\cdot)$ for $t\in[0,T]$ be the unique entropy solution of (4.4) with
flux function (4.18) and initial data $M^{0}\in\mathscr{R}_{p}$ for
$p\in[1,\infty)$. Then $M(t,\cdot)\in\mathscr{R}_{p}$.

###### Proof.

Let us denote by $m_{p}(\rho(t))$ the $p$th moment of
$\rho(t)\in\mathscr{P}_{p}(\mathbb{R})$. The case $p=1$ follows almost
immediately from the triangle inequality and (4.22). Indeed, we have

$m_{1}(\rho(t))=d_{\mathscr{W}_{1}}(\rho(t),\delta_{0})\leq
d_{\mathscr{W}_{1}}(\rho(0),\delta_{0})+d_{\mathscr{W}_{1}}(\rho(t),\rho(0))=m_{1}(\rho(0))+\lVert
M(t)-M(0)\rVert_{L^{1}(\mathbb{R})},$

and the result follows.

For $p>1$ we have to work a bit more, making use of that the probability
measure $\rho$ solves the continuity equation (1.1a). First, pick $\phi\in
C^{\infty}_{\mathrm{c}}(\mathbb{R})$ and introduce the test function
$\varphi(t,x)=\chi^{\varepsilon}(t)\phi^{\prime}(x)$ for (4.15), where
$\chi^{\varepsilon}$ is as in the proof of Lemma 4.10. Letting $\varepsilon\to
0+$ we obtain

$-\int_{\mathbb{R}}\phi^{\prime}(x)(M(t,x)-M(s,x))\mathop{}\\!\mathrm{d}x+\int_{s}^{t}\int_{\mathbb{R}}\phi^{\prime\prime}(x)\mathcal{U}(r,M(r,x))\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}r=0,$

and integrating by parts we arrive at

$\displaystyle\int_{\mathbb{R}}\phi(x)\mathop{}\\!\mathrm{d}\rho(t)$
$\displaystyle=\int_{\mathbb{R}}\phi(x)\mathop{}\\!\mathrm{d}\rho(s)+\int_{s}^{t}\int_{\mathbb{R}}\phi^{\prime}(x)\mathcal{U}^{\prime}_{M}(r,x)\mathop{}\\!\mathrm{d}\rho(r)\mathop{}\\!\mathrm{d}r$
$\displaystyle\leq\int_{\mathbb{R}}\phi(x)\mathop{}\\!\mathrm{d}\rho(s)+(p-1)\int_{s}^{t}\int_{\mathbb{R}}\left\lvert\frac{1}{p}\phi^{\prime}(x)\right\rvert^{\frac{p}{p-1}}\mathop{}\\!\mathrm{d}\rho(r)\mathop{}\\!\mathrm{d}r+\int_{s}^{t}\int_{\mathbb{R}}\left\lvert\mathcal{U}^{\prime}_{M}(r,x)\right\rvert^{p}\mathop{}\\!\mathrm{d}\rho(r)\mathop{}\\!\mathrm{d}r,$

having applied Young’s inequality. Now, define $\phi_{R}(x)\coloneqq(\lvert
x\rvert\wedge R)^{p}$ such that $\phi^{\prime}_{R}(x)=p\lvert
x\rvert^{p-1}\operatorname*{sgn}(x)$ for $\lvert x\rvert<R$ and
$\phi^{\prime}_{R}(x)=0$ for $\lvert x\rvert>R$. Since $\phi^{\prime}_{R}$ is
compactly supported we can find a sequence of smooth approximations
$(\phi^{\varepsilon}_{R})^{\prime}$ such that
$(\phi^{\varepsilon}_{R})^{\prime}(x)=\phi^{\prime}_{R}(x)$ for $\lvert
x\rvert\leq R-\varepsilon$, $(\phi^{\varepsilon}_{R})^{\prime}(x)=0$ for
$\lvert x\rvert\geq R$, and $(\phi^{\varepsilon}_{R})^{\prime}(x)$ tends to
$\phi_{R}^{\prime}(x)$ from below (above) for $R-\varepsilon<x<R$
($-R<x<\varepsilon-R$). In consequence, the primitive
$\phi^{\varepsilon}_{R}(x)=\int_{-\infty}^{x}(\phi^{\varepsilon}_{R})^{\prime}(y)\mathop{}\\!\mathrm{d}y$
tends to $\phi_{R}(x)-R^{p}\in C_{\mathrm{c}}(\mathbb{R})$ pointwise from
below. By the dominated convergence theorem it then follows that

$\displaystyle\int_{\mathbb{R}}(\phi_{R}(x)-R^{p})\mathop{}\\!\mathrm{d}\rho(t)$
$\displaystyle\leq\int_{\mathbb{R}}(\phi_{R}(x)-R^{p})\mathop{}\\!\mathrm{d}\rho(s)+(p-1)\int_{s}^{t}\int_{\mathbb{R}}\phi_{R}\chi_{(-R,R)}(x)\mathop{}\\!\mathrm{d}\rho(r)\mathop{}\\!\mathrm{d}r$
$\displaystyle\quad+\int_{s}^{t}\int_{\mathbb{R}}\left\lvert\mathcal{U}^{\prime}_{M}(r,x)\right\rvert^{p}\mathop{}\\!\mathrm{d}\rho(r)\mathop{}\\!\mathrm{d}r,$

and since $\rho(t)\in\mathscr{P}$ for $t\in[0,T]$ we are led to

$\int_{\mathbb{R}}\phi_{R}(x)\mathop{}\\!\mathrm{d}\rho(t)\leq\int_{\mathbb{R}}\phi_{R}(x)\mathop{}\\!\mathrm{d}\rho(s)+(p-1)\int_{s}^{t}\int_{\mathbb{R}}\phi_{R}(x)\mathop{}\\!\mathrm{d}\rho(r)\mathop{}\\!\mathrm{d}r+\int_{s}^{t}\lVert\mathcal{U}^{\prime}_{M}(r,\cdot)\rVert_{L^{p}(\mathbb{R},\rho_{r})}^{p}\mathop{}\\!\mathrm{d}r.$

The Grönwall inequality then yields

$\int_{\mathbb{R}}\phi_{R}(x)\mathop{}\\!\mathrm{d}\rho(t)\leq\left(\int_{\mathbb{R}}\phi_{R}(x)\mathop{}\\!\mathrm{d}\rho_{0}+\int_{0}^{t}\lVert\mathcal{U}^{\prime}_{M}(s,\cdot)\rVert_{L^{p}(\mathbb{R},\rho(s))}^{p}\mathop{}\\!\mathrm{d}s\right)\mathrm{e}^{(p-1)t},$

and letting $R\to\infty$ we use once more the dominated convergence theorem to
conclude. ∎

###  4.3. Entropy solutions of the Euler system

Now we want to follow the ideas of [36, 37] and use the entropy solutions of
(4.4) to define solutions of (1.1) by applying a two-dimensional extension of
the $BV$-chain rule Corollary 4.11. That is, following the proof therein, we
can prove a slight modification of [37, Theorem 2.2].

######  Theorem 4.13.

Let $p\geq 2$ and assume that $M\in
C([0,T];L^{1}_{\mathrm{loc}}(\mathbb{R}))$, $M\in\mathscr{R}$ for $t\in(0,T)$
and that it satisfies (4.4) in distributions
$\mathscr{D}^{\prime}((0,T)\times\mathbb{R})$ with $\mathcal{U}(t,m)$ given by
(4.18). According to Proposition 4.11 we let

$\mathcal{U}^{\prime}_{M}\partial_{x}M(t,\cdot)=\partial_{x}\mathcal{U}(t,M)\enspace\text{
for every }t\in(0,T).$

Then we have $M$, $\mathcal{U}(t,M)\in
BV_{\mathrm{loc}}((0,T)\times\mathbb{R})$, $\mathcal{U}^{\prime}_{M}\in
L^{1}(\lvert\nabla M\rvert)$ and

(4.23) $\nabla\mathcal{U}(t,M)=\mathcal{U}_{M}^{\prime}\nabla
M+\left(\sum_{i=1}^{n}b_{i}(t)\mathcal{F}_{i}(M),\>0\right)=\left(-(\mathcal{U}_{M}^{\prime})^{2},\>\mathcal{U}_{M}^{\prime}\right)\partial_{x}M+\left(\sum_{i=1}^{n}b_{i}(t)\mathcal{F}_{i}(M),\>0\right),$

where $\nabla=(\partial_{t},\partial_{x})$ denotes the $BV$-gradient.

Let us denote by $\mathscr{T}_{p}(\mathbb{R})$ the tangent bundle of the
Wasserstein space $\mathscr{P}_{p}(\mathbb{R})$, i.e.,

$\mathscr{T}_{p}(\mathbb{R})=\left\\{(\mu,v)\colon\mu\in\mathscr{P}_{p}(\mathbb{R})\text{
and }v\in L^{p}(\mathbb{R},\mu)\right\\},$

which we recognize as the generalization of (3.24).

Now we can give the following existence proof of solutions to (1.1), which can
be compared with [37, Theorem 1.3].

######  Theorem 4.14.

Let $p\in[2,\infty)$, $(\rho_{0},v_{0})\in\mathscr{T}_{p}(\mathbb{R})$ and
denote by $M^{0}$ the right-continuous cumulative distribution function of
$\rho_{0}$. Let $M\in\mathscr{R}_{p}$ be the corresponding entropy solution of
(4.4) with flux function (4.18). Define $\rho(t,x)\coloneqq\partial_{x}M(t,x)$
and $v(t,x)\coloneqq\mathcal{U}_{M}^{\prime}(t,x)$ as in Theorem 4.13. Then
$(\rho(t,\cdot),v(t,\cdot))\in\mathscr{T}_{p}(\mathbb{R})$ for $t\in(0,T)$ and
$(\rho,v)$ is a distributional solution of (1.1) for

$f[\rho]=\sum_{i=1}^{n}b_{i}(t)\partial_{x}\mathcal{F}_{i}(M).$

Moreover, we have $\rho(t)\in\mathscr{P}_{p}(\mathbb{R})$ and $v(t,\cdot)\in
L^{p}(\mathbb{R},\rho(t))$ for every $t>0$ and

  1. (i)

$\lVert v(t,\cdot)\rVert_{L^{p}(\mathbb{R},\rho(t))}\leq\sum_{i=0}^{n}\lvert
B_{i}(t)\rvert\lVert F_{i}\rVert_{L^{p}(0,1)}$ for a.e. $t>0$,

  2. (ii)

(4.24) $d_{\mathscr{W}_{p}}(\rho(t),\rho(s))\leq\lvert
t-s\rvert\sum_{i=0}^{n}\lVert F_{i}\rVert_{L^{p}(0,1)}\lVert
B_{i}\rVert_{L^{\infty}(0,T)},\quad 0\leq s\leq t\leq T.$

###### Proof.

Since $M$ satisfies (4.4) in the distributional sense, Theorem 4.13 yields
that (4.2a) is satisfied in the same sense. Differentiating with respect to
$x$ in the sense of distributions shows that (1.1a) is satisfied. For the
momentum equation (1.1b) we again find from Theorem 4.13 that

$\displaystyle\partial_{t}(\rho v)$
$\displaystyle=\partial_{t}(\partial_{x}\mathcal{U}(t,M))=\partial_{x}(\partial_{t}\mathcal{U}(t,M))=\partial_{x}\left(v\partial_{t}M+\sum_{i=1}^{n}b_{i}(t)\mathcal{F}_{i}(M)\right)$
$\displaystyle=\partial_{x}(-v^{2}\rho)+\sum_{i=1}^{n}b_{i}(t)\partial_{x}\mathcal{F}_{i}(M).$

Indeed, we have shown that the following momentum equation is satisfied in the
distributional sense,

$\partial_{t}(\rho v)+\partial_{x}(\rho
v^{2})=\left(\sum_{i=1}^{n}b_{i}(t)\mathcal{F}_{i,M}^{\prime}(t,x)\right)\rho\quad\text{
in }\mathscr{D}^{\prime}((0,T)\times\mathbb{R}).$

Now, the convergence $\rho(t,\cdot)\to\rho_{0}$ in $\mathscr{P}(\mathbb{R})$
follows from the convergence of $M(t,\cdot)$ to $M^{0}$ in $L^{1}(\mathbb{R})$
in (4.22): indeed, by (4.21) this implies convergence in the 1-Wasserstein
metric, which again implies narrow convergence, see, e.g, [47, Theorem 7.12].
We must also show $\rho v(t,\cdot)\to\rho_{0}v_{0}$ in
$\mathscr{M}(\mathbb{R})$, and from (4.23) we have

$\lim\limits_{t\to
0+}\int_{\mathbb{R}}\varphi(x)v(t,x)\mathop{}\\!\mathrm{d}\rho(t,x)=-\lim\limits_{t\to
0+}\int_{\mathbb{R}}\varphi^{\prime}(x)\mathcal{U}(t,M(t,x))\mathop{}\\!\mathrm{d}x=\int_{\mathbb{R}}\varphi^{\prime}(x)\mathcal{V}^{0}(M^{0}(x))\mathop{}\\!\mathrm{d}x.$

It remains to show that the distributional derivative of
$\mathcal{V}^{0}(M^{0})$ is $v_{0}\rho_{0}$; however, this is in a sense how
$\mathcal{V}^{0}$ was defined. Indeed, using again the the push-forward, we
have

$\mathcal{V}^{0}(M^{0}(x))=\int_{0}^{M^{0}(x)}v_{0}\circ
X^{0}(m)\mathop{}\\!\mathrm{d}m=\int_{0}^{1}\chi_{(-\infty,x]}v_{0}\circ
X^{0}(m)\mathop{}\\!\mathrm{d}m=\int_{(-\infty,x]}v_{0}(x)\mathop{}\\!\mathrm{d}\rho_{0}(x),$

and so we have a distributional solution of (1.1). Statement (i) follows from
Corollary 4.11 applied to $\mathcal{U}(t,M)$. For the remaining statement we
note that $\rho(t)\in C([0,T];\mathscr{P}_{1}(\mathbb{R}))$ follows from $M\in
C([0,T];\mathscr{R}_{1})$, while $\rho(t)\in\mathscr{P}_{p}(\mathbb{R})$ for
$t\in[0,T]$ by Lemma 4.12. Therefore,
$\rho(t)\colon[0,T]\to\mathscr{P}_{p}(\mathbb{R})$ is a narrowly continuous
curve satisfying the continuity equation (1.1a) for $v$ with $\lVert
v(t,\cdot)\rVert_{L^{p}(\mathbb{R},\rho(t))}\in L^{1}(0,T)$, and so by [1,
Theorem 8.3.1] it is also an absolutely continuous curve with
$\mathscr{W}_{p}$-metric derivative satisfying
$\lvert\rho^{\prime}(t)\rvert\leq\lVert
v(t,\cdot)\rVert_{L^{p}(\mathbb{R},\rho(t))}$ for a.e. $t\in(0,T)$. Combining
this with (i) we can integrate to obtain (4.24). ∎

######  Example 4.15 (Examples 4.3 and 4.4 revisited).

For the flux function 4.9, the forcing term in Theorem 4.14 becomes

$1\cdot\mathcal{A}_{M}^{\prime}(t,x)\rho=-\left(\frac{\alpha}{2}\left(M(t,x)+M(t,x-)-1\right)+\beta\right)\rho,$

which is exactly the Euler–Poisson forcing. On the other hand, with linear
damping as in (4.11) it becomes

$\displaystyle\left(-\gamma\mathrm{e}^{-\gamma
t}\cdot(\mathcal{V}^{0})^{\prime}_{M}(t,x)+\mathrm{e}^{-\gamma
t}\cdot\mathcal{A}_{M}^{\prime}(t,x)\right)\rho$
$\displaystyle=-\gamma\left(\mathrm{e}^{-\gamma
t}(\mathcal{V}^{0})_{M}^{\prime}(t,x)+\frac{1-\mathrm{e}^{-\gamma
t}}{\gamma}\mathcal{A}_{M}^{\prime}(t,x)\right)\rho+\mathcal{A}_{M}^{\prime}(t,x)\rho$
$\displaystyle=-\gamma v\rho+\mathcal{A}_{M}^{\prime}(t,x)\rho,$

which is the Euler–Poisson forcing with linear damping.

######  Example 4.16 (The Riemann problem for a single Dirac mass).

Let us now consider the Riemann problem for our scalar conservation law (4.4),
(4.9) with $\beta=0$, that is, initial data of the form

$M^{0}(x)=\begin{cases}M_{l},&x<0,\\\ M_{r},&x\geq 0.\end{cases}$

This corresponds to a Dirac mass of size $M_{r}-M_{l}$ for the initial density
$\rho_{0}$, located at $x=0$. Assume that this mass has initial velocity
$v^{0}$, so that $\mathcal{V}^{0}(m)=\mathcal{V}^{0}(M_{l})+v^{0}m$ for
$m\in[M_{l},M_{r}]$. On the same interval, the full flux function is then
$\mathcal{U}(t,\theta)=\mathcal{V}^{0}(M_{l})+\int_{M_{l}}^{\theta}(v^{0}-t\frac{\alpha}{2}(2m-1)\mathop{}\\!\mathrm{d}m$.
If $\alpha\geq 0$ then this is concave on the interval for all $t\geq 0$, and
its corresponding lower convex envelope is

$\mathcal{U}_{\smallsmile}(t,m)=\mathcal{V}^{0}(M_{l})+\left(v^{0}-t\frac{\alpha}{2}(M_{l}+M_{r}-1)\right)(m-M_{l}).$

Therefore, for $\alpha\geq 0$, the entropic solution is a shock wave
connecting the states $M_{l}$ and $M_{r}$ with velocity $\dot{x}(t)$
satisfying the Rankine–Hugoniot-condition , i.e.,

$M(t,x)=\begin{cases}M_{l},&x<v^{0}t-\frac{\alpha}{4}t^{2}(M_{l}+M_{r}-1),\\\
M_{r},&x\geq v^{0}t-\frac{\alpha}{4}t^{2}(M_{l}+M_{r}-1).\end{cases}$

That is, the Dirac mass continues moving along a parabolic, or linear if
$M_{l}=0$ and $M_{r}=1$, trajectory; that is, particles remain particles.

On the other hand, if $\alpha<0$ then the flux function is strictly convex on
$[M_{l},M_{r}]$ for all time, meaning
$\mathcal{U}_{\smallsmile}(t,m)=\mathcal{U}(t,m)$. For fixed $t$ we can
consider dervative of $m\mapsto\mathcal{U}(t,m)$, which we denote by
$\mathcal{U}^{\prime}_{\smallsmile}(t,m)$. Since
$m\mapsto\mathcal{U}^{\prime}_{\smallsmile}(t,m)$ is monotone in $m$ we may
define its inverse $v\mapsto(\mathcal{U}^{\prime}_{\smallsmile})^{-1}(t,v)$.
The left and right constant states will travel with respective velocities
$\mathcal{U}^{\prime}(t,M_{l})=v^{0}-t\frac{\alpha}{2}(2M_{l}-1)$ and
$\mathcal{U}^{\prime}(t,M_{r})>\mathcal{U}^{\prime}(t,M_{l})$. The remaining
middle section will be a continuous transition from $m=M_{l}$ to $m=M_{r}$,
where the velocity corresponding to $m$ is $\mathcal{U}^{\prime}(t,m)$.
Integrating we find $x=v^{0}t-\frac{1}{4}t^{2}\alpha(2m-1)$, which we can
invert to find the value of $M(t,x)$ in the middle section, and we find the
solution

$M(t,x)=\begin{cases}M_{l},&x<v^{0}t-\frac{1}{4}t^{2}\alpha(2M_{l}-1),\\\
\frac{x-v^{0}t-\frac{1}{4}\alpha t^{2}}{\frac{1}{2}\alpha
t^{2}},&v^{0}t-\frac{1}{4}t^{2}\alpha(2M_{l}-1)\leq
x<v^{0}t-\frac{1}{4}t^{2}\alpha(2M_{r}-1)\\\ M_{r},&x\geq
v^{0}t-\frac{1}{4}t^{2}\alpha(2M_{r}-1).\end{cases}$

That is, the Dirac mass is smoothed out over time with a parabolic rarefaction
fan. Compare these examples with the second order diffusion of a Dirac mass in
[7, Equation (1.14)], as well as the corresponding first-order solutions [4,
Equations (2.23)–(2.24)].

From Example 4.15 we see that Theorem 4.13 is all well and good when the
forcing term in (4.23) turns out to match the forcing term for the original
Euler system. For an entropy solution of (4.4) to yield a distributional
solution $(\rho,v)=(\partial_{x}M,\mathcal{U}^{\prime}_{M})$ of (1.1), we see
by applying the $BV$-gradient of (4.23) that this will be the case if

(4.25) $\partial_{x}\frac{\partial\mathcal{U}(t,M)}{\partial
t}=f[\partial_{x}M]\quad\text{in }\mathscr{D}^{\prime},$

which is indeed the compatibility condition we read from the flux function
(4.18) and the force distribution appearing in the proof of Theorem 4.14. To
ensure that (4.25) holds, we can define entropy solutions of the Euler system
as follows.

######  Definition 4.17 (Entropy solution of the Euler system).

We define the distributional solution of (1.1) coming from an entropy solution
$M$ of the scalar conservation law (4.4) in Theorem 4.14 to be an entropy
solution of the Euler system (1.1) if the flux function $\mathcal{U}(t,m)$
additionally satisfies

(4.26)
$\mathcal{U}(t,m)=\mathcal{V}^{0}(m)+\int_{0}^{m}\int_{0}^{t}F[M_{s}^{-1}](\omega)\mathop{}\\!\mathrm{d}\omega,$

where $M_{t}^{-1}$ is the generalized inverse of $M(t,\cdot)$, and $F[X]$ is
defined as in (3.16).

Combining Lemmas 4.1 and 4.2 with the relation (3.16), it follows that (4.26)
implies (4.25). We note that unless one is in a case where $F[M^{-1}_{t}]$
simplifies, such as for Euler–Poisson, (4.26) as it stands depends on the
generalized inverse of $M$, and in order to write it as a flux function
depending only on $M$, if possible, would require some additional information
about the evolution of solutions of the system, e.g., that mass is preserved,
which is used in the simplification for Euler–Poisson, and the evolution of
the center of mass, see Section 7.2. In general, it would then seem that the
Lagrangian solutions of Section 3 are simpler to work with, as one can work
directly with the monotone rearrangement $X$ appearing in $F[X]$.

##  5\. Equivalence of solution concepts

In this section we will show that the Lagrangian solutions coming from
$L^{2}$-gradient flow in Section 3 and the entropy solutions in Section 4 are
equivalent, and uniquely defined. Indeed, from their expositions we know that
each solution concept gives rise to a uniquely defined distributional solution
of the Euler–Poisson system (1.3) under suitable hypotheses on the forcing
term. Our contribution is to show that they are equivalent already from their
definitions.

###  5.1. Projections and shock admissibility

As a preparation for the main statement, we present some auxiliary results
which contain the core of our argument.

######  Lemma 5.1 (Projection on $\mathscr{H}_{X}$ and the Radon–Nikodym
derivative).

Let $X\in\mathscr{K}$ and $W\in L^{2}(\Omega)$ be given. From these we can
define the right-continuous generalized inverse $M(x)$ of $X(m)$ and any
primitive $\mathcal{W}(m)$ such that $\mathcal{W}^{\prime}=W$. Then we have
the following identity,

$\operatorname{P}_{\mathscr{H}_{X}}W=\mathcal{W}_{M}^{\prime}(X),$

where $\mathscr{H}_{X}$ is defined in 3.10 and $\mathcal{W}^{\prime}_{M}(x)$
is the Radon–Nikodym derivative of $\partial_{x}\mathcal{W}(M(x))$ with
respect to $\partial_{x}M(x)$.

###### Proof.

From Lemma 4.2 we have that $\mathcal{W}^{\prime}_{M}(x)$ is given by

(5.1) $\mathcal{W}^{\prime}_{M}(x)=\begin{cases}W(M(x)),&[[M(x)]]=0\\\
\displaystyle\frac{\mathcal{W}(M(x))-\mathcal{W}(M(x-))}{M(x)-M(x-)},&[[M(x)]]\neq
0,\end{cases}$

where $[[M(x)]]=M(x)-M(x-)$. As in Section 3.2, we can consider the open set
$\Omega_{X}$ as consisting of countably many disjoint maximal intervals, and
if $m\notin\Omega_{X}$, then $M(t,X(t,m))=m$. On the other hand, if
$m\in(m_{l},m_{r})$ for a maximal interval $(m_{l},m_{r})\subset\Omega_{X}$,
then by the right-continuity of $X$ and $M$ we have $M(X(m))=m_{r}$ and
$M(X(m)-)=m_{l}$. Combining these properties with the expression for
$\mathcal{W}^{\prime}_{M}$, we find exactly
$\operatorname{P}_{\mathscr{H}_{X}}W$ defined by (3.11). ∎

Lemma 5.1 provides a connection between projection onto $\mathscr{H}_{X}$ and
the Rankine–Hugoniot condition (4.16). In particular, if
$\dot{X}(t)=\operatorname{P}_{\mathscr{H}_{X(t)}}U(t)$ for some $U(t)\in
L^{2}(\Omega)$ with
$\mathcal{U}(t,m)=\int_{0}^{m}U(t,\omega)\mathop{}\\!\mathrm{d}\omega$ we have

(5.2)
$(\operatorname{P}_{\mathscr{H}_{X(t)}}U)(m)=\mathcal{U}_{M}^{\prime}(t,X(t,m)),$

which for $m\in\Omega_{X(t)}$ corresponds exactly to the Rankine–Hugoniot
condition for $M$ and flux function $\mathcal{U}(t,M)$.

Now, let $X(t)\in\mathscr{K}$ be a Lagrangian solution of the differential
inclusion (3.21) for which at time $t$ we have $\Omega_{X(t)}\neq\emptyset$.
Then there is some maximal interval $(m_{l},m_{r})\subset\Omega_{X(t)}$ where
$X(t,m)$ is ‘flat’, and so its generalized inverse $M(t,x)$ has a jump of size
$m_{r}-m_{l}$ at $x=X(t,m)$ for $m\in(m_{l},m_{r})$. Then we observe from
Lemma 5.1 that having $\dot{X}=\operatorname{P}_{\mathscr{H}_{X(t)}}U$
corresponds exactly to the Rankine–Hugoniot condition (4.16) for $M$ and flux
function $\mathcal{U}$ given by the primitive of $U$.

######  Lemma 5.2 (Projection on $T_{X}\mathscr{K}$ and the Oleĭnik
E-condition).

Let $X\in\mathscr{K}$ and $U\in L^{2}(\Omega)$ be given, and define $M$ and
$\mathcal{U}$ as in Lemma 5.1. Then $U-\operatorname{P}_{\mathscr{H}_{X}}U\in
N_{X}\mathscr{K}$, or equivalently
$\operatorname{P}_{\mathscr{H}_{X}}U=\operatorname{P}_{T_{X}\mathscr{K}}U$, if
and only if $\operatorname{P}_{\mathscr{H}_{X}}U$ satisfies

(5.3)
$(\operatorname{P}_{\mathscr{H}_{X}}U)(m)\leq\frac{\mathcal{U}(m)-\mathcal{U}(m_{l})}{m-m_{l}}$

for any $m\in(m_{l},m_{r})$, a maximal interval of $\Omega_{X}$. In
particular, if $\dot{X}(t)=\operatorname{P}_{\mathscr{H}_{X(t)}}U(t)$ for some
$U(t)\in L^{2}(\Omega)$, then for $m\in\Omega_{X(t)}$, (5.3) is exactly the
Oleĭnik E-condition (4.17) for $M(t,x)$ and flux function
$\mathcal{U}(t,m)=\int_{0}^{m}U(t,\omega)\mathop{}\\!\mathrm{d}\omega$.

###### Proof.

Define
$\Xi(m)=\int_{0}^{m}(U-\operatorname{P}_{\mathscr{H}_{X}}U)(\omega)\mathop{}\\!\mathrm{d}\omega$
and observe that on any maximal interval $(m_{l},m_{r})\subset\Omega_{X}$,
$\operatorname{P}_{\mathscr{H}_{X}}U$ by its definition (3.11) equals the
average of $U$ over the same interval, which shows that $\Xi(m)=0$ for any
$m\in\Omega\setminus\Omega_{X}$. On the other hand, for any $m$ contained in a
maximal interval $(m_{l},m_{r})$ we have

$\Xi(m)=\Xi(m)-\Xi(m_{l})=\mathcal{U}(m)-\mathcal{U}(m_{l})-(m-m_{l})\operatorname{P}_{\mathscr{H}_{X}}U,$

and so $\Xi(m)\geq 0$ if and only if (5.3) holds. The claim is then a
consequence of Lemma 3.1. If $X(t,\cdot),U(t,\cdot)\in L^{2}(\Omega)$ were to
depend on time, we see that the corresponding inequality would still hold. ∎

Observe that, strictly speaking, (5.3) only gives the rightmost inequality in
(4.17). However, we can obtain the other inequality,

(5.4)
$(\operatorname{P}_{\mathscr{H}_{X}}U)(m)\geq\frac{\mathcal{U}(m_{r})-\mathcal{U}(m)}{m_{r}-m},$

by considering $\Xi(m_{r})-\Xi(m)$ for $m\in(m_{l},m_{r})$ together with the
fact that $\Xi(m_{r})=0$.

For a right-continuous probability distribution function $M(x)$ corresponding
to $\mu\in\mathscr{P}$ we may then apply Lemma 4.2 to the Kružkov entropy-
entropy flux pair $\eta_{k}(M)$ and $q_{k}(t,M)$ from (4.13) to find

(5.5) $\displaystyle\eta_{k,M}^{\prime}(x)$
$\displaystyle=\begin{cases}\operatorname*{sgn}(M(x)-k),&\mu(\\{x\\})=0,\\\\[5.69054pt]
\displaystyle\frac{\lvert M(x)-k\rvert-\lvert
M(x-)-k\rvert}{\mu(\\{x\\})},&\mu(\\{x\\})\neq 0\end{cases}$
$\displaystyle=\begin{cases}\operatorname*{sgn}(M(x)-k),&k\notin(M(x-),M(x)),\\\\[5.69054pt]
\displaystyle\frac{M(x)+M(x-)-2k}{M(x)-M(x-)},&k\in(M(x-),M(x))\end{cases}$

and

(5.6) $\displaystyle q_{k,M}^{\prime}(t,x)$
$\displaystyle=\begin{cases}\operatorname*{sgn}(M(x)-k)U(t,M(x)),&\mu(\\{x\\})=0,\\\\[5.69054pt]
\displaystyle\frac{[[\operatorname*{sgn}(M-k)\left(\mathcal{U}(t,M)-\mathcal{U}(t,k)\right)]](x)}{\mu(\\{x\\})},&\mu(\\{x\\})\neq
0,\end{cases}$
$\displaystyle=\begin{cases}\operatorname*{sgn}(M(x)-k)U(t,M(x)),&M(x-)=M(x),\\\\[5.69054pt]
\displaystyle\operatorname*{sgn}(M(x)-k)\frac{\mathcal{U}(t,M(x))-\mathcal{U}(t,M(x-))}{M(x)-M(x-)},&k\notin(M(x-),M(x))\neq\emptyset,\\\\[11.38109pt]
\displaystyle\frac{\mathcal{U}(t,M(x))+\mathcal{U}(t,M(x-))-2\mathcal{U}(t,k)}{M(x)-M(x-)},&k\in(M(x-),M(x)),\end{cases}$

where we as short-hand notation write $[[f(t,M)]](x)\coloneqq
f(t,M(x))-f(t,M(x-))$ for a function
$f\colon[0,T]\times\mathbb{R}\to\mathbb{R}$. Note that we can apply Lemma 5.1
to (4.13) to obtain the identities

$\operatorname{P}_{\mathscr{H}_{X}}\left(\operatorname*{sgn}(m-k)\right)=\eta^{\prime}_{k,M}(X),\qquad\operatorname{P}_{\mathscr{H}_{X}}\left(\operatorname*{sgn}(m-k)U(t,m)\right)=q^{\prime}_{k,M}(t,X).$

Combining the above results we can deduce the following result, which should
be compared with the distributional inequality defining the Kružkov entropy
condition (4.12).

######  Corollary 5.3.

Let $X\in\mathscr{K}$ and $U(t,\cdot)\in L^{2}(\Omega)$ be given, and define
$M$ and $\mathcal{U}(t,\cdot)$ as in Lemma 5.2. Assume furthermore that
$U-\operatorname{P}_{\mathscr{H}_{X}}U\in N_{X}\mathscr{K}$, then we have the
inequality

(5.7)
$q_{k,M}^{\prime}(t,X)\leq\eta^{\prime}_{k,M}(X)\mathcal{U}_{M}^{\prime}(t,X).$

###### Proof.

Following the proof of Lemma 5.1 while comparing (5.1), (5.5) and (5.6) we see
that this holds as an equality for the cases where
$k\notin(M(X(m)-),M(X(m)))$. For the remaining case, we consider
$k\in(m_{l},m_{r})=(M(X(m)-),M(X(m)))$, and by (5.6) we have

$\displaystyle q^{\prime}_{k,M}(t,X(m))$
$\displaystyle=\frac{\mathcal{U}(t,m_{r})+\mathcal{U}(t,m_{l})-2\mathcal{U}(t,k)}{m_{r}-m_{l}}\leq\frac{m_{r}-k}{m_{r}-m_{l}}(\operatorname{P}_{\mathscr{H}_{X}}U)(m)-\frac{k-m_{l}}{m_{r}-m_{l}}(\operatorname{P}_{\mathscr{H}_{X}}U)(m)$
$\displaystyle=\frac{m_{r}+m_{l}-2k}{m_{r}-m_{l}}(\operatorname{P}_{\mathscr{H}_{X}}U)(m)=\eta^{\prime}_{k,M}(X(m))\mathcal{U}^{\prime}_{M}(t,X(m)),$

where we have combined (5.2), (5.3) and (5.4). ∎

By a similar reasoning, based on the formulas for the Radon-Nikodym
derivatives in (5.1), (5.5) and (5.6), we can characterize entropy solutions
of (4.4) in Definition 4.5 in the space $BV_{\mathrm{loc}}$ by the Oleĭnik
E-condition (4.17), see [19, Section 4.5] for a similar result.

######  Lemma 5.4.

Assume $M$ is a weak solution of (4.4) with a flux function of the form
$\mathcal{U}(t,m)=\int_{0}^{m}U(t,\omega)\mathop{}\\!\mathrm{d}\omega$ for
$U(t)\in L^{2}(\Omega)$. Then it is an entropy solution if and only if it
satisfies the Oleĭnik E-condition (4.17) along its discontinuities, i.e.,
where $M(t,x)>M(t,x-)$.

###### Proof.

Being a weak solution, $M$ satisfies
$\partial_{t}M=-\partial_{x}\mathcal{U}(t,M)=-\mathcal{U}^{\prime}_{M}(t,x)\partial_{x}M$
in distributions. Then the Krǔzkov entropy inequality can be rewritten as
$\partial_{t}\eta+\partial_{x}q\leq 0$ in $\mathscr{D}^{\prime}$, that is,

$\displaystyle 0$
$\displaystyle\leq-\int_{0}^{T}\int_{\mathbb{R}}\varphi(t,x)\eta^{\prime}_{k,M}(t,x)\partial_{t}M(t,x)\mathop{}\\!\mathrm{d}t-\int_{0}^{T}\int_{\mathbb{R}}\varphi(t,x)q^{\prime}_{k,M}(t,x)\partial_{x}M(t,x)\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}}\varphi(t,x)\left(\eta_{k,M}^{\prime}(t,x)\mathcal{U}_{M}^{\prime}(t,x)-q_{k,M}^{\prime}(t,x)\right)\partial_{x}M(t,x)\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int_{\\{x\colon\rho(t,\\{x\\})>0\\}}\varphi(t,x)\left(\eta_{k,M}^{\prime}(t,x)\mathcal{U}_{M}^{\prime}(t,x)-q_{k,M}^{\prime}(t,x)\right)\mathop{}\\!\mathrm{d}\rho(t)\mathop{}\\!\mathrm{d}t$

for any nonnegative $\varphi\in
C^{\infty}_{\mathrm{c}}([0,T)\times\mathbb{R})$, where we have combined (5.1)
for $\mathcal{U}(t,M)$ with (5.5) and (5.6) to deduce the identity
$\eta_{k,M}^{\prime}(t,x)\mathcal{U}_{M}^{\prime}(t,x)=q_{k,M}^{\prime}(t,x)$
whenever $\rho(t,\\{x\\})=0$ or $k\notin(M(t,x-),M(t,x))$. That is, the
Krǔzkov condition reduces to a condition on the discontinuities of
$M(t,\cdot)$. Assume now there is such a discontinuity at $x=\bar{x}$, i.e.,
$m_{r}\coloneqq M(t,\bar{x})>M(t,\bar{x}-)\eqqcolon m_{l}$, and let
$k\in(m_{l},m_{r})$. Then the Krǔzkov condition implies for a.e. $t$ that

$\frac{m_{r}+m_{l}-2k}{m_{r}-m_{l}}\frac{\mathcal{U}(t,m_{r})-\mathcal{U}(t,m_{l})}{m_{r}-m_{l}}\geq\frac{\mathcal{U}(t,m_{r})+\mathcal{U}(t,m_{l})-2\mathcal{U}(t,k)}{m_{r}-m_{l}}$

for any $k\in(m_{l},m_{r})$, which is equivalent to the Oleĭnik E-condition
(4.17), cf. [25, Section 2.1]. Therefore, we see that the Krǔzkov condition is
satisfied if and only if the Oleĭnik E-condition above holds for every such
discontinuity for a.e. $t$. ∎

A final useful observation concerns integrating orthogonal functions, i.e.,
for $X\in\mathscr{K}$ we have

$\int_{\Omega}U(m)W(m)\mathop{}\\!\mathrm{d}m=0,\qquad
U\in\mathscr{H}_{X},\quad W\in\mathscr{H}_{X}^{\perp}.$

In particular, for any $W\in L^{2}(\Omega)$ we have that
$W-\operatorname{P}_{\mathscr{H}_{X}}W\in\mathscr{H}_{X}^{\perp}$, while for
any smooth $\varphi\colon\mathscr{K}\to\mathbb{R}$, $\varphi\circ
X\in\mathscr{H}_{X}$, and so

(5.8) $\int_{\Omega}(\varphi\circ
X)W\mathop{}\\!\mathrm{d}m=\int_{\Omega}(\varphi\circ
X)(\operatorname{P}_{\mathscr{H}_{X}}W)\mathop{}\\!\mathrm{d}m.$

###  5.2. Equivalence theorem and proof

We now state our main results.

######  Theorem 5.5 (Lagrangian solutions are entropy solutions).

Let $(X(t),V(t))$ for $t\in[0,T]$ be a Lagrangian solution of (3.21) in the
sense of Definition 3.6. Then $M(t,\cdot)$, defined as the right-continuous
generalized inverse of $X(t)$, is an entropy solution of (4.4) for $t\in[0,T]$
with flux function given by the primitive of the prescribed velocity $U$.

######  Theorem 5.6 (Entropy solutions are Lagrangian solutions).

Let $M(t,x)$ for $t\in[0,T]$ be an entropy solution of the scalar conservation
law (4.4) in the sense of Definition 4.5 with flux function (4.18). Then,
$X(t)$, defined as the right-continuous generalized inverse of $M(t,\cdot)$,
satisfies the differential inclusion (3.21a) for
$U=\frac{\partial\mathcal{U}}{\partial m}$. If $M$ additionally provides an
entropy solution of the Euler system (1.1) in the sense of Definition 4.17,
then (3.21b) is also satisfied, and $X$ is a Lagrangian solution of (3.21) for
$t\in[0,T]$.

Now, given initial data $(\rho_{0},v_{0})\in\mathscr{T}_{2}$, we want to use
the Lagrangian and entropy solution concepts to derive distributional
solutions of (1.1). Clearly, the Lagrangian initial data $(X^{0},V^{0})$ and
the initial data $M^{0}$ and flux function $\mathcal{U}^{0}$ for the
conservation law (4.4) are in one-to-one correspondence through $X^{0},M^{0}$
being generalized inverses of one another and
$\mathcal{U}^{0}=\int^{m}_{0}V^{0}(\omega)\mathop{}\\!\mathrm{d}\omega$. If
$f[\rho]$ furthermore is densely Lipschitz-continuous, then the corresponding
$F[X]\colon\mathscr{K}\to L^{2}(\Omega)$ is also Lipschitz and the prescribed
velocity $U$ in (3.20) is uniquely defined. Based on the previous theorems and
the uniqueness of each solution concept in this case, we deduce the following
corollary.

######  Corollary 5.7 (Equivalence of solutions to the Euler system).

Let $(\rho_{0},v_{0})\in\mathscr{T}_{2}$ be initial data for (1.1), and assume
we have two sets of solutions
$(\check{\rho}_{t},\check{v}_{t}),(\hat{\rho}_{t},\hat{v}_{t})\in\mathscr{T}_{2}$
for $t\in[0,T]$ provided by respectively the Lagrangian and entropy solution
concept, see Definitions 3.6 and 4.17. Assume $f[\rho]$ is densely Lipschitz-
continuous. Then
$(\check{\rho}_{t},\check{v}_{t})=(\hat{\rho}_{t},\hat{v}_{t})$ in
$\mathscr{T}_{2}$.

###### Proof of Theorem 5.5.

Assume we have a Lagrangian solution of (3.21); in particular, by Lemma 3.7 we
have $\rho=X_{\\#}\mathfrak{m}$ and there exists a unique map $v\in
L^{2}(\mathbb{R},\rho)$ such that for $t\in\mathcal{T}$ we have
$V(t,\cdot)=v\circ X(t,\cdot)\in\mathscr{H}_{X(t)}$. To give an idea of the
proof strategy, we first show that the Lagrangian solution is a weak solution
of (4.4).

Let $\varphi\in C^{\infty}_{\mathrm{c}}([0,T)\times\mathbb{R})$ be the test
function we want to use in the weak formulation (4.15), and define
$\Phi(t,x)=-\int_{x}^{\infty}\varphi(t,y)\mathop{}\\!\mathrm{d}y\in
C_{\mathrm{b}}^{\infty}([0,T)\times\mathbb{R})$ with
$\lim\limits_{x\to\infty}\Phi(t,x)=0$. Then, the chain rule implies

(5.9) $0-\lim\limits_{t\to
0+}\Phi(t,X(t,m))=\int_{0}^{T}\left(\Phi_{t}(t,X(t,m))+\Phi_{x}(t,X(t,m))\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X(t,m)\right)\mathop{}\\!\mathrm{d}t.$

Since $X(t)$ converges strongly to $X^{0}$ in $L^{2}(\Omega)$ by definition of
the Lagrangian solution, it follows that

$\lim\limits_{t\to
0+}\int_{0}^{1}\Phi(t,X(t,m))\mathop{}\\!\mathrm{d}m=\int_{0}^{1}\Phi(0,X^{0}(m))\mathop{}\\!\mathrm{d}m=\int_{\mathbb{R}}\varphi(0,x)\partial_{x}M^{0}(x)=-\int_{\mathbb{R}}\varphi(0,x)M^{0}(x)\mathop{}\\!\mathrm{d}x.$

Then, integrating (5.9) over $\Omega=(0,1)$ we obtain

$\displaystyle\int_{\mathbb{R}}\varphi(0,x)M^{0}(x)\mathop{}\\!\mathrm{d}x$
$\displaystyle=\int_{0}^{T}\int_{0}^{1}\left(\Phi_{t}(t,X(t,m))+V(t,m)\Phi_{x}(t,X(t,m))\right)\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int_{0}^{1}\left(\Phi_{t}(t,X(t,m))+\mathcal{U}_{M}^{\prime}(t,X(t,m))\Phi_{x}(t,X(t,m))\right)\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}}\left(\Phi_{t}(t,x)+\mathcal{U}_{M}^{\prime}(t,x)\Phi_{x}(t,x)\right)\partial_{x}M(t,x)\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}}\left(\Phi_{t}(t,x)\partial_{x}M+\Phi_{x}(t,x)\partial_{x}\mathcal{U}(t,M)\right)\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\left[\Phi_{t}(t,x)M(t,x)+\Phi_{x}(t,x)\mathcal{U}(t,M(t,x))\right]_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad-\int_{0}^{T}\int_{\mathbb{R}}\left(\Phi_{xt}(t,x)M(t,x)+\Phi_{xx}(t,x)\mathcal{U}(t,M(t,x))\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$
$\displaystyle=-\int_{0}^{T}\int_{\mathbb{R}}\left(\varphi_{t}(t,x)M(t,x)+\varphi_{x}(t,x)\mathcal{U}(t,M(t,x))\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t.$

We conclude that (4.15) is satisfied and this is a weak solution of (4.4). In
the above, the second identity is a consequence of Lemmas 3.7 and 5.1, in
particular that $V(t)=\operatorname{P}_{\mathscr{H}_{X(t)}}U(t)$ for
$t\in\mathcal{T}$, where $\mathcal{T}$ is dense in $(0,\infty)$, together with
(5.2).

To show that it is also an entropy solution, note that we may subtract zero in
the form of $\int_{0}^{T}\int_{\mathbb{R}}\varphi_{t}\lvert
k\rvert\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$ on the left-hand side
to deal with boundary terms as $x\to-\infty$ when integrating by parts;
indeed, for

$\int_{(-R,R]}\varphi_{t}(\lvert M-k\rvert-\lvert
k\rvert)\mathop{}\\!\mathrm{d}x=\left[\Phi_{t}(t,x)(\lvert
M(t,x)-k\rvert-\lvert
k\rvert)\right]_{x=-R}^{R}-\int_{(-R,R]}\Phi_{t}\partial_{x}\lvert M-k\rvert$

the boundary terms vanish as $R\to\infty$. Then, recalling (5.5) and (5.6), we
find

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\varphi_{t}\lvert
M-k\rvert+\varphi_{x}\operatorname*{sgn}(M-k)(\mathcal{U}(t,M)-\mathcal{U}(t,k))\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\lvert
k\rvert\int_{\mathbb{R}}\varphi(0,x)\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad=-\int_{0}^{T}\int_{\mathbb{R}}\left(\Phi_{t}\eta_{M}^{\prime}+\Phi_{x}q_{M}^{\prime}\right)(t,x)\partial_{x}M\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad=-\int_{0}^{T}\int_{\Omega}\left(\Phi_{t}\eta_{M}^{\prime}+\Phi_{x}q_{M}^{\prime}\right)(t,X(m))\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad\geq-\int_{0}^{T}\int_{\Omega}\eta_{M}^{\prime}(X(m))\left(\Phi_{t}(t,X(m))+\mathcal{U}_{M}^{\prime}(t,X(m))\Phi_{x}(t,X(m))\right)\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad=-\int_{0}^{T}\int_{\Omega}\operatorname*{sgn}(m-k)\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}\Phi(t,X(m))\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad=\int_{\Omega}\operatorname*{sgn}(m-k)\Phi(0,X^{0}(m))\mathop{}\\!\mathrm{d}m,$

where we have used (5.7) and (5.8). Applying (5.8) once more we find

$\int_{0}^{1}\operatorname*{sgn}(m-k)\Phi(0,X^{0}(m))\mathop{}\\!\mathrm{d}m=\int_{0}^{1}\eta_{M}^{\prime}(0,X^{0}(m))\Phi(0,X^{0}(m))\mathop{}\\!\mathrm{d}m=\int_{\mathbb{R}}\eta_{M}^{\prime}(0,x)\Phi(0,x)\partial_{x}M^{0}(x),$

and integrating by parts we have

$\int_{\mathbb{R}}\eta_{M}^{\prime}(0,x)\Phi(0,x)\partial_{x}M^{0}(x)=-\int_{\mathbb{R}}(\lvert
M^{0}(x)-k\rvert-\lvert k\rvert)\varphi(0,x)\mathop{}\\!\mathrm{d}x.$

Combining the above relations we arrive at the desired inequality

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\varphi_{t}\lvert
M-k\rvert+\varphi_{x}\operatorname*{sgn}(M-k)(\mathcal{U}(t,M)-\mathcal{U}(t,k))\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\lvert
k\rvert\int_{\mathbb{R}}\varphi(0,x)\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad=-\int_{0}^{T}\int_{0}^{1}\left(\Phi_{t}\eta_{M}^{\prime}+\Phi_{x}q_{M}^{\prime}\right)(t,X(t,m))\mathop{}\\!\mathrm{d}m\mathop{}\\!\mathrm{d}t\geq-\int_{\mathbb{R}}\varphi(0,x)\lvert
M(0,x)-k\rvert\mathop{}\\!\mathrm{d}x+\lvert
k\rvert\int_{\mathbb{R}}\varphi(0,x)\mathop{}\\!\mathrm{d}x,$

which we rearrange to find (4.14). ∎

###### Proof of Theorem 5.6.

Assume we have an entropy solution $M(t,x)$ of (4.4); in particular we have
$\rho=\partial_{x}M$ and $\rho
v=\partial_{x}\mathcal{U}(t,M)=\mathcal{U}_{M}^{\prime}(t,x)\partial_{x}M$,
meaning $v(t,x)=\mathcal{U}_{M}^{\prime}(t,x)$.

We then introduce the right-continuous generalized inverse $X(t,m)$ of
$M(t,x)$ through $X_{\\#}\mathfrak{m}=\rho$, as well as the velocity
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}X(t,m)=V(t,m)\coloneqq
v(t,X(t,m))=\mathcal{U}_{M}^{\prime}(t,X(t,m))$ for $t\geq 0$. Recall from
Lemma 5.1 that
$\mathcal{U}^{\prime}_{M}(t,X(t,m))=\operatorname{P}_{\mathscr{H}_{X(t)}}U(t)$.
At any time $t$, the generalized inverse $X(t,m)$ must be a nondecreasing
function of $m$, meaning $X(t,\cdot)\in\mathscr{K}$, where we recall the
convex cone (3.4). Moreover, $X$ is a locally Lipschitz curve
$X\colon[0,T]\to\mathscr{K}$ due to (4.24).

We must now show that $U(t)-\operatorname{P}_{\mathscr{H}_{X(t)}}U(t)\in
N_{X(t)}\mathscr{K}$ for a.e. $t$. However, since $M$ is an entropy solution,