On a novel gradient flow structure for the aggregation equation

Abstract

The aggregation equation arises naturally in kinetic theory in the study of granular media, and its interpretation as a 2-Wasserstein gradient flow for the nonlocal interaction energy is well-known. Starting from the spatially homogeneous inelastic Boltzmann equation, a formal Taylor expansion reveals a link between this equation and the aggregation equation with an appropriately chosen interaction potential. Inspired by this formal link and the fact that the associated aggregation equation also dissipates the kinetic energy, we present a novel way of interpreting the aggregation equation as a gradient flow, in the sense of curves of maximal slope, of the kinetic energy, rather than the usual interaction energy, with respect to an appropriately constructed transportation metric on the space of probability measures.

1 Introduction

In this work we propose a novel, rigorous interpretation of the one-dimensional aggregation equation

$$\begin{aligned} \partial _t f_t = \partial _v {(}{f_t}\, \partial _v W* {f_t}{)},\quad W(v):=c \left| v\right| ^3, \end{aligned}$$

(1.1)

where the probability measure $f_t$ describes the distribution of velocities of the system at time $t>0$. Here, $c>0$ is some constant to be specified later. Equation (1.1) has been considered in [6, 7, 15, 21] as a kinetic model for the evolution of a granular medium undergoing inelastic collisions. As we shall see in Sect. 1.3, such an equation can, indeed, be formally derived from the inelastic and spatially homogeneous Boltzmann equation.

More recently, Equation (1.1) has been studied as a nonlocal interaction equation with an attractive interaction kernel in, e.g., [8, 14] and references therein, which can be obtained as the mean-field limit of a set of interacting particles, [11], or as a zero inertia limit [20]. In this context, the interaction between individuals is described in terms of their relative positions rather than their relative velocities (i.e., relabelling ‘v’ by ‘x’ in (1.1)). Moreover, it is well-known that the nonlocal interaction equation can be viewed as a 2-Wasserstein gradient flow of the nonlocal interaction energy, [2].

This paper focuses on the kinetic description provided in [7]. We show that (1.1) is a gradient flow of the kinetic energy with respect to a metric that can be understood as a generalisation of the 2-Wasserstein distance, inspired by the approach in [16, 18] and motivated by the formal link with the inelastic Boltzmann equation.

In recent years, gradient flow structures have been proposed for several kinetic equations: for the homogeneous (elastic) Boltzmann equation [18], the linear Boltzmann equation [4], and the homogeneous Landau equation [3, 12]. See also [1] for a different gradient flow description of the inhomogeneous granular medium equation. Recently, the authors of [13] made a connection between the gradient flow structures of the (homogeneous) Boltzmann and Landau equations. These results indicate that an appropriate gradient flow structure can link the inelastic Boltzmann equation and the aggregation equation.

In the remainder of the introduction, we give a formal sketch of the main ideas and the intuition behind our approach, with the inelastic spatially homogeneous Boltzmann equation acting as the starting point of our discussions. We commence by introducing some necessary notation and other preliminary notions in Sect. 1.1. Then, in Sect. 1.2, we discuss the inelastic homogeneous Boltzmann equation. Moreover, we propose a formal gradient flow structure for this equation with the kinetic energy as the natural energy functional. This is important in order to draw the connection with the aggregation equation (1.1), via a formal Taylor expansion which we describe in Sect. 1.3. As a consequence, we can obtain the gradient flow structure of equation (1.1) in Sect. 1.4. We conclude the introduction in Sect. 1.5 with a discussion of the main results and an outline of the rest of the manuscript.

1.1 Notation and preliminaries

We use the notation ${{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ to denote the set ${\{}{(}v,v_*{)}\in {{\mathbb {R}}}^2: v \ne v_*{\}}$. This set often acts as our state space since it is impossible for particles to collide if they move at the same velocity and in the same direction. Furthermore, we denote by $L^p(\Omega ,\mu )$, $p \ge 1$, the Lebesgue spaces on some measure space $(\Omega ,\mu )$ and by $L^p(\Omega )$, $p \ge 1$, the standard Lebesgue spaces when $\Omega$ is a smooth Euclidean subdomain¹ and $\mu$ is the Lebesgue measure. In the same setting, we denote by $C^k(\Omega )$ the space of k-times continuously differentiable real-valued functions on $\Omega$ and $C_c^k(\Omega )$ (resp. $C_0^k(\Omega )$, $C_b^k(\Omega )$) the subspace of $C^k(\Omega )$ functions that are compactly supported (resp. vanishing at infinity, with bounded derivatives up to order k).

We denote by ${{\mathcal {P}}}(\Omega )$ the set of Borel probability measures on $\Omega$, and we write ${{\mathcal {M}}}(\Omega )$ (resp. ${{\mathcal {M}}}^+(\Omega )$) to denote finite (resp. non-negative) Radon measures on $\Omega$, where $\Omega$ is some Euclidean subdomain. Besides, for $p\ge 1$, we denote by

$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}_p(\Omega )&= \Bigg \{ f \in {{\mathcal {P}}}(\Omega ): m_p(f):=\!\int _\Omega |v|^p\!\,\,\text {d}f(v) < \infty \Bigg \}, \\ {{\mathcal {P}}}_p^{\textrm{cm}}(\Omega )&= \Bigg \{ f\in {{\mathcal {P}}}_p(\Omega ): \!\int _\Omega v \!\,\,\text {d}f (v) = 0 \Bigg \}. \end{aligned} \end{aligned}$$

Additionally, will denote by $d_p$, $p\ge 1$, the p-Wasserstein distance, [25]. For two sequences, ${\{}f_n{\}}_n \subset {{\mathcal {P}}}(\Omega )$ and ${\{}U_n{\}}_n \subset {{\mathcal {M}}}(\Omega )$ as well as two elements $f \in {{\mathcal {P}}}(\Omega )$ and $U\in {{\mathcal {M}}}(\Omega )$, we write $f_n \rightarrow f \in {{\mathcal {P}}}(\Omega )$ if, by duality with continuous and bounded functions, $g\in C_b(\Omega )$, there holds

$$\begin{aligned} \int _{\Omega } g \!\,\,\text {d}f_n \rightarrow \int _{\Omega } g \!\,\,\text {d}f , \end{aligned}$$

as $n\rightarrow \infty$. In this case, we say ${\{}f_n{\}}_n$ converges narrowly or weakly to f. Moreover, we write $U_n \rightarrow U$ in ${{\mathcal {M}}}(\Omega )$ if, by duality with continuous functions that vanish at infinity, $g\in C_0(\Omega )$, there holds

$$\begin{aligned} \int _{\Omega } g \!\,\,\text {d}U_n \rightarrow \int _{\Omega } g \!\,\,\text {d}U , \end{aligned}$$

as $n\rightarrow \infty$. When satisfied, we say ${\{}U_n{\}}_n$ converges weakly-$^*$ to U.

Likewise, we write $U_n \rightarrow ^c U$ in ${{\mathcal {M}}}(\Omega )$ if, by duality with continuous functions with compact support, $g\in C_c(\Omega )$, there holds

$$\begin{aligned} \int _{\Omega } g \!\,\,\text {d}U_n \rightarrow \int _{\Omega } g \!\,\,\text {d}U , \end{aligned}$$

as $n\rightarrow \infty$. In this case, the induced topology is the vague topology.

1.2 The inelastic Boltzmann equation & decay of the kinetic energy

We consider the time evolution of the velocity distribution, $f_t$, of a system of particles that undergo inelastic collisions with coefficient of restitution $e \in [0,1)$. Throughout this paper, we shall denote by $v,v_*$, the pre-collisional velocities and by $v',v'_*$, the post-collisional velocities, respectively, which can be computed using the following two laws: the reduction of the relative velocity of the particles due to the inelastic collisions

$$\begin{aligned} v'-v_*'&= -e(v-v_*), \end{aligned}$$

and the conservation of momentum, i.e.,

$$\begin{aligned} v' + v'_*&= v+ v_*. \end{aligned}$$

The limit $e\rightarrow 1$ corresponds to elastic collisions, while $e=0$ models sticky collisions. Solving for the post-collisional velocities, $v',v'_*$, we obtain

$$\begin{aligned} \left\{ \begin{array}{rl} v' \!\!\! &{}= \dfrac{1- e}{2}v + \dfrac{1+ e}{2}v_*, \\[1em] v'_* \!\!\! &{}= \dfrac{1+ e}{2}v + \dfrac{1- e}{2}v_*. \end{array} \right. \end{aligned}$$

We now define the weak form of the Boltzmann equation. We refer to the appendix for a formal derivation of the equation from a simple gain-loss argument.

Definition 1.1

(Nonlocal gradient and weak form for the inelastic Boltzmann equation) We define the nonlocal gradient of a function $\varphi \in C^0({{\mathbb {R}}})$ as follows

$$\begin{aligned} {\overline{\nabla }} \varphi (v,v_*) := \frac{\varphi (v')+ \varphi (v'_*) -\varphi (v) -\varphi (v_*)}{\left| v-v_*\right| ^2}(v-v_*), \end{aligned}$$

(1.2)

for $(v, v_*)\in {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$, i.e., ${\overline{\nabla }} \varphi : {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\rightarrow {{\mathbb {R}}}$. A curve $f: [0,T] \rightarrow {{\mathcal {P}}}({{\mathbb {R}}})$ is a weak solution of the inelastic Boltzmann equation with collision kernel $\sigma =\sigma (|v|)$ provided that for all $\varphi \in C_c^\infty ({{\mathbb {R}}})$ and almost all $t\in [0,T]$, it holds

$$\begin{aligned} {\langle }\varphi , \partial _t {f_t}{\rangle } = \frac{1}{2}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma ( \left| v-v_*\right| ) {(}v-v_*{)}{\overline{\nabla }}\varphi (v,v_*) \!\,\,\text {d}{f_t}(v) \!\,\,\text {d}{f_t}(v_*). \end{aligned}$$

(1.3)

The choice of (1.2) is made such that it has the units of inverse velocity and such that it generalises to higher dimensions in a straightforward manner. By considering its negative adjoint in the weighted space $L^2({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup },\sigma )$, we obtain a divergence acting on nonlocal fluxes $U\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ such that

$$\begin{aligned} \int \varphi (v) \!\,\,\text {d}({{\overline{\nabla }}} \cdot U)(v) = -\frac{1}{2} \iint \sigma (\left| v-v_*\right| ) {{\overline{\nabla }}} \varphi (v,v_*) \!\,\,\text {d}{U(v,v_*)} . \end{aligned}$$

(1.4)

In this sense, we obtain that the weak form (1.3) can be cast into the form of a nonlocal continuity equation

$$\begin{aligned} \partial _t f_t + {{\overline{\nabla }}} \cdot U_t = 0, \end{aligned}$$

where the associated flux, $U_t$, is given by

$$\begin{aligned} \!\,\,\text {d}U_t(v,v_*) = (v-v_*) \!\,\,\text {d}f_t(v) \!\,\,\text {d}f_t(v_*) . \end{aligned}$$

(1.5)

1.2.1 Decay of the kinetic energy

For a given velocity distribution, f, we define the kinetic energy as follows

$$\begin{aligned} {{\mathcal {E}}}(f) := \frac{1}{2} \int _{{\mathbb {R}}}v^2 \!\,\,\text {d}f(v) \, . \end{aligned}$$

(1.6)

Due to the fact that collisions between particles are inelastic, one would expect that the post-collisional kinetic energy is less than the pre-collisional energy. In fact, one can see that the post-collisional kinetic energy is related to the pre-collisional kinetic energy via

$$\begin{aligned} \left| v'\right| ^2 +\left| v'_*\right| ^2 = \frac{1+ e^2}{2}\big (\left| v\right| ^2 + \left| v_*\right| ^2\big ) + \big (1-e^2\big )v v_*, \end{aligned}$$

(1.7)

for $e \in [0,1)$. We now use the weak formulation, (1.3), to show that the kinetic energy decays along a solution of the inelastic Boltzmann equation. By noting that, $\frac{\delta {{\mathcal {E}}}}{\delta f} = \frac{1}{2} v^2$, we use (1.7) to obtain

$$\begin{aligned} (v-v_*){{\overline{\nabla }}}\left( { \left| v\right| ^2}\right)&= \left| v'\right| ^2 +\left| v'_*\right| ^2 - \left| v\right| ^2 - \left| v_*\right| ^2 = - \frac{1-e^2}{2} \left| v-v_*\right| ^2, \end{aligned}$$

which, upon substituting $\varphi =\frac{\delta {{\mathcal {E}}}}{\delta f}$ into (1.3), yields

$$\begin{aligned} \frac{\!\,\,\text {d}{}}{\!\,\,\text {d}{t}}{{\mathcal {E}}}({f_t})&= -\frac{1-e^2}{8}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma ( \left| v-v_*\right| )\left| v-v_*\right| ^2 \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*) \le 0 \, . \end{aligned}$$

For the specific case of Maxwell molecules, that is $\sigma (|a|)=|a|$, we have

$$\begin{aligned} \frac{\!\,\,\text {d}{}}{\!\,\,\text {d}{t}}{{\mathcal {E}}}({f_t})&= -\frac{1-e^2}{8}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| v-v_*\right| ^3 \!\,\,\text {d}({f_t}\otimes {f_t})(v,v_*). \end{aligned}$$

(1.8)

Remark 1.2

From (1.8), we can heuristically obtain Haff’s law by considering the evolution of a family of local equilibria $m_\eta$ such that ${{\mathcal {E}}}(m_\eta )= \eta$. One then obtains an equation for $\eta$ of the form

$$\begin{aligned} \frac{\mathop {}\!\text {d}^{} }{\mathop {}\!\text {d} t^{}}\eta (t) = \frac{\mathop {}\!\text {d}^{} }{\mathop {}\!\text {d} t^{}} {{\mathcal {E}}}(m_{\eta (t)})\lesssim -\eta (t)^{\frac{3}{2}}, \end{aligned}$$

which leads to

$$\begin{aligned} \eta (t) \lesssim \frac{1}{1+t^2}. \end{aligned}$$

Hence, the solutions converge on an algebraic time scale to a Dirac measure. A rigorous proof of this convergence can be found in [22, 23]. From the decay of the kinetic energy in (1.8), it becomes, indeed, clear that the system loses kinetic energy in the long run, i.e., it cools down. This leads to the formation of a Dirac measure as time goes to infinity, which is at the same time a minimiser of (1.6) in the space of probability measures with a fixed centre of mass. Hence, the only stationary states of the system are Dirac measures.

1.2.2 Identification of a novel gradient structure

From our analysis we know that the system is driven by the kinetic energy (1.6), whose first variation $\frac{\delta {{\mathcal {E}}}}{\delta f} = \frac{v^2}{2}$ can be identified in the flux (1.5) by re-expressing it as

$$\begin{aligned} \!\,\,\text {d}U_t(v,v_*) =- \frac{4}{1-e^2} {{\overline{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*) \!\,\,\text {d}{(} f_t \otimes f_t{)}(v,v_*) \, . \end{aligned}$$

In this way, we can reformulate the homogeneous inelastic Boltzmann equation in its weak form (1.3) as

$$\begin{aligned} {\langle }\varphi , \partial _t f_t {\rangle } =- \frac{2}{1-e^2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma (|v-v_*|) {{\overline{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*) {{\overline{\nabla }}}\varphi (v,v_*) \!\,\,\text {d}(f_t \otimes f_t) (v, v_*), \end{aligned}$$

which by the definition of the divergence from (1.4) becomes

$$\begin{aligned} \partial _t f_t = \frac{4}{1-e^2} {{\overline{\nabla }}} \cdot \bigg ({ f \otimes f \ {{\overline{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f}}\bigg ), \end{aligned}$$

whence we can identify the kinetic relation, also called Onsager operator, between forces² and fluxes as

$$\begin{aligned} K_f \psi = -\frac{4}{1-e^2}{{\overline{\nabla }}} \cdot {(} f \otimes f \ {{\overline{\nabla }}} \psi {)}, \end{aligned}$$

which in the weak form becomes

$$\begin{aligned} {\langle }\varphi , K_f \psi {\rangle } =\frac{2}{1-e^2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma (\left| v-v_*\right| ) {\overline{\nabla }} \varphi {\overline{\nabla }} \psi \!\,\,\text {d}(f \otimes f)(v, v_*). \end{aligned}$$

(1.9)

Remark 1.3

(The Onsager operator for elastic Boltzmann and physical kernels) In particular, we observe that $K_f$ is only defined for $e\in [0,1)$ and becomes meaningless in the elastic limit $e\rightarrow 1$. Nevertheless, it has structural similarities to the Onsager operator introduced in [18] for the homogeneous elastic Boltzmann equation.

1.3 Formal derivation of the aggregation equation

This section is dedicated to a formal derivation of the aggregation equation from the inelastic Boltzmann equation. To this end, we consider the weak formulation of the inelastic Boltzmann equation, (1.3). For $v'$ close to $v_*$ and $v'_*$ close to v, i.e., for almost elastic collisions, i.e., $e\approx 1$, by (1.2), we have

$$\begin{aligned} {\overline{\nabla }} \varphi \sim \frac{1- e}{2}{(}\varphi '(v_*) - \varphi '(v){)} + {\mathcal {O}}\bigg ({\left| \frac{1-e}{2}\right| ^2 |v-v_*|}\bigg ). \end{aligned}$$

(1.10)

Substituting this into (1.3) and disregarding all higher order terms, we obtain

$$\begin{aligned} {\langle }\varphi , \partial _t f_t{\rangle }= \bigg ({\frac{1- e}{4}}\bigg ) \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma ( \left| v-v_*\right| )(v-v_*)( \varphi '(v_*) - \varphi '(v)) \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*). \end{aligned}$$

(1.11)

Letting the function $\Sigma :{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ be such that $\partial _v\Sigma (v)=\sigma (|v|)\, v$, the above equation simplifies to

$$\begin{aligned} {\langle }\varphi , \partial _t f_t{\rangle }= \bigg ({\frac{1- e}{4}}\bigg ) \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \partial _v \Sigma (v-v_*) (\varphi '(v_*) - \varphi '(v)) \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*). \end{aligned}$$

(1.12)

Unsymmetrising in v and $v_*$ yields

$$\begin{aligned} \begin{aligned} {\langle }\varphi , \partial _t f_t{\rangle }&= - \bigg ({\frac{1- e}{2}}\bigg ) \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \partial _v \Sigma (v-v_*) \varphi '(v) \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*)\\&= - \int _{{{\mathbb {R}}}} \varphi '(v) \int _{{{\mathbb {R}}}}\bigg ({\frac{1- e}{2}}\bigg ) \partial _v \Sigma (v-v_*) \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*). \end{aligned} \end{aligned}$$

(1.13)

Choosing

$$\begin{aligned} W(v)=\frac{1-e}{2} \Sigma (v), \end{aligned}$$

it is immediate to see that (1.13) is the weak formulation of the aggregation equation

$$\begin{aligned} \partial _t f_t = \partial _v {(}{f_t}\, \partial _v W* {f_t}{)}. \end{aligned}$$

(1.14)

Note that for the physical kernel, $\sigma (\left| v\right| )=\left| v\right|$, the interaction potential for the aggregation equation becomes

$$\begin{aligned} W: {{\mathbb {R}}}&\rightarrow {{\mathbb {R}}}_+,\quad v \mapsto \frac{1-e}{6} \left| v\right| ^3 \, . \end{aligned}$$

(1.15)

Furthermore, we stress that this expansion relies on the fact $e<1$ as otherwise the evolution is trivial, i.e., ${\langle }\varphi ,\partial _t f_t{\rangle }=0$ in (1.11).

1.4 Formal gradient flow structure of the aggregation equation

As previously mentioned, the aggregation equation can be cast into a 2-Wasserstein gradient flow framework (cf. e.g. [2, 14]) for the nonlocal interaction energy

$$\begin{aligned} {{\mathcal {W}}}(f) = \frac{1}{2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} W(v-v_*) \!\,\,\text {d}(f\otimes f)(v, v_*) \, , \end{aligned}$$

which is dissipated along the flow, (1.14), in such a way that

$$\begin{aligned} \frac{\!\,\,\text {d}}{\!\,\,\text {d}{t}}{{\mathcal {W}}}({f_t}) = - \int _{{\mathbb {R}}}\left| \partial _v W * {f_t}\right| ^2 \!\,\,\text {d}{f_t}(v). \end{aligned}$$

As demonstrated in Sect. 1.3, the aggregation equation can be formally derived from the inelastic Boltzmann equation. It is therefore not unreasonable to expect that the aggregation equation is also a gradient flow for the kinetic energy defined in (1.6). To this end, we study its dissipation along the flow of equation (1.12). For convenience, we introduce the notation

$$\begin{aligned} \sigma _e(|v-v_*|) := \frac{1-e}{4} |v-v_*|, \end{aligned}$$

(1.16)

which we shall use throughout this work. Setting $\varphi (v)=v^2/2$ we have

$$\begin{aligned} \frac{\!\,\,\text {d}}{\!\,\,\text {d}{t}}{{\mathcal {E}}}({f_t})= - \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| v-v_*\right| ^2 \sigma _e(v-v_*) \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*) =: - {{\mathcal {D}}}(f_t) \le 0 \, , \end{aligned}$$

(1.17)

where ${{\mathcal {D}}}: {{\mathcal {P}}}({{\mathbb {R}}}) \rightarrow [0,+\infty ]$ is the so-called dissipation functional. Thus, the kinetic energy is a Lyapunov function for the dynamics of the aggregation equation.

The preceding computation reveals an energy-dissipation structure of the aggregation equation with respect to the kinetic energy, cf. (1.17), which suggests there may exist an appropriate metric for which (1.14) is a gradient flow of ${{\mathcal {E}}}(f)$. Next, we identify the Onsager operator for this metric and, using the new formalism, derive the weak form of the aggregation equation. More precisely, (1.11) becomes

$$\begin{aligned} - {\langle } \varphi , \partial _t f_t{\rangle }&=-\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(\left| v-v_*\right| )(v-v_*){(}\varphi '(v_*) - \varphi '(v){)} \!\,\,\text {d}({f_t}\otimes {f_t})(v,v_*)\nonumber \\&=\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(\left| v-v_*\right| )\left( {\partial _v \frac{\delta {{\mathcal {E}}}}{\delta f}(v_*) - \partial _v \frac{\delta {{\mathcal {E}}}}{\delta f}(v)} \right) {\varphi '(v_*) - \varphi '(v)} \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*) \nonumber \\&={\langle }\varphi , K^\textrm{agg}_{{f_t}} \textrm{D}{{\mathcal {E}}}{\rangle }, \end{aligned}$$

(1.18)

where $\sigma _e$ is as in (1.16). Then, we can read off the appropriate Onsager operator in its weak form

$$\begin{aligned} {\langle }\varphi , K^\textrm{agg}_{f} \psi {\rangle } \!=\! \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) {(}\varphi '(v_*) - \varphi '(v){)} \, {(}\psi '(v_*) - \psi '(v){)}\!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*) \, , \end{aligned}$$

(1.19)

where $f\in {{\mathcal {P}}}({{\mathbb {R}}})$, $\varphi \in C_c^1({{\mathbb {R}}})$ is a test function, and $\psi \in C_c^1({{\mathbb {R}}})$ a driving vector field.

By virtue of (1.19), we note that the Onsager operator induces a positive-definite (${\langle }\varphi , K_f^\textrm{agg} \varphi {\rangle } \ge 0$), bilinear form which is structurally similar to the operator in (1.9). To make this connection more evident, we rewrite the expression in (1.19) in terms of the gradient defined in the following definition, ${{\widetilde{\nabla }}}$. The similarity with the Onsager operator of Sect. 1.2 is in particular seen since, up to a multiplicative constant, one can be obtained from the other by replacing ${\widetilde{\nabla }}$ by ${\overline{\nabla }}$ or vice-versa, cf.  (1.10), i.e., ${{\overline{\nabla }}} \varphi \approx \frac{1-e}{2} {{\widetilde{\nabla }}} \varphi$.

Definition 1.4

(Nonlocal-local gradient) For any function $\varphi \in C^1({{\mathbb {R}}})$ we define its nonlocal-local gradient ${\widetilde{\nabla }} \varphi : {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\rightarrow {{\mathbb {R}}}$ by

$$\begin{aligned} {\widetilde{\nabla }}\varphi (v,v_*) =\varphi '(v_*)-\varphi '(v), \qquad \text {for all } (v,v_*)\in {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }. \end{aligned}$$

(1.20)

Using this definition, we revisit (1.19), which now reads

$$\begin{aligned} {\langle }\varphi , K^{\textrm{agg}}_{f} \psi {\rangle } = \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) {{\widetilde{\nabla }}} \varphi (v,v_*) \, {{\widetilde{\nabla }}} \psi (v,v_*) \!\,\,\text {d}(f \otimes f)(v, v_*). \end{aligned}$$

(1.21)

Based on this definition, (1.17) can be written as

$$\begin{aligned} \frac{\!\,\,\text {d}}{\!\,\,\text {d}{t}}{{\mathcal {E}}}({f_t})= - \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {{\widetilde{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f}\right| ^2 \sigma _e(v-v_*) \!\,\,\text {d}({f_t}\otimes {f_t})(v, v_*) =: - {{\mathcal {D}}}(f_t) \le 0. \end{aligned}$$

(1.22)

Remark 1.5

(Connection to graphs) From Definition 1.4, we can read a continuous graph structure $({{\mathbb {R}}},{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$, where ${{\mathbb {R}}}$ is the set of vertices and ${{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ that of edges, equipped with an operator ${{\widetilde{\nabla }}}:C^1({{\mathbb {R}}})\rightarrow C({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ connecting test functions on vertices to test functions on edges. This gives rise to the negative dual operator, which we interpret as a divergence ${{\widetilde{\nabla }}}\cdot : {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })\rightarrow {{\mathcal {M}}}({{\mathbb {R}}})$ connecting a flux on the edge set ${{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ to an infinitesimal change of state, i.e., a tangential direction (see Definition 2.1). Moreover, note that the driving force field ${{\widetilde{\nabla }}}\delta _f {{\mathcal {E}}}(v,v_*)$ is in our case not just a difference of potential values at $\delta _f{{\mathcal {E}}}(v)$ and $\delta _f{{\mathcal {E}}}(v_*)$, as it is the case for simple graph gradients (see e.g. [19]), but rather a difference of rates $(\delta _f{{\mathcal {E}}})'$. It is in this sense that ${{\widetilde{\nabla }}}$ is nonlocal-local.

1.5 Outline and results

In this paper, we show that the kinetic energy (1.6) is not merely a Lyapunov functional for the aggregation equation as was shown in (1.22). Indeed, the aggregation equation can be cast into a rigorous metric gradient flow setting where a dynamical transport cost induces the metric in the spirit of [5, 16], and the kinetic energy acts as the driving energy functional.

The variational description we propose provides a promising setting to make rigorous the link with the inelastic spatially homogeneous Boltzmann equation, i.e., to rigorously derive the aggregation of particles from the inelastic spatially homogeneous Boltzmann equation, as was formally shown in [7]. This investigation is kept for future work, along with an extension of our results to more general and singular collision kernels, as well as to higher dimensions, following, e.g., [24].

We start by introducing a generalised notion of the continuity equation based on the aforementioned nonlocal-local operators, ${{\widetilde{\nabla }}}$, and its formal negative adjoint, ${{\widetilde{\nabla }}} \cdot$ (cf. Definition 2.1). This consists of a pair ${\{}(f_t, U_t){\}}_{t\in [0,T]}\subset {{\mathcal {P}}}({{\mathbb {R}}})\times {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ satisfying, in a suitable measure-valued sense (see Definition 2.2), the equation

$$\begin{aligned} \partial _t f_t +{\widetilde{\nabla }}\cdot U_t=0, \qquad \text{ on } [0,T]\times {{\mathbb {R}}}. \end{aligned}$$

(CE)

Using the definition of the Onsager operator in (1.21), we then introduce an action-density functional, ${{\mathcal {A}}}:{{\mathcal {P}}}({{\mathbb {R}}})\times {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })\rightarrow [0,\infty ]$, which gives rise to a dynamical transport cost, $d_{{\mathcal {A}}}(\mu _0,\mu _1)$, by minimising the total action of a curve ${\{}(f_t, U_t){\}}_{t\in [0,1]}$ connecting two measures $\mu _0,\mu _1\in {{\mathcal {P}}}({{\mathbb {R}}})$ and satisfying (CE), cf. Theorem 2.19.

Moreover, in this metric setting, we are able to provide a characterisation of weak solutions to the aggregation equation in the form (1.18) as curves of maximal slope. To this end, we define along any curve ${\{}(f_t, U_t){\}}_{t\in [0,T]}$ of finite action staisfying (CE) the so-called De Giorgi functional

$$\begin{aligned} {{\mathcal {G}}}_T(f) = {{\mathcal {E}}}(f_T)- {{\mathcal {E}}}(f_0) + \frac{1}{2} \int _0^T {{\mathcal {A}}}(f_t, U_t) \!\,\,\text {d}t + \frac{1}{2} \int _0^T {{\mathcal {D}}}(f_t) \!\,\,\text {d}t \ge 0 \end{aligned}$$

where the non-negativity is the consequence of a suitable chain rule (see Lemma 3.3). The weak solutions to (1.18) are found to be elements of the zero locus of the De Giorgi functional, i.e., ${{\mathcal {G}}}_T(f)=0$. Conversely, any element of the zero locus of the De Giorgi functional is necessarily a weak solution to the aggregation equation (see Theorem 3.6). Finally, we prove that curves of maximal slope are stable with respect to convergence of the initial measures $\mu _0^n \rightarrow \mu _0$ such that ${{\mathcal {E}}}(\mu _0^n)\rightarrow {{\mathcal {E}}}(\mu _0)$ (cf. Theorem 3.8). This allows us to prove the existence of gradient flow solutions based on a finite-dimensional particle approximation (see Theorem 3.9).

2 The nonlocal-local continuity equation and the collision metric

2.1 A nonlocal-local continuity equation

For the subsequent analysis, we study arbitrary curves, ${\{}f_t{\}}_{t \in [0,T]} \subset {{\mathcal {P}}}({{\mathbb {R}}})$, in the set of probability measures induced by a driving field, $\psi _t$, connecting two probability measures $f_0,f_T \in {{\mathcal {P}}}({{\mathbb {R}}})$. By (1.19) and (1.21), we have

$$\begin{aligned} {\langle }\varphi ,\partial _t {f_t}{\rangle }&= -{\langle }\varphi , K^\textrm{agg}_{{f_t}} \psi _t{\rangle } \\&=- \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(v) {f_t}(v_*)\sigma _e(|v-v_*|) \widetilde{\nabla }\varphi (v,v_*) {\widetilde{\nabla }}\psi _t(v,v_*) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*}, \end{aligned}$$

which we take as the basis for the definition of a nonlocal-local continuity equation (CE). To this end, we first define an appropriate divergence as the formal adjoint of the nonlocal-local gradient from Definition 1.4.

Definition 2.1

(Nonlocal-local divergence) For any $U\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$, its nonlocal-local divergence ${\widetilde{\nabla }}\cdot U \in {{\mathcal {M}}}({{\mathbb {R}}})$ is defined as negative dual with weight $\sigma _e$ of ${\widetilde{\nabla }}$, i.e., for all $\varphi \in C^1_c({{\mathbb {R}}})$ it holds

$$\begin{aligned} \int _{{\mathbb {R}}}\varphi (v) \!\,\,\text {d}{({\widetilde{\nabla }} \cdot U)}(v)&= -\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}{\widetilde{\nabla }}\varphi (v,v_*)\sigma _e(|v-v_*|)\!\,\,\text {d}U(v,v_*)\\&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \varphi '(v) \sigma _e(|v-v_*|) {(} \!\,\,\text {d}U(v,v_*) - \!\,\,\text {d}U(v_*,v){)} . \end{aligned}$$

Now, we can define the nonlocal-local continuity equation.

Definition 2.2

(Weak solution to (CE)) A pair $\{({f_t},U_t)\}_{t\in [0,T]}$ is called (weak) solution of the nonlocal-local continuity equation (CE) on [0, T] if there exist two families of measures $\{f_t\}_{t\in [0,T]}\subset {{\mathcal {P}}}({{\mathbb {R}}})$ and $\{U_t\}_{t\in [0,T]}\subset {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ such that the map $t \mapsto f_t$ (resp. $t \mapsto U_t$) is measurable with respect to the weak-$^*$ topology on finite Radon measures and they satisfy the following integrability condition

$$\begin{aligned} \int _0^T\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}\sigma _e(|v-v_*|)\!\,\,\text {d}|U_t|(v,v_*)\!\,\,\text {d}{t}<+\infty , \end{aligned}$$

(2.1)

along with the weak form of the nonlocal-local continuity equation (CE) for every $C_c^1((0,T)\times {{\mathbb {R}}})$

$$\begin{aligned} \int _0^T \int _{{\mathbb {R}}}\partial _t \varphi _t(v) \!\,\,\text {d}{f_t}(v) \!\,\,\text {d}{t} + \int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) {\widetilde{\nabla }}\varphi _t(v,v_*) \!\,\,\text {d}{U_t(v, v_*)}\!\,\,\text {d}{t} =0. \end{aligned}$$

(2.2)

We denote by ${\textrm{CE}}_T(\mu _0)$ the class of solutions ${\{}(f_t,U_t){\}}_{t\in [0,T]}$ of the nonlocal-local continuity equation on [0, T] starting at $\mu _0$, and we write ${\textrm{CE}}_T(\mu _0,\mu _T)$ for solutions connecting $\mu _0$ with $\mu _T$. We will drop the subscript T whenever $T=1$.

Note that the second term in the weak formulation (2.2) of the (CE) is well-defined under the integrability condition (2.1), since $|{\widetilde{\nabla }}\varphi _t(v,v_*)|\le 2\Vert \partial _v \varphi _t(\cdot )\Vert _{C^0({{\mathbb {R}}})}$, for all $t\in [0,T]$.

Remark 2.3

(Strong form of (CE)) Note that, for $U_t\ll f_t\otimes f_t$ and $f_t\ll \!\,\,\text {d}v$ for any $t\in [0,T]$, after an integration by parts in v of (2.2), we arrive at

$$\begin{aligned} {\langle }\varphi ,\partial _t {f_t}{\rangle }&=- \int _{{\mathbb {R}}}\varphi (v)2\, \partial _v \bigg ({ \int _{{\mathbb {R}}}{f_t}(v) {f_t}(v_*)\sigma _e(|v-v_*|) \widetilde{\nabla }\psi _t(v,v_*) \!\,\,\text {d}{v_*}}\bigg ) \!\,\,\text {d}{v} . \end{aligned}$$

(2.3)

From (2.3), we have that a couple, $({f_t}, \psi _t)$, consisting of the curve, ${f_t}$, and the driving field, $\psi _t$, satisfies the strong form of the nonlocal-local continuity equation provided that

$$\begin{aligned} \partial _t {f_t}+ 2\partial _v \int _{{\mathbb {R}}}{f_t}(v){f_t}(v_*)\sigma _e{(}\left| v-v_*\right| {)} {{\widetilde{\nabla }}} \psi _t \!\,\,\text {d}{v_*} = 0, \end{aligned}$$

where ${{\widetilde{\nabla }}} \psi = \psi '(v_*) - \psi '(v)$, as in Definition 1.4. In the following, we will always use the weak formulation in the sense of Definition 2.2.

As a matter of fact, the integrability condition, (2.1), allows us to infer additional time regularity in that we can prove the existence of a continuous representative for weak solutions to the nonlocal-local continuity equation as stated in the following proposition.

Proposition 2.4

(Continuous representative) Let $\{(f_t,U_t)\}_{t\in [0,T]}$ be a solution to the (CE) in the sense of Definition 2.2. Then, there exists a narrowly continuous curve $[0,T]\ni t\mapsto {\widetilde{f}}_t\in {{\mathcal {P}}}({{\mathbb {R}}})$ such that $f_t={\widetilde{f}}_t$ for ${\mathcal {L}}^1$-a.e. $t\in (0,T)$ and, for any test function $\varphi \in C_c^1({{\mathbb {R}}})$, there holds

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int \varphi (v) \textrm{d} {{\widetilde{f}}}_t(v) = \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {{\widetilde{\nabla }}} \varphi (v,v_*) \sigma _e(|v-v_*|) \textrm{d} U_t(v, v_*). \end{aligned}$$

(2.4)

Proof

Let $\{(f_t,U_t)\}_{t\in [0,T]}$ be a solution in the sense of Definition 2.2 and $\varphi \in C_c^1((0,T)\times {{\mathbb {R}}})$ be a test function. Following the argument of [2, Lemma 8.1.2] or [17, Lemma 3.1] by setting $V(t):=\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}\sigma _e(|v-v_*|) \!\,\,\text {d}{| U_t|}(v,v_*)$, we arrive at

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}}&\varphi _{t_2}(v)\!\,\,\text {d}{{\widetilde{f}}_{t_2}(v)}-\int _{{\mathbb {R}}}\varphi _{t_1}(v)\!\,\,\text {d}{{\widetilde{f}}_{t_1}(v)}\\&=\int _{t_1}^{t_2} \int _{{\mathbb {R}}}\partial _t \varphi _t(v) \!\,\,\text {d}{f_t}(v) \!\,\,\text {d}{t} + \int _{t_1}^{t_2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }}\varphi _t(v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U_t}(v,v_*)\!\,\,\text {d}{t}, \end{aligned} \end{aligned}$$

(2.5)

for any $0\le t_1 < t_2 \le T$. In order to obtain the expression claimed in the statement of the proposition, let us choose a sequence of test functions that are in product form and whose time-component is an approximation of the indicator on an interval $(t_1,t_2)$ with $0<t_1<t_2<T$, i.e.,

$$\begin{aligned} \varphi ^\varepsilon (t,v) = \psi ^\varepsilon (t) \phi (v), \end{aligned}$$

where ${{\,\textrm{supp}\,}}\psi ^\varepsilon = [t_1-\varepsilon ,t_2+\varepsilon ]$ such that $\psi ^\varepsilon (t)= 1$ for $t\in [t_1,t_2]$ and $\psi ^\varepsilon \in C_c^1([0,T]), \phi \in C_c^1({{\mathbb {R}}})$. We may, for instance, choose the following approximating sequence

$$\begin{aligned} \psi ^\varepsilon (t) = \left\{ \begin{array}{ll} 0, &{} t\in (-\infty ,t_1- \varepsilon ), \\ \varepsilon ^{-1}(t-t_1+\varepsilon ), &{} t\in (t_1 - \varepsilon , t_1),\\ 1, &{} t\in (t_1, t_2), \\ \varepsilon ^{-1}(t_2 + \varepsilon - t), &{} t\in (t_2, t_2 + \varepsilon ), \\ 0, &{} t\in (t_2 + \varepsilon , \infty ). \end{array} \right. \end{aligned}$$

Upon substituting $\varphi _\varepsilon (t,x)$ into  (2.5), we obtain

$$\begin{aligned} \begin{aligned} \int _{t_1-\varepsilon }^{t_2+\varepsilon } \int _{{\mathbb {R}}}\partial _t \varphi ^\varepsilon (t,v) \!\,\,\text {d}{f_t}(v) \!\,\,\text {d}{t} + \int _{t_1-\varepsilon }^{t_2+\varepsilon } \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }}\varphi ^\varepsilon (t,v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U_t}(v,v_*)\!\,\,\text {d}{t}=0, \end{aligned} \end{aligned}$$

whence

$$\begin{aligned}&\left| \frac{1}{\varepsilon }\left( {\int _{t_1-\varepsilon }^{t_1}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \phi (v) \!\,\,\text {d}{f_t^n}(v)\!\,\,\text {d}{t} - \int _{t_2}^{t_2+\varepsilon }\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \phi (v) \!\,\,\text {d}{f_t^n}(v)\!\,\,\text {d}{t}}\right) \right| \\&\quad \le \int _{t_1 - \varepsilon }^{t_2+ \varepsilon } \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {\widetilde{\nabla }} \phi (v,v_*)\right| \sigma _e(|v-v_*|) \!\,\,\text {d}{\left| U_t^n\right| }(v,v_*)\!\,\,\text {d}{t}\\&\quad \le 2\left\| \phi '\right\| _{C^0({{\mathbb {R}}})} \int _{t_1-\varepsilon }^{t_2+ \varepsilon } \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) \!\,\,\text {d}|U_t^n|(v,v_*)\!\,\,\text {d}{t}, \end{aligned}$$

where (2.1) ensures that the right-hand side is $L^1$-integrable which then acts as the modulus of absolute continuity. Letting $\varepsilon \rightarrow 0$, we have

$$\begin{aligned}&\left| \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \phi (v) \!\,\,\text {d}{f_{t_1}^n}(v)\!\,\,\text {d}{t} - \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \phi (v) \!\,\,\text {d}{f_{t_2}^n}(v)\!\,\,\text {d}{t}\right| \\&\qquad \le 2\left\| \phi '\right\| _{C^0({{\mathbb {R}}})} \int _{t_1}^{t_2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) \!\,\,\text {d}|U_t^n|(v,v_*)\!\,\,\text {d}{t}, \end{aligned}$$

implying the narrow continuity of ${{\widetilde{f}}}_t$. $\square$

Remark 2.5

(Extension of test function class) In view of (2.4) and the integrability condition on the flux we can choose $\varphi \in \textrm{Lip}({{\mathbb {R}}})$ as test-function class.

We now show two peculiar properties of the solutions to the nonlocal-local continuity equation.

Proposition 2.6

(Preservation of centre of mass and bounded first moments) Let $f_0\in {{\mathcal {P}}}({{\mathbb {R}}})$ be such that $\int v \!\,\,\text {d}f_0(v)<\infty$. Then, any ${\{}(f_t,U_t){\}}_{t\in [0,T]}\in {\textrm{CE}}_T(f_0)$ preserves the centre of mass, that is for all $t\in [0,T]$ it holds

$$\begin{aligned} \int _{{\mathbb {R}}}v\,\, \textrm{d} f_t(v) = \int _{{\mathbb {R}}}v\,\, \textrm{d} f_0(v). \end{aligned}$$

Likewise, if $f_0\in {{\mathcal {P}}}({{\mathbb {R}}})$ is such that $\int |v|\,\,\textrm{d} f_0(v)<\infty$, then any ${\{}(f_t,U_t){\}}_{t\in [0,T]}\in {\textrm{CE}}_T(f_0)$ satisfies for all $t\in [0,T]$ the bound

$$\begin{aligned} \left| \frac{ \textrm{d}}{\textrm{d}t} \int _{{\mathbb {R}}}|v| \textrm{d} f_t( v)\right| \le 2 \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}\sigma _e(|v-v_*|)\textrm{d} |U_t|(v,v_*). \end{aligned}$$

(2.6)

Proof

Let $R>0$ and let us consider the function $\varphi _R:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ defined as

$$\begin{aligned} \varphi _R(v) = \left\{ \begin{array}{ll} 0, &{} v\in (-\infty , -2R), \\ -2R-v, &{} v\in (-2R, -R),\\ v, &{} v\in (-R, R), \\ 2R-v, &{} v\in (R, 2R) \\ 0, &{} v\in (2R, \infty ). \end{array} \right. \end{aligned}$$

(2.7)

Note that

$$\begin{aligned} \left| {{\widetilde{\nabla }}} \varphi _R(v,v_*)\right| \le 2 , \qquad \text {for almost all}\,\, (v,v_*)\in {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }; \end{aligned}$$

while, at the same time

$$\begin{aligned} \left| {{\widetilde{\nabla }}} \varphi _R(v,v_*)\right| = 0, \qquad \text {for}\,\, (v,v_*) \in [-R,R]^2 . \end{aligned}$$

By Remark 2.5, this is an admissible test function in (2.4) and we can estimate

$$\begin{aligned} \biggl |\int \varphi _R(v) \!\,\,\text {d}f_t(v)&- \int \varphi _R(v) \!\,\,\text {d}f_0(v)\biggr | \\&= \biggl | \int _0^t \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {(} \varphi '_R(v_*) - \varphi '_R(v){)} \sigma _e(|v-v_*|) \!\,\,\text {d}U_s(v, v_*) \!\,\,\text {d}{s} \biggr | \\&\le 2 \int _0^t \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\setminus [-R,R]^2} \sigma _e(|v-v_*|) \!\,\,\text {d}\left| U_s\right| (v,v_*) \!\,\,\text {d}s \rightarrow 0, \quad \text {as } R\rightarrow \infty . \end{aligned}$$

Since $\int \varphi _R(v) \!\,\,\text {d}f_0(v) \rightarrow \int v \!\,\,\text {d}f_0(v) \in {{\mathbb {R}}}$, this concludes the proof of the preservation of the centre of mass. The bound for the first moment, follows from a similar construction, by choosing $|\varphi _R|$, with $\varphi _R$ as in (2.7), to be the test function in (2.4). Indeed, we note $\left| {{\widetilde{\nabla }}}\left| \varphi _R\right| (v,v_*)\right| \le 2$, for almost all $(v,v_*)\in {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$. Hence, for any $0\le s < t\le T$ we have

$$\begin{aligned} \biggl |\int \left| \varphi _R(v)\right| \!\,\,\text {d}f_t(v) - \int \left| \varphi _R(v)\right| \!\,\,\text {d}f_s(v)\biggr | \le 2 \int _s^t \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) \!\,\,\text {d}\left| U_s\right| (v,v_*) \!\,\,\text {d}s. \end{aligned}$$

Then, we obtain the bound (2.6) after dividing by $t-s$, letting $t\rightarrow s$, and noting that $\left| \varphi _R(v)\right| \rightarrow \left| v\right|$ as $R\rightarrow \infty$. $\square$

In the following proposition, we provide a sufficient condition for the existence of a weak solution to the nonlocal-local continuity equation. In particular, any curve that is absolutely continuous with respect to 2-Wasserstein distance, denoted by $d_2$, connecting two probability measures $\mu _0$ and $\mu _T$, and preserving the centre of mass, is also a weak solution to (CE).

Proposition 2.7

(Existence of weak solutions) Let $\mu _0,\mu _T \in {{\mathcal {P}}}({{\mathbb {R}}})$ be with equal centre of mass, i.e., $\int v\,\, \mathrm{{d}} \mu _0(v) = \int v \,\, \mathrm{{d}} \mu _T(v)$, and $d_2(\mu _0,\mu _T)<\infty$. Then, there exists $\{(f_t, U_t)\}_{t\in [0,T]} \in {\textrm{CE}}_T(\mu _0, \mu _T)$.

Proof

Since $d_2(\mu _0,\mu _T)<\infty$, there exists an absolutely continuous curve $f_t: [0,T] \rightarrow {{\mathcal {P}}}({{\mathbb {R}}})$ connecting $\mu _0$ and $\mu _T$ preserving the centre of mass and a vector field $V \in {L}^2(0,T; {L}^2({{\mathbb {R}}}, \!\,\,\text {d}{f_t}))$ such that the flux $\,\,\text {d}{C_t} =V_t \!\,\,\text {d}{f_t}$ satisfies for a.e. $t\in [0,T]$

$$\begin{aligned} \frac{\!\,\,\text {d}}{\!\,\,\text {d}t} \int _{{\mathbb {R}}}\varphi (v) \!\,\,\text {d}{f_t}(v) =\int _{{{\mathbb {R}}}} \partial _v \varphi (v) \!\,\,\text {d}{C_t}(v) , \end{aligned}$$

for all $\varphi \in C_c^1({{\mathbb {R}}})$. Note that we may simply take the 2-Wasserstein geodesic as such a curve. By a similar argument as in the proof of Proposition 2.6 using the test-function (2.7), from the preservation of the centre of mass we obtain that $C_t$ has mean zero, that is for a.e. $t\in [0,T]$ it holds $\int _{{{\mathbb {R}}}} \!\,\,\text {d}C_t = 0$. The well-posedness of the weak form follows by noting that

$$\begin{aligned} \int _0^T \int _{{\mathbb {R}}}\!\,\,\text {d}{|C_t|}(v)\!\,\,\text {d}{t} = \int _0^T \int _{{\mathbb {R}}}|V_t| \!\,\,\text {d}f_t(v) \!\,\,\text {d}{t} \le T^{\frac{1}{2}} \left\| V\right\| _{{L}^2(0,T; L^2({{\mathbb {R}}}, \!\,\,\text {d}f_t))} < \infty . \end{aligned}$$

(2.8)

We define for all $t\in [0,T]$ the flux $U_t \in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ by

$$\begin{aligned} \!\,\,\text {d}{U_t}(v,v_*):= \frac{1}{2\sigma _e(\left| v-v_*\right| )}{(} \!\,\,\text {d}{f_t}(v) \!\,\,\text {d}{C_t}(v_*) -\!\,\,\text {d}{C_t}(v) \!\,\,\text {d}{f_t}(v_*){)}. \end{aligned}$$

We can check that the resulting pair satisfies $(f_t, U_t)_{t\in [0,T]} \in {\textrm{CE}}_T(\mu _0,\mu _T)$. First, we check the weak form (2.4) for which we take any $\varphi \in C_c^1({{\mathbb {R}}})$ and obtain

$$\begin{aligned} \frac{\!\,\,\text {d}}{\!\,\,\text {d}{t}}\int _{{\mathbb {R}}}\varphi (v) \!\,\,\text {d}{{f_t}}(v) \!\,\,\text {d}{t}&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }}\varphi (v,v_*) \sigma _e(|v-v_*|) \!\,\,\text {d}{U_t(v, v_*)} \\&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}(\varphi '(v_*)-\varphi '(v))\frac{1}{2}{(} \!\,\,\text {d}{C_t}(v_*) \!\,\,\text {d}{f_t}(v)- \!\,\,\text {d}{C_t}(v) \!\,\,\text {d}{f_t}(v_*){)} \\&= \int _{{{\mathbb {R}}}}\partial _v \varphi (v) \!\,\,\text {d}{C_t}(v), \end{aligned}$$

where we have used the fact that $\int _{{\mathbb {R}}}\!\,\,\text {d}{C_t}(v)=0$. Second, we check the integrability condition (2.1) and bound

$$\begin{aligned} \int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|)\textrm{d}| U_t|(v, v_*)\, \textrm{d} t&= \frac{1}{2} \int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {(} \!\,\,\text {d}f_t(v) \!\,\,\text {d}|C_t|(v_*) + \!\,\,\text {d}|C_t|(v) \!\,\,\text {d}f_t(v_*){)} \!\,\,\text {d}{t} \\&\le \int _0^T \int _{{{\mathbb {R}}}} \!\,\,\text {d}| C_t|(v) < \infty , \end{aligned}$$

by the bound (2.8). $\square$

2.2 The action-density functional and its properties

This section is dedicated to introducing the action-density functional which plays a crucial role in the subsequent analysis. We start by considering the auxiliary function $\alpha :{{\mathbb {R}}}_+ \times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}_+$ given by

$$\begin{aligned} \alpha ( s, u):= {\left\{ \begin{array}{ll} \frac{u^2}{s},&{} \text {if } s>0, \\ 0, &{} \text {if } u= 0,\\ +\infty , &{} \text {if } u \ne 0, s = 0. \end{array}\right. } \end{aligned}$$

(2.9)

We observe that $\alpha$ is jointly convex, lower semicontinuous, and 1-homogeneous.

Following the strategy of [16,17,18,19], we define the action-density functional.

Definition 2.8

(Action-density functional) For any $f\in {{\mathcal {P}}}({{\mathbb {R}}})$ and $U\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$, set $|\lambda |=f\otimes f+|U|\in {{\mathcal {M}}}^+({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$. We define the action-density functional by

$$\begin{aligned} {{\mathcal {A}}}(f,U):=\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}\alpha \biggl (\frac{\!\,\,\text {d}{f\otimes f}}{\!\,\,\text {d}{|\lambda |}},\frac{\!\,\,\text {d}{U}}{\!\,\,\text {d}{|\lambda |}}\biggr )\sigma _e(\left| v-v_*\right| )\!\,\,\text {d}{|\lambda |}(v,v_*) \, , \end{aligned}$$

where the function $\alpha$ is defined as in (2.9).

Note that the above definition is independent of the choice of $|\lambda |$ as long as $f\otimes f+|U|\ll |\lambda |$. In the next lemma, we see that the flux of any couple, (f, U), with finite action-density, takes a specific form.

Lemma 2.9

Let $f \in {{\mathcal {P}}}({{\mathbb {R}}})$ and $U\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ be such that ${{\mathcal {A}}}(f,U)<+\infty$. Then, there exists a Borel function ${\hat{U}}:{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\rightarrow {{\mathbb {R}}}$ such that

$$\begin{aligned} \textrm{d}{U}(v,v_*)= {\hat{U}}(v,v_*)\textrm{d}{{(}f \otimes f{)}}(v,v_*) \,, \end{aligned}$$

and the action-density is given by

$$\begin{aligned} {{\mathcal {A}}}(f,U)&=\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}|{\hat{U}}|^2(v,v_*) \sigma _e(\left| v-v_*\right| )\,\textrm{d}{{(}f \otimes f{)}}(v, v_*) \, . \end{aligned}$$

In particular, if $f\ll {{\mathcal {L}}}$ then $U\ll {{\mathcal {L}}}\otimes {{\mathcal {L}}}$, as well.

Proof

Let $f \in {{\mathcal {P}}}({{\mathbb {R}}})$, $U \in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$, and $|\lambda |\in {{\mathcal {M}}}^+({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ be as in Definition  2.8 such that ${{\mathcal {A}}}(f, U) < \infty$. Then, setting $\mu :=f\otimes f$, the action functional can be written as

$$\begin{aligned} {{\mathcal {A}}}(f, U)&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \alpha \left( {\frac{\!\,\,\text {d}{\mu }}{\!\,\,\text {d}{\left| \lambda \right| }}, \frac{\!\,\,\text {d}{U}}{\!\,\,\text {d}{\left| \lambda \right| }}}\right) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{\left| \lambda \right| } = \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \alpha {(}{\widetilde{\mu }}, {\widetilde{U}}{)} \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{\left| \lambda \right| } \, , \end{aligned}$$

where ${\widetilde{\mu }}, {\widetilde{U}}$ are the Radon–Nikodym derivatives of $\mu ,U$, respectively, with respect to $\left| \lambda \right|$. In order to be able to use the 1-homogeneity of the kernel, $\alpha$, we show that $U \ll \mu$. To this end, let $N\subset {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ be a $(\sigma _e\mu )$-null set, i.e., ${{\widetilde{\mu }}}(v, v_*) = 0$, for $v,v_*\in N$, $\sigma _e\left| \lambda \right|$-a.e. in ${{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$. Since the action of (f, U) is finite, we conclude, by definition of $\alpha$, cf. (2.9), that ${{\widetilde{U}}}(v, v_*) = 0$, $\sigma _e\left| \lambda \right|$-a.e., which, in turn, implies $U \ll \mu$. Upon an application of the chain rule we obtain

$$\begin{aligned} \frac{\!\,\,\text {d}{U}}{\!\,\,\text {d}{\left| \lambda \right| }} = \frac{\!\,\,\text {d}{U}}{\!\,\,\text {d}{\mu }} \frac{\!\,\,\text {d}{\mu }}{\!\,\,\text {d}{\left| \lambda \right| }} =: {{\hat{U}}} {{\widetilde{\mu }}}. \end{aligned}$$

Substituting this expression into the action density above in conjunction with the homogeneity of order one, we obtain

$$\begin{aligned} {{\mathcal {A}}}(f,U)&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{{\hat{U}}}}|^2 {{\widetilde{\mu }}} \,\sigma _e(|v-v_*|) \!\,\,\text {d}{\left| \lambda \right| } = \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{{\hat{U}}}}|^2\,\sigma _e(|v-v_*|) \!\,\,\text {d}{\mu } \\&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{{\hat{U}}}}|^2 \sigma _e(|v-v_*|) \!\,\,\text {d}{{(}f\otimes f{)}}(v,v_*), \end{aligned}$$

which concludes the proof. $\square$

Proposition 2.10

(Antisymmetric fluxes have lower action) Let $f \in {{\mathcal {P}}}({{\mathbb {R}}})$ and $U \in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ be such that ${{\mathcal {A}}}(f, U) < \infty$. Then, there exists an antisymmetric³ measure $U^{\textrm{as}} \in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$, $U^{\textrm{as}} \ll \mu$, such that

$$\begin{aligned} {{\mathcal {A}}}(f, U^{\textrm{as}})\le {{\mathcal {A}}}(f, U),\quad \text{ and } \quad {\widetilde{\nabla }} \cdot U^{\textrm{as}}= {\widetilde{\nabla }}\cdot U. \end{aligned}$$

Proof

We define ${\hat{U}}^{\textrm{as}}: {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\rightarrow {{\mathbb {R}}}$ to be

$$\begin{aligned} {\hat{U}}^{\textrm{as}}(v,v_*) := \frac{1}{2}{(}{\hat{U}}(v,v_*)- {\hat{U}}(v,v_*){)}, \end{aligned}$$

where ${\hat{U}}$ is as defined in the statement of Lemma 2.9. This defines a measure, $U^{\textrm{as}} \in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$, via the relation

$$\begin{aligned} \!\,\,\text {d}{U^{\textrm{as}}}(v,v_*):={\hat{U}}^{\textrm{as}}(v,v_*) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*). \end{aligned}$$

The proof then follows by substitution. We have that

$$\begin{aligned} {{\mathcal {A}}}(f, U^{\textrm{as}})&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{\hat{U}}{^{\textrm{as}}}|^2} (v,v_*) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*) \\&= \frac{1}{2}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{\hat{U}}}|^2(v,v_*) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*)\\&\quad - \frac{1}{2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\hat{U}}(v,v_*) {\hat{U}}(v_*,v) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*). \end{aligned}$$

Applying Young’s inequality, we obtain

$$\begin{aligned} {{\mathcal {A}}}(f, U^{\textrm{as}})&\le \frac{1}{2}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{\hat{U}}}|^2(v,v_*) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*) \\&\quad + \frac{1}{4} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{\hat{U}}}|^2(v,v_*) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*) \\&\quad +\frac{1}{4} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{\hat{U}}}|^2(v_*,v) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*) \\&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |{{\hat{U}}}|^2(v,v_*) \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{(}f \otimes f{)}(v,v_*) \\&= {{\mathcal {A}}}(f, U). \end{aligned}$$

Finally, we can check that, for any test function $\varphi \in C^\infty _c({{\mathbb {R}}})$, it holds that

$$\begin{aligned}&\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }} \varphi (v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U^{\textrm{as}}}(v,v_*) \\&= \frac{1}{2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }} \varphi (v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{(}U(v,v_*) - U(v_*,v){)} \\&=\frac{1}{2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }} \varphi (v,v_*)\sigma _e(|v-v_*|)\!\,\,\text {d}{U(v,v_*)} - \frac{1}{2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }} \varphi (v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U(v_*,v)} \\&=\frac{1}{2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }} \varphi (v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U(v,v_*)} +\frac{1}{2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }} \varphi (v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U(v,v_*)} \\&=\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {\widetilde{\nabla }} \varphi (v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U}(v,v_*), \end{aligned}$$

where in the penultimate step we have used the fact that ${\widetilde{\nabla }}\varphi (v,v_*)=-{\widetilde{\nabla }}\varphi (v_*,v)$ from Definition 1.4. Using Definition 2.1, the result follows. $\square$

Proposition 2.11

(Lower semicontinuity of the action density) The action-density functional is lower semicontinuous with respect to the weak-$^*$ convergence in ${{\mathcal {P}}}({{\mathbb {R}}})\times {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })\subset {{\mathcal {M}}}({{\mathbb {R}}}\times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$.

Proof

Let us consider $\{f_n\}_{n\in {{\mathbb {N}}}}\subset {{\mathcal {P}}}({{\mathbb {R}}})$ and $\{U_n\}_{n\in {{\mathbb {N}}}}\subset {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ such that

$$\begin{aligned} f_n\rightarrow f, \quad \text{ in } {{\mathcal {P}}}({{\mathbb {R}}}), \end{aligned}$$

as well as

$$\begin{aligned} U_n\rightarrow U, \quad \text{ in } {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }). \end{aligned}$$

Obviously, convergence in ${{\mathcal {P}}}({{\mathbb {R}}})$ of $\{f_n\}_{n\in {{\mathbb {N}}}}$ implies that $\{f_n\otimes f_n\}_{n\in {{\mathbb {N}}}}$ converges weakly-$^*$ in ${{\mathcal {P}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$. Let us define the function $g:{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\times ({{\mathbb {R}}}_+ \times {{\mathbb {R}}})\rightarrow {{\mathbb {R}}}$ as

$$\begin{aligned} g((v,v_*), (s,u))=\alpha (s, u)\sigma _e(\left| v-v_*\right| ), \end{aligned}$$

which is lower semicontinuous in all its variables, jointly convex, and 1-positive homogeneous in (s, u). Then, [10, Theorem 3.4.3] implies the action is weakly-$^*$ sequentially lower semicontinuous in ${{\mathcal {M}}}({{\mathbb {R}}}\times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$. $\square$

Proposition 2.12

(Convexity of the action density) Let $f^i\in {{\mathcal {P}}}({{\mathbb {R}}})$ and $U^i\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ for $i=0,1$. For any $\tau \in [0,1]$, such that $f_\tau :=(1-\tau )f^0+\tau f^1$ and $U_\tau :=(1-\tau ) U^0+\tau U^1$ it holds

$$\begin{aligned} {{\mathcal {A}}}(f_\tau ,U_\tau ) \le (1-\tau ) {{\mathcal {A}}}(f^0,U^0) + \tau {{\mathcal {A}}}(f^1, U^1). \end{aligned}$$

Proof

Let us set $\mu ^i:=f^i\otimes f^i$ and consider $|\lambda |\in {{\mathcal {M}}}^+({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ such that $\,\,\text {d}{\mu ^i}={\widetilde{\mu }}^i \!\,\,\text {d}{|\lambda |}$ and $\,\,\text {d}{U^i}={\widetilde{U}}^i\!\,\,\text {d}{|\lambda |}$, cf. Definition 2.8, for instance. As consequence we have $\,\,\text {d}{\mu _\tau }={\widetilde{\mu }}_\tau \!\,\,\text {d}{|\lambda |}$ and $\,\,\text {d}{U_\tau }={\widetilde{U}}_\tau \!\,\,\text {d}{|\lambda |}$, where

$$\begin{aligned}&{{\widetilde{\mu }}}_\tau :=(1-\tau ){\widetilde{\mu }}^0+\tau {\widetilde{\mu }}^1,\\&{\widetilde{U}}_\tau :=(1-\tau ){\widetilde{U}}^0 + \tau {\widetilde{U}}^1. \end{aligned}$$

The result follows by using the convexity of the function $\alpha$:

$$\begin{aligned} {{\mathcal {A}}}(f_\tau ,U_\tau )&=\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}\alpha \left( {{\widetilde{\mu }}}_\tau ,{{\widetilde{U}}}_\tau \right) \sigma _e(\left| v-v_*\right| )\,\!\,\,\text {d}{|\lambda |}(v,v_*)\\&\le (1-\tau )\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}\alpha \left( {\widetilde{\mu }}^0,{\widetilde{U}}^0\right) \sigma _e(\left| v-v_*\right| )\,\!\,\,\text {d}{|\lambda |}(v,v_*)\\&\quad +\tau \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}\alpha \left( {\widetilde{\mu }}^1,{\widetilde{U}}^1\right) \sigma _e(\left| v-v_*\right| )\,\!\,\,\text {d}{|\lambda |}(v,v_*)\\&=(1-\tau ){{\mathcal {A}}}(f^0,U^0)+\tau {{\mathcal {A}}}(f^1,U^1). \end{aligned}$$

$\square$

2.3 Curves of finite action

This section is dedicated to revisiting (CE) introduced in Definition 2.2 and presenting some of its properties.

Lemma 2.13

(Curves of finite action) Let ${\{}(f_t,U_t){\}}_{t\in [0,T]}$ be a solution to the nonlocal-local continuity equation in the sense of Definition 2.2 with initial datum $\mu _0 \in {{\mathcal {P}}}({{\mathbb {R}}})$ not necessarily satisfying the integrability condition  (2.1), but satisfying $\int _0^T {{\mathcal {A}}}(f_t,U_t) \textrm{d}{t} < \infty$ and $\int _{{\mathbb {R}}}|v|\, \textrm{d} \mu _0(v)< \infty$, then ${\{}(f_t,U_t){\}}_{t\in [0,T]}\in {\textrm{CE}}_T(\mu _0)$.

In particular, if $\mu _0\in {{\mathcal {P}}}_1({{\mathbb {R}}})$, then $f_t \in {{\mathcal {P}}}_1({{\mathbb {R}}})$ and the following estimate holds for all $t\in [0,T]$

$$\begin{aligned} \left| \frac{\textrm{d}}{\textrm{d}t} m_1(f_t)^{\frac{1}{2}}\right| \le \bigg ({\frac{1-e}{2}}\bigg )^{\frac{1}{2}} {{\mathcal {A}}}(f_t,U_t)^{\frac{1}{2}}. \end{aligned}$$

(2.10)

Proof

The proof follows by applying the bound (2.6) in Proposition 2.6 for which we further need to bound, for almost every $t\in [0,T]$, the total variation norm of the flux by a suitable Cauchy-Schwarz inequality:

$$\begin{aligned} \frac{1}{2} \left| \frac{\mathop {}\!\text {d}^{} }{\mathop {}\!\text {d} t^{}} m_1(f_t)\right|&\le \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) \!\,\,\text {d}|U_t|(v,v_*) \\&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) |{{\hat{U}}}_t(v,v_*)| \!\,\,\text {d}(f_t\otimes f_t)(v,v_*) \\&\le {{\mathcal {A}}}(f_t,U_t)^{\frac{1}{2}} \bigg ({\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) \!\,\,\text {d}(f_t\otimes f_t)(v,v_*) }\bigg )^{\frac{1}{2}} \\&\le \bigg ({\frac{1-e}{4}}\bigg )^{\frac{1}{2}} {{\mathcal {A}}}(f_t,U_t)^{\frac{1}{2}} \bigg ({\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} (|v| + |v_*|)\!\,\,\text {d}(f_t\otimes f_t)(v,v_*)}\bigg )^{\frac{1}{2}} \\&\le \bigg ({\frac{1-e}{2}}\bigg )^{\frac{1}{2}} m_1(f_t)^{\frac{1}{2}} {{\mathcal {A}}}(f_t,U_t)^{\frac{1}{2}} . \end{aligned}$$

$\square$

In the next result, we associate to a given curve $(U_t)_{t\in [0,T]}$ a measure $U\in {{\mathcal {M}}}{(}{[0,T}{]} \times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }{)}$ by setting $\,\,\text {d}U(t,v,v_*)=\!\,\,\text {d}U_t(v,v_*)\!\,\,\text {d}{t}$, for $(t,v,v_*)\in [0,T]\times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$.

Proposition 2.14

(Compact subsets of ${\textrm{CE}}_T$) Let ${\{}(f_t^n,U_t^n)_{t\in [0,T]}{\}}_{n \in {{\mathbb {N}}}} \subset {\textrm{CE}}_T(f_0^n,f_T^n)$ and assume there exists a constant $0<C<\infty$ such that

$$\begin{aligned} \sup _{n \in {{\mathbb {N}}}}\int _0^T {{\mathcal {A}}}(f_t^n,U_t^n) \,\, \textrm{d}{t}< C , \qquad \text {and} \qquad \sup _{n \in {{\mathbb {N}}}} \int |v|\,\, \textrm{d}(f_0^n+ f_T^n)(v) < C \, . \end{aligned}$$

(2.11)

Then, there exists ${\{}(f_t,U_t){\}}_{t\in [0,T]} \in {\textrm{CE}}_T(f_0,f_T)$, and, for all $t \in [0,T]$, along a subsequence (not relabelled)

$$\begin{aligned} f_t^n&\rightarrow f_t, \quad \text { in } {{\mathcal {P}}}({{\mathbb {R}}}), \quad \text {as well as}\quad U^n \rightarrow ^c U, \quad \text { in } {{\mathcal {M}}}_\textrm{loc}{(}{[0,T}{]} \times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }{)}. \end{aligned}$$

Moreover, the action is lower semicontinuous along the above subsequences $\{f^n\}_n$ and $\{U^n\}_n$, i.e.,

$$\begin{aligned} \liminf _{n\rightarrow \infty }\int _0^T{{\mathcal {A}}}(f_t^n,U_t^n)\textrm{d}t \ge \int _0^T{{\mathcal {A}}}(f_t,U_t)\textrm{d}t. \end{aligned}$$

Proof

We first show that the total variation measure $\left| U^n\right|$ is bounded on compact sets. We let $I \times K \subset {[0,T}{]} \times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ be compacts. It is then relatively straightforward to see that

$$\begin{aligned} \left| U^n\right| {(}I \times K{)} \le \int _I \left| U_t^n\right| (K) \!\,\,\text {d}{t} \le \int _I \int _K |{\hat{U}}_t^n(v, v_*)|\!\,\,\text {d}{{(}f_t^n \otimes f_t^n{)}}(v,v_*) \!\,\,\text {d}{t} \, , \end{aligned}$$

where for the last inequality we have used finiteness of the action and the result of Lemma 2.9, which states that $U_t^n$ has a density with respect to $f_t^n\otimes f_t^n$. Upon applying the Cauchy–Schwartz inequality, we obtain the following bound

$$\begin{aligned} \begin{aligned} \left| U^n\right| {(}I \times K{)}&\le \bigg ({\int _I \int _K |{\hat{U}}_t^n(v,v_*)|^2 \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{{(}f_t^n \otimes f_t^n{)}}(v,v_*) \!\,\,\text {d}{t}}\bigg )^{\frac{1}{2}}\\&\qquad \times \bigg ({\int _I \int _K \frac{ \!\,\,\text {d}{{(}f_t^n \otimes f_t^n{)}}(v,v_*)}{\sigma _e(\left| v-v_*\right| )} \!\,\,\text {d}{t}}\bigg )^{\frac{1}{2}} \\&\le \bigg ({\frac{1-e}{2} C_K |I|}\bigg )^{\frac{1}{2}} , \end{aligned} \end{aligned}$$

(2.12)

where $C_K=C \sup _{(v,v_*)\in K} \sigma _e(\left| v-v_*\right| )^{-1}< \infty$ with C as in (2.11), since $\sigma _e$ is continuous and positive on ${{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$. Since $I\times K$ was arbitrary, it is clear from the above estimate that we can obtain uniform local control on the total variation of the measures $U^n \in {{\mathcal {M}}}{(}{[0,T}{]} \times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }{)}$. Thus by Prokhorov’s theorem there exists a measure $U \in {{\mathcal {M}}}{(}{[0,T}{]} \times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }{)}$ such that $U^n \rightarrow ^c U$, i.e., tested against $C_c([0,T]\times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$.

We now note that $U \in {{\mathcal {M}}}_{\textrm{loc}}({[0,T}{]} \times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ can be disintegrated with respect to the Lebesgue measure on ${[0,T}{]}$. Indeed, consider for any compact set, $K\subset {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$, the measure $\lambda ^K:= \pi _{\#}^K U \in {{\mathcal {M}}}{(}{[0,T}{]}{)}$, where $\pi ^K: {[0,T}{]} \times K \rightarrow {[0,T}{]}$ is the projection map defined as $\pi ^K(t,x):=t$, for $x\in K$. By the definition of the pushforward we have for any measurable $I\subset [0,T]$ from (2.12) the estimate

$$\begin{aligned} \lambda ^K(I) = U(I \times K) \le \left( {\frac{1-e}{2} C_K |I|}\right) ^{\frac{1}{2}}. \end{aligned}$$

Thus, $\lambda ^K$ is absolutely continuous with respect for the Lebesgue measure on I, for any $K\subset {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ compact. Additionally, for any $\varphi \in C_c({[0,T}{]}\times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ choose $K\subset {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ such that ${{\,\textrm{supp}\,}}\varphi \subset [0,T]\times K$. By the disintegration theorem, cf. [2, Theorem 5.3.1], we have the existence of a family ${\{}\mu _t^K{\}}_{t\in [0,T]}$ such that $\,\,\text {d}U = \!\,\,\text {d}\mu _t^K \!\,\,\text {d}\lambda ^K$. In particular

$$\begin{aligned} \int _0^T&\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \varphi (t,v,v_*)\sigma _e(|v-v_*|)\!\,\,\text {d}{U}(t,v,v_*)\\&=\int _0^T \left( {\int _{{\{}t{\}} \times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \varphi (t,v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{\mu _t^K}(v,v_*)}\right) \!\,\,\text {d}\lambda ^K(t)\\&=\int _0^T \int _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \varphi (t,v,v_*)\sigma _e(|v-v_*|) \!\,\,\text {d}{U_t^K}(v,v_*) \!\,\,\text {d}t, \end{aligned}$$

where $U_t^K:= \frac{\!\,\,\text {d}{\lambda ^K}}{\!\,\,\text {d}t} \mu _t^K$ and $\mu _t^K \in {{\mathcal {M}}}(K)$ is the parametrised family of measures arising from the disintegration theorem.

We readily observe that integrating (2.4) over $[t_1,t_2]$ gives for any $\psi \in C_c^1({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$

$$\begin{aligned} \begin{aligned} \bigg |&\int _{{{\mathbb {R}}}} \psi (v) \!\,\,\text {d}{f_{t_1}^n}(v)- \int _{{{\mathbb {R}}}} \psi (v) \!\,\,\text {d}{f_{t_2}^n}(v)\bigg | \le \int _{t_1}^{t_2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {\widetilde{\nabla }} \psi (v,v_*)\right| \sigma _e(|v-v_*|) \!\,\,\text {d}{|U_t^n|}\!\,\,\text {d}{t}\\&\le \int _{t_1}^{t_2} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {\widetilde{\nabla }} \psi (v,v_*)\right| \sigma _e(|v-v_*|) \left| {\hat{U}}_t^n(v, v_*)\right| \!\,\,\text {d}{{(}f_t^n \otimes f_t^n{)}}(v,v_*)\!\,\,\text {d}t \\&\le \int _{t_1}^{t_2} {{\mathcal {A}}}(f_t^n,U_t^n)^{\frac{1}{2}} \bigg ({ \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {\widetilde{\nabla }} \psi (v,v_*)\right| ^2 \sigma _e(|v-v_*|) \!\,\,\text {d}{{(}f_t^n \otimes f_t^n{)}}(v,v_*)}\bigg )^{\frac{1}{2}} \!\,\,\text {d}t \\&\le \bigg ({\frac{1-e}{4}}\bigg )^{\frac{1}{2}} \int _{t_1}^{t_2} {{\mathcal {A}}}(f_t^n,U_t^n)^{\frac{1}{2}} \bigg ({ \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {(}\psi '(v)-\psi '(v_*){)}^2 {(} |v| + |v_*|{)} \!\,\,\text {d}f_t^n(v) \!\,\,\text {d}f_t^n(v_*) }\bigg )^{\frac{1}{2}} \!\,\,\text {d}{t} \\&\le C \left\| \psi '\right\| _{\infty } | t_2-t_1|^{\frac{1}{2}}, \end{aligned}\end{aligned}$$

(2.13)

according to Eq. (2.12), having used the definition of $\sigma _e$, cf. (1.16) and applied the stability of the first moment (2.10) from Lemma 2.13, which also ensures that $(f_t^n, U_t^n)_{t\in [0,T]}\in {\textrm{CE}}(f_0^n,f_T^n)$. Passing to the supremum in $\psi$ among all Lipschitz functions with Lipschitz constant 1, we recover the 1/2-Hölder continuity in the 1-Wasserstein distance, i.e.,

$$\begin{aligned} d_1(f_{t_2}^n, f_{t_1}^n) \le C |t_2-t_1|^{\frac{1}{2}}, \end{aligned}$$

uniformly in $n\in {{\mathbb {N}}}$. An application of the generalised Arzela-Ascoli theorem concludes the proof of convergence of the densities, see [2, Sect. 3]. In particular, we have that the limiting curve is absolutely continuous in time with values in probability measures and hence $(f_t,U_t)_{t\in [0,T]}\in {\textrm{CE}}(f_0,f_T)$. Finally, the lower semicontinuity property is a consequence of Proposition 2.11. $\square$

2.4 The collision metric

In this section, we define and prove properties for an extended metric coming from the nonlocal-local continuity equation. We start with the definition of the collision transportation cost.

Definition 2.15

Let $\mu _0,\mu _1\in {{\mathcal {P}}}({{\mathbb {R}}})$. The collision transportation cost is defined by

$$\begin{aligned} d_{{{\mathcal {A}}}}(\mu _0, \mu _1)^2:= \inf \left\{ \int _0^1{{\mathcal {A}}}(f_t,U_t)\,\!\,\,\text {d}{t}: (f_t,U_t)_{t\in [0,1]}\in {\textrm{CE}}(\mu _0,\mu _1)\right\} . \end{aligned}$$

(2.14)

Note that the minimisation problem above is well defined as consequence of the direct method of calculus of variations by means of Proposition 2.14, whenever the action is bounded, i.e., $\int _0^1{{\mathcal {A}}}(f_t,U_t)\,\!\,\,\text {d}{t}<\infty$. Moreover, by observing that $\alpha$ defined in (2.9) is 2-homogeneous in the second variable, we can apply the same reparametrisation argument used in [16, Theorem 5.4] to obtain the following result.

Lemma 2.16

(Reparametrisation) For any $T>0$, $\mu _0,\mu _1\in {{\mathcal {P}}}({{\mathbb {R}}})$ it holds

$$\begin{aligned} d_{{{\mathcal {A}}}}(\mu _0,\mu _1)=\inf \left\{ \int _0^T {{\mathcal {A}}}(f_t,U_t)^\frac{1}{2}\,\textrm{d}{t}: (f_t,U_t)_{t\in [0,T]}\in {\textrm{CE}}_T(\mu _0,\mu _1)\right\} . \end{aligned}$$

In the following proposition we see under which conditions the infimum in Eq. (2.14) is a minimum.

Proposition 2.17

Let $\mu _0,\mu _1\in {{\mathcal {P}}}({{\mathbb {R}}})$ such that $d_{{\mathcal {A}}}:=d_{{\mathcal {A}}}(\mu _0,\mu _1)<+\infty$. Then the infimum in Eq. (2.14) is attained by a curve $(f_t,U_t)_{t\in [0,1]}\in {\textrm{CE}}(\mu _0,\mu _1)$ such that

$$\begin{aligned} {{\mathcal {A}}}(f_t,U_t)=d_{{\mathcal {A}}}^2(\mu _0,\mu _1), \end{aligned}$$

for a.e. $t\in [0,1]$. Such a curve is a constant speed geodesic for $d_{{\mathcal {A}}}$, i.e.,

$$\begin{aligned} d_{{\mathcal {A}}}(f_s,f_t)=|t-s|d_{{\mathcal {A}}}(\mu _0,\mu _1), \end{aligned}$$

for all $s,t\in [0,1]$.

Proof

If $d_{{\mathcal {A}}}$ is finite, which holds when $\int _0^1 {{\mathcal {A}}}(f_t,U_t) \!\,\,\text {d}{t} <\infty$ for some $(f_t,U_t)_{t\in [0,1]} \in {\textrm{CE}}(\mu _0,\mu _1)$, the infimum in Eq. (2.14) is attained as a consequence of Proposition 2.14 by means of the direct method of calculus of variations. Thus, there exists a minimising curve $(f_t^*,U_t^*)_{t\in [0,1]}\in {\textrm{CE}}(\mu _0,\mu _1)$. By the reparametrisation result in Lemma 2.16 and the Jensen’s inequality, we obtain

$$\begin{aligned} \int _0^1{{\mathcal {A}}}(f_t^*,U_t^*)^\frac{1}{2}\!\,\,\text {d}{t} \ge d_{{\mathcal {A}}}(\mu _0,\mu _1) = \left( \int _0^1{{\mathcal {A}}}(f_t^*,U_t^*) \!\,\,\text {d}{t}\right) ^{\frac{1}{2}} \ge \int _0^1 {{\mathcal {A}}}(f_t^*,U_t^*)^\frac{1}{2} \!\,\,\text {d}{t}, \end{aligned}$$

whence $d_{{\mathcal {A}}}^2(\mu _0, \mu _1)={{\mathcal {A}}}(f_t^*,U_t^*)$, for almost every $t\in [0,1]$. Moreover, we obtain

$$\begin{aligned} d_{{\mathcal {A}}}(f_s,f_t) = \int _s^t {{\mathcal {A}}}(f_r^*,U_r^*)^\frac{1}{2}\!\,\,\text {d}{r} = |t-s|d_{{\mathcal {A}}}(\mu _0,\mu _1), \end{aligned}$$

for all $s,t\in [0,1]$, which concludes the proof. $\square$

Given the preservation of the centre of mass and the stability of the first moment along curves of finite action implied by Proposition 2.6, it makes sense to restrict the collision transport cost to certain subspaces. Let us note the metric $d_{{\mathcal {A}}}$ can be compared with $d_1$, the 1-Wasserstein distance.

Proposition 2.18

(Comparison with $d_1$) Let $\mu _0,\mu _1\in {{\mathcal {P}}}_1({{\mathbb {R}}})$. There exists a constant $C=C(e)$ such that

$$\begin{aligned} d_1(\mu _0,\mu _1) \le C {(}m_1(\mu _0) + d_{{{\mathcal {A}}}}(\mu _0,\mu _1){)} d_{{\mathcal {A}}}(\mu _0,\mu _1). \end{aligned}$$

Proof

The proof is obtained along the lines of the estimate (2.13), and using (2.10). $\square$

Theorem 2.19

The collision transport cost defined in  (2.14) is an extended metric on ${{\mathcal {P}}}({{\mathbb {R}}})$. The map $(\mu _0,\mu _1)\mapsto d_{{\mathcal {A}}}(\mu _0,\mu _1)$ is lower semicontinuous with respect to the convergence in ${{\mathcal {P}}}({{\mathbb {R}}})$. Moreover, the topology induced by $d_{{\mathcal {A}}}$ is stronger then the $d_1$-topology.

Proof

Let us assume that $d_{{\mathcal {A}}}(\mu _0,\mu _1)=0$. By Proposition 2.17 there exists a curve $(f_t,U_t)_{t\in [0,T]}\in {\textrm{CE}}(\mu _0,\mu _1)$ such that ${{\mathcal {A}}}(f_t, U_t)=0$ for a.e. $t\in [0,1]$, which implies $U_t=0$ for a.e. $t\in [0,1]$. Thus, from Eq. (2.4) we obtain $\mu _0=\mu _1$. The opposite implication is trivial. The symmetry of $d_{{\mathcal {A}}}$ follows from the fact that $\alpha (\cdot , u)=\alpha (\cdot , -u)$. In order to prove the triangle inequality we notice that solutions to ${\textrm{CE}}$ can be concatenated. Indeed, if $(f^i, U^i)\in {\textrm{CE}}_{T_i}(\mu _0^i,\mu ^i_{T_i})$ for $i=1,2$ such that $\mu _{T_1}^1=\mu ^2_0$, then

$$\begin{aligned} f_t:={\left\{ \begin{array}{ll} f_t^1 \quad &{}\text{ if } 0\le t\le T_1\\ f_{t-T_1}^2 \quad &{}\text{ if } T_1\le t\le T_1+T_2 \end{array}\right. };\quad U_t:={\left\{ \begin{array}{ll} U_t^1 \quad &{}\text{ if } 0\le t\le T_1\\ U_{t-T_1}^2 \quad &{}\text{ if } T_1\le t\le T_1+T_2 \end{array}\right. } \end{aligned}$$

belongs to ${\textrm{CE}}_{T_1+T_2}(\mu _0^1,\mu _{T_2}^2)$ by using Eq. (2.5). This observation and Lemma 2.16 imply the triangle inequality. The lower semicontinuity property is a consequence of Proposition 2.14, while Proposition 2.18 gives that the topology induced by $d_{{\mathcal {A}}}$ is stronger than that of $d_1$. $\square$

Let us recall the definition of absolutely continuous curves in a metric space. A curve $[0,T]\ni t\mapsto f_t\in {{\mathcal {P}}}({{\mathbb {R}}})$ is said to be 2-absolutely continuous with respect to $d_{{{\mathcal {A}}}}$ if there exists $m\in L^2(0,T)$ such that

$$\begin{aligned} d_{{{\mathcal {A}}}}(f_{t_0},f_{t_1})\le \int _{t_0}^{t_1}m(t)\!\,\,\text {d}{t}, \quad \text{ for } \text{ all } \quad 0<t_0\le t_1<T. \end{aligned}$$

(2.15)

In this case, we write $f\in {{\,\textrm{AC}\,}}(0,T;({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$. For any $f\in {{\,\textrm{AC}\,}}(0,T;({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ the quantity

$$\begin{aligned} |f'|(t)=\lim _{h\rightarrow 0}\frac{d_{{{\mathcal {A}}}}(f_{t+h},f_t)}{h} \end{aligned}$$

is well-defined for a.e. $t\in [0,T]$ and is called metric derivative of f at t. Moreover, the function $t\rightarrow |f'|(t)$ belongs to $L^2(0,T)$ and it satisfies $|f'|(t)\le m(t)$ for a.e. $t\in [0,T]$, i.e., $f'$ is the minimal integrand satisfying (2.15). The length of a curve $f\in {{\,\textrm{AC}\,}}(0,T;({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ is defined by $L(f):=\int _0^T|f'|(t)\!\,\,\text {d}{t}$.

Given the above results we can easily obtain the following characterisation, as in [16, Theorem 5.17]. The proof is then omitted.

Proposition 2.20

(Metric velocity) A curve $\{f_t\}_{t\in [0,T]}\subset {{\mathcal {P}}}({{\mathbb {R}}})$ belongs to the space ${{\,\textrm{AC}\,}}(0,T;({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ if and only if there exists a family of flux $\{U_t\}_{t\in [0,T]}$ such that ${\{}(f_t,U_t){\}}_{t\in [0,T]}\in {\textrm{CE}}_T$ with

$$\begin{aligned} \int _0^T{{\mathcal {A}}}(f_t,U_t)^\frac{1}{2} \textrm{d}{t} < \infty . \end{aligned}$$

In particular, $\textrm{d} U_t(v,v_*) = {{\hat{U}}}_t(v,v_*) \textrm{d}(f_t\otimes f_t)(v,v_*)$ for a measurable family ${{\hat{U}}}: [0,T]\times {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\rightarrow {{\mathbb {R}}}$. In this case, the metric derivative is bounded as in $|f'|^2(t)\le {{\mathcal {A}}}(f_t,U_t)$ for a.e. $t\in [0,T]$. In addition, there exists a unique $\{{\widetilde{U}}_t\}_{t\in [0,T]}$ such that $(f_t, {{\widetilde{U}}}_t)_{t\in [0,T]}\in {\textrm{CE}}_T$ and

$$\begin{aligned} |f'|^2(t)={{\mathcal {A}}}(f_t,{\widetilde{U}}_t), \qquad \text {for a.e. } t\in [0,T]. \end{aligned}$$

(2.16)

Corollary 2.21

(Tangent space) Let ${\{}(f_t,U_t){\}}_{t\in [0,T]} \in {\textrm{CE}}_T$ such that the curve $f\in {{\,\textrm{AC}\,}}(0,T;({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$. The flux U satisfies (2.16) if and only if $U_t\in T_f{{\mathcal {P}}}({{\mathbb {R}}})$ for a.e. $t\in [0,T]$, where

$$\begin{aligned} \begin{aligned} T_f{{\mathcal {P}}}({{\mathbb {R}}})=\bigl \{&U\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }):{{\mathcal {A}}}(f,U)<\infty , \, {{\mathcal {A}}}(f,U)\le {{\mathcal {A}}}(f,U+w),\\&\text{ for } \text{ any } w\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }), \text{ s.t. } {{\widetilde{\nabla }}}\cdot w=0\bigr \}. \end{aligned} \end{aligned}$$

(2.17)

Proof

According to Proposition 2.20 the metric derivative satisfies $|f'|^2(t)\le {{\mathcal {A}}}(f_t,U_t)$ for a.e. $t\in [0,T]$. Therefore, the only flux satisfying (2.16) is that of minimal action. Let $t\in [0,T]$ such that ${{\mathcal {A}}}(f_t,U_t)<+\infty$. As proved in Proposition 2.10, the flux, ${\widetilde{U}}_t$, of minimal action has to be antisymmetric, ${\widetilde{U}}_t\in {{\mathcal {M}}}^{\textrm{as}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$, and by assumption satisfy the nonlocal-local continuity equation. In particular,

$$\begin{aligned} {\widetilde{U}}_t=\mathop {\textrm{argmin}}\limits _{U\in {{\mathcal {M}}}^{\textrm{as}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })}\{{{\mathcal {A}}}(f_t,U):\widetilde{\nabla }\cdot U_t={{\widetilde{\nabla }}}\cdot U\}. \end{aligned}$$

(2.18)

Note that the set $\{U\in {{\mathcal {M}}}^{as}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }):{{\widetilde{\nabla }}}\cdot U_t={{\widetilde{\nabla }}}\cdot U\}$ is closed with respect to the weak-$^*$ convergence, and sublevel sets of the functional ${{\mathcal {M}}}^{\textrm{as}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })\ni U \mapsto {{\mathcal {A}}}(f,U)$, for any $f \in {{\mathcal {P}}}({{\mathbb {R}}})$, are locally weakly-$^*$ relatively compact by arguing as in Proposition 2.14, since for any compact set $K\subset {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$ it holds

$$\begin{aligned} |U|(K)\le {{\mathcal {A}}}(f_t,U)^\frac{1}{2} \sup _{K}\sigma _e(|v-v_*|)^{-1}. \end{aligned}$$

Moreover, note that the functional ${{\mathcal {M}}}^{\textrm{as}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }) \ni U \mapsto {{\mathcal {A}}}(f,U)$, for any $f \in {{\mathcal {P}}}({{\mathbb {R}}})$, is strictly convex according to Lemma 2.9. Therefore, the flux in (2.18) is uniquely determined. $\square$

In the previous corollary we have a Lagrangian formulation of the tangent space $T_f{{\mathcal {P}}}({{\mathbb {R}}})$, which can be further characterised in terms of tangent velocity fields.

Proposition 2.22

Let $f\in {{\mathcal {P}}}({{\mathbb {R}}})$. Then, it holds that $U\in T_f{{\mathcal {P}}}({{\mathbb {R}}})$ if and only if $U\in {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ such that ${{\mathcal {A}}}(f,U)<\infty$ and, for a measurable ${\hat{U}}:{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\rightarrow {{\mathbb {R}}}$, it holds

$$\begin{aligned} {\hat{U}}\in \overline{\{{{\widetilde{\nabla }}}\varphi :\varphi \in C_c^\infty ({{\mathbb {R}}})\}}^{L^2({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup },\sigma _e\textrm{d}{(f\otimes f)})}. \end{aligned}$$

Proof

If the action ${{\mathcal {A}}}(f,U)<\infty$, Lemma 2.9 provides the existence of a measurable ${\hat{U}}:{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }\rightarrow {{\mathbb {R}}}$ such that $\,\,\text {d}{U}(v,v_*)={\hat{U}}(v,v_*)\!\,\,\text {d}{(f\otimes f)}(v,v_*)$, for any $(v,v_*)\in {{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }$, whence

$$\begin{aligned} {{\mathcal {A}}}(f,U)=\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}|{\hat{U}}(v,v_*)|^2\sigma _e(|v-v_*|)\!\,\,\text {d}{(f\otimes f)}(v,v_*)=\Vert {\hat{U}}\Vert ^2_{L^2(\sigma _e\!\,\,\text {d}{(f\otimes f)})}. \end{aligned}$$

As consequence of the above relation between U and ${\hat{U}}$, the nonlocal divergence ${{\widetilde{\nabla }}}\cdot U$ can be re-written in terms of ${\hat{U}}$, for any $\varphi \in C_c^\infty ({{\mathbb {R}}})$, as

$$\begin{aligned}{} & {} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}{{\widetilde{\nabla }}}\varphi (v,v_*)\sigma _e(|v-v_*|)\!\,\,\text {d}U(v,v_*)\\{} & {} \qquad =\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}{{\widetilde{\nabla }}}\varphi (v,v_*){\hat{U}}(v,v_*)\sigma _e(|v-v_*|)\!\,\,\text {d}{(f\otimes f)}(v,v_*). \end{aligned}$$

Thus, the characterisation (2.17) can be equivalently stated as

$$\begin{aligned} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}|{\hat{U}}|^2\sigma _e(|\cdot -\cdot |)\!\,\,\text {d}{(f\otimes f)}\le \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}|{\hat{U}}+W|^2\sigma _e(|\cdot -\cdot |)\!\,\,\text {d}{(f\otimes f)}, \end{aligned}$$

for all $W\in L^2({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup },\sigma _e\!\,\,\text {d}{(f\otimes f)})$ such that

$$\begin{aligned} \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}{{\widetilde{\nabla }}}\varphi (v,v_*)W(v,v_*)\sigma _e(|v-v_*|)\!\,\,\text {d}{(f\otimes f)}(v,v_*)=0 \qquad \text {for all}\,\, \varphi \in C_c^\infty ({{\mathbb {R}}}). \end{aligned}$$

Therefore, ${\hat{U}}$ belongs to the closure of $\{{{\widetilde{\nabla }}}\varphi :\varphi \in C_c^\infty ({{\mathbb {R}}})\}$ in $L^2{(}{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup },\sigma _e\!\,\,\text {d}{(f\otimes f)}{)}$. $\square$

3 The aggregation equation in a new light

This section focuses on the aggregation equation (1.14), with a cubic interaction potential (1.15). As discussed in Sect. 1.3, (1.14) can be formally derived from the inelastic spatially homogeneous Boltzmann equation by Taylor-expanding the test function in its weak formulation. In this process, we notice that the collision kernel obtained from the cubic interaction, W, is precisely the modulus function. This suggests that we interpret (1.14) as nonlocal-local continuity equation, as explained in Sect. 2.1, driven by the potential obtained from the kinetic energy (1.6).

More precisely, in this Section, we consider the (CE) driven by the kinetic energy (1.6). In addition to the definition of weak solutions to (CE) (see Definition 2.2), we require the curve to have finite kinetic energy, which is a natural requirement.

Definition 3.1

(Weak solution) A curve ${\{}f_t{\}}_{t\in [0,T]} \subset {{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}})$ is a weak solution to (1.14) if, for the flux ${\{}U_t^{{\mathcal {E}}}{\}}_{t\in [0,T]} \subset {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup })$ given by

$$\begin{aligned} \!\,\,\text {d}U_t^{{\mathcal {E}}}(v,v_*)=-{{\widetilde{\nabla }}}\frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*)\!\,\,\text {d}(f_t\otimes f_t)(v,v_*), \end{aligned}$$

(3.1)

the pair ${\{}(f_t,U_t^{{\mathcal {E}}}){\}}_{t\in [0,T]}$ satisfies the nonlocal-local continuity equation (CE) in the sense of Definition 2.2.

In order to achieve a new gradient flow formulation of the equation above as steepest descent of the kinetic energy with respect to the collision metric defined in Sect. 2.4, we follow [2] and use the concept of curve of maximal slope with respect to a specific strong upper gradient, which is the square root of the dissipation functional, cf. (3.3) below. To motivate this, we consider the decay of the kinetic energy along a curve $f\in {{\,\textrm{AC}\,}}([0,T];({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ which is a solution of the nonlocal-local continuity equation (2.2), i.e., there exists a flux $\,\,\text {d}U_t = {{\hat{U}}}_t \!\,\,\text {d}(f\otimes f)$ such that the pair $\{(f_t,U_t)\}_{t\in [0,T]}$ is a weak solution in the sense of Definition 2.2. Formally applying the chain rule, we have

$$\begin{aligned} {{\mathcal {E}}}(f_T)- {{\mathcal {E}}}(f_0)&= \int _0^T \!\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \! {\widetilde{\nabla }}\frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*) {{\hat{U}}}_t(v,v_*) \sigma _e(|v-v_*|) \!\,\,\text {d}(f \otimes f)(v, v_*) \!\,\,\text {d}{t}. \end{aligned}$$

(3.2)

After an application of Young’s inequality to both the inner integrals with weight $\sigma _e \!\,\,\text {d}(f \otimes f)$, we observe

$$\begin{aligned} \int _0^T \!\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }}&\! {\widetilde{\nabla }}\frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*) {{\hat{U}}}_t(v,v_*) \sigma _e(|v-v_*|) \!\,\,\text {d}(f \otimes f)(v,v_*)\!\,\,\text {d}{t}, \\&\ge - \frac{1}{2} \int _0^T \left| {{\hat{U}}}_t(v,v_*)\right| ^2 \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}(f_t\otimes f_t)(v,v_*) \!\,\,\text {d}t \\&\quad - \frac{1}{2} \int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {\widetilde{\nabla }}\frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*)\right| ^2 \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}(f_t\otimes f_t)(v,v_*) \!\,\,\text {d}t \\&= -\frac{1}{2} \int _0^T {{\mathcal {A}}}(f_t, U_t) \!\,\,\text {d}t - \frac{1}{2} \int _0^T {{\mathcal {D}}}(f_t) \!\,\,\text {d}t, \end{aligned}$$

where the dissipation is defined by

$$\begin{aligned} {{\mathcal {D}}}(f):= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| v - v_*\right| ^2 \sigma _e(\left| v-v_*\right| ) \!\,\,\text {d}{{(}f \otimes f{)}}(v, v_*), \end{aligned}$$

(3.3)

cf. also (1.17), in the context of the formal derivation. This motivates our definition of gradient flow solutions as curves $f\in {{\,\textrm{AC}\,}}([0,T];({{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ in the zero locus of the De Giorgi functional

$$\begin{aligned} {{\mathcal {G}}}_T(f) := {{\mathcal {E}}}(f_T)- {{\mathcal {E}}}(f_0) + \frac{1}{2} \int _0^T {{\mathcal {A}}}(f_t, U_t) \!\,\,\text {d}t + \frac{1}{2} \int _0^T {{\mathcal {D}}}(f_t) \!\,\,\text {d}t . \end{aligned}$$

(3.4)

Based on the preceding computations we introduce our notion of gradient flow solutions as curves of maximal slope.

Definition 3.2

(Curves of maximal slope) A curve $f \in {{\,\textrm{AC}\,}}([0,T],({{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ is a curve of maximal slope if ${{\mathcal {G}}}_T(f)=0$.

In order to show that weak solutions to (3.1) are curves of maximal slope and to mathematically justify the definition of the De Giorgi functional (3.4), we need to rigorously derive the chain rule in (3.2). In particular, the chain rule implies that the square root of the dissipation functional ${{\mathcal {D}}}$, defined in (3.3), is a strong upper-gradient for ${{\mathcal {E}}}$ with respect to the extended metric $d_{{\mathcal {A}}}$ (cf. [2, Definition 1.2.1]).

3.1 The chain rule and characterisation of weak solutions

Lemma 3.3

(Stability and chain rule) Let $T>0$ and ${\{}(f_t,U_t){\}}_{t\in [0,T]} \in {\textrm{CE}}_T(\mu _0)$ for some $\mu _0 \in {{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}})$. Assume that

$$\begin{aligned} \int _0^T {{\mathcal {A}}}(f_t, U_t)^\frac{1}{2} \textrm{d}{t}< \infty , \qquad \text {and} \qquad \int _0^T \! {{\mathcal {A}}}(f_t, U_t)^\frac{1}{2} {{\mathcal {D}}}(f_t)^\frac{1}{2} \textrm{d}{t} <\infty \, , \end{aligned}$$

(3.5)

where ${{\mathcal {A}}}: {{\mathcal {P}}}({{\mathbb {R}}}) \times {{\mathcal {M}}}({{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }) \rightarrow (-\infty ,+\infty ]$ is the action, as defined in Definition 2.8, and ${{\mathcal {D}}}: {{\mathcal {P}}}({{\mathbb {R}}}) \rightarrow (-\infty ,+\infty ]$ is the dissipation defined in (3.3).

Then, the following properties hold:

• (a) $\sup _{t \in [0,T]} {{\mathcal {E}}}(f_t) < \infty$.

• (b) For any $0\le s \le t \le T$

  $$\begin{aligned} {{\mathcal {E}}}(f_t)- {{\mathcal {E}}}(f_s)&= \int _s^t \!\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \! {\widetilde{\nabla }}\frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*) \sigma _e(|v-v_*|) \textrm{d} U_\tau (v,v_*) \, \textrm{d}{\tau }. \end{aligned}$$

Proof

We define a globally Lipschitz approximation of $|v|^2/2$ which we can use as a test function in the weak formulation of (CE) by Remark 2.5. Let

$$\begin{aligned} \varphi _R(v) := \left\{ \begin{array}{ll} \displaystyle v^2/2, &{} v\in [0,R],\\ R^2/2 + R (v-R), &{} v \in [R, \infty ), \end{array} \right. \end{aligned}$$

(3.6)

and extend it to ${{\mathbb {R}}}$ by setting $\varphi _R(v) = \varphi _R(-v)$ for $v\in (-\infty , 0)$. Note, that this choice of test function also satisfies the following condition

$$\begin{aligned} \left| \frac{\varphi _R'(v) - \varphi _R'(v_*)}{v-v_*}\right| \le 1, \end{aligned}$$

which we will exploit in the subsequent analysis. For any weak solution of (CE), ${\{}(f_t, U_t){\}}_{t\in [0,T]}$, there holds (2.4), i.e.,

$$\begin{aligned} \int _{{\mathbb {R}}}\varphi (v) \!\,\,\text {d}{{\widetilde{f}}}_{T}(v) - \int _{{\mathbb {R}}}\varphi (v) \!\,\,\text {d}{{\widetilde{f}}}_{0}(v) = \int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {{\widetilde{\nabla }}} \varphi (v,v_*) \sigma _e(|v-v_*|) \!\,\,\text {d}U_t(v,v_*)\!\,\,\text {d}{t}, \end{aligned}$$

for any regular test function, $\varphi \in C_c^1({{\mathbb {R}}})$. In particular, choosing $\varphi = \varphi _R$, with $\varphi _R$ as in (3.6), we have

$$\begin{aligned} \int _{{\mathbb {R}}}\varphi _R(v) \!\,\,\text {d}{{\widetilde{f}}}_{T}(v) \!-\! \int _{{\mathbb {R}}}\varphi _R(v) \!\,\,\text {d}{{\widetilde{f}}}_{0}(v) = \int _0^T\!\! \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {{\widetilde{\nabla }}} \varphi _R(v,v_*) \sigma _e(|v-v_*|) \!\,\,\text {d}U_t(v,v_*)\!\,\,\text {d}t, \end{aligned}$$

(3.7)

where we can estimate the right-hand side as follows:

$$\begin{aligned}&\int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {{\widetilde{\nabla }}} \varphi _R (v,v_*) \sigma _e(|v-v_*|) \!\,\,\text {d}{U_t}(v,v_*)\!\,\,\text {d}{t} \\&= \int _0^T\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {{\widetilde{\nabla }}} \varphi _R (v,v_*) \sigma _e(|v-v_*|) {\hat{U}}_t(v,v_*) \!\,\,\text {d}(f_t\otimes f_t)(v,v_*)\!\,\,\text {d}{t} \\&\le \int _0^T \bigg ({\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \sigma _e(|v-v_*|) |{{\hat{U}}}_t(v,v_*)|^2 \!\,\,\text {d}(f_t\otimes f_t)(v,v_*)}\bigg )^{\frac{1}{2}}\\&\qquad \times \bigg ({\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {{\widetilde{\nabla }}} \varphi _R\right| ^2 \sigma _e(|v-v_*|) \!\,\,\text {d}(f_t\otimes f_t)(v,v_*)}\bigg )^{\frac{1}{2}} \!\,\,\text {d}t\\&=\int _0^T {{\mathcal {A}}}(f_t, U_t)^{\frac{1}{2}} \bigg ({\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {{\widetilde{\nabla }}} \varphi _R\right| ^2 \sigma _e(|v-v_*|) \!\,\,\text {d}(f_t \otimes f_t)(v,v_*)}\bigg )^{\frac{1}{2}} \!\,\,\text {d}t\\&=\int _0^T {{\mathcal {A}}}(f_t, U_t)^{\frac{1}{2}} \bigg ({\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| \frac{\varphi _R'(v) - \varphi _R'(v_*)}{v-v_*}\right| ^2 |v-v_*|^2 \sigma _e(|v-v_*|) \!\,\,\text {d}(f_t \otimes f_t)(v, v_*)}\bigg )^{\frac{1}{2}} \!\!\!\!\,\,\text {d}t\\&\le \int _0^T {{\mathcal {A}}}(f_t, U_t)^\frac{1}{2} {{\mathcal {D}}}(f_t)^\frac{1}{2}\!\,\,\text {d}t. \end{aligned}$$

Hence, the right-hand side is uniformly integrable and due to the pointwise convergence of $\varphi _R$ we may pass to the limit $R\rightarrow \infty$ in the weak form, (3.7), due to Lebesgue’s dominated convergence theorem. Hence we get

$$\begin{aligned} {{\mathcal {E}}}(f_T)- {{\mathcal {E}}}(f_0)&= \int _0^T \!\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \! {\widetilde{\nabla }}\frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*) \sigma _e(|v-v_*|) \!\,\,\text {d}{{\hat{U}}}_t(v,v_*) \, \!\,\,\text {d}f(v) \!\,\,\text {d}f(v_*)\!\,\,\text {d}{t}, \end{aligned}$$

as claimed in the statement.

As the test function $\varphi _R$ in (3.6) has linear growth at infinity, we can use it in the weak formulation in (2.4) by Remark 2.5, i.e.,

$$\begin{aligned} \begin{aligned} \frac{\!\,\,\text {d}{}}{\!\,\,\text {d}{t}} \int _{{{\mathbb {R}}}} \varphi _R(v) \!\,\,\text {d}{f_t}(v)&= - \frac{1-e}{4}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |v-v_*| {{\widetilde{\nabla }}}\varphi _R (v,v_*) (v_* - v) \!\,\,\text {d}{f_t}(v) \!\,\,\text {d}{f_t}(v_*)\, . \end{aligned} \end{aligned}$$

(3.8)

By expanding the definition of ${{\widetilde{\nabla }}}\varphi _R$ from (1.20) and using the short-hand notation

$$\begin{aligned} \!\,\,\text {d}g(v,v_*) := |v-v_*| {(}\varphi _R'(v_*)-\varphi _R'(v){)} (v_* - v) \!\,\,\text {d}(f\otimes f)(v, v_*), \end{aligned}$$

we have

$$\begin{aligned} \frac{\!\,\,\text {d}{}}{\!\,\,\text {d}{t}} \int _{{{\mathbb {R}}}} \varphi _R(v) \!\,\,\text {d}{f_t}(v)&= - \frac{1-e}{4}\left( {\mathcal {I}}_1 + \ldots + {\mathcal {I}}_9\right) , \end{aligned}$$

with

$$\begin{aligned} {\mathcal {I}}_1&= \frac{1}{2} \int _{-\infty }^{-R} \int _{-\infty }^{-R} \!\,\,\text {d}g(v,v_*), \quad {\mathcal {I}}_2 = \frac{1}{2} \int _{-\infty }^{-R} \int _{-R}^R \!\,\,\text {d}g(v,v_*), \\ {\mathcal {I}}_3&= \frac{1}{2} \int _{-\infty }^{-R} \int _{R}^\infty \!\,\,\text {d}g(v,v_*), \end{aligned}$$

and

$$\begin{aligned} {\mathcal {I}}_4 = \frac{1}{2} \int _{-R}^{R} \int _{-\infty }^{-R} \!\,\,\text {d}g(v,v_*), \quad {\mathcal {I}}_5 = \frac{1}{2} \int _{-R}^{R} \int _{-R}^R \!\,\,\text {d}g(v,v_*), \quad {\mathcal {I}}_6 = \frac{1}{2} \int _{-R}^{R} \int _{R}^\infty \!\,\,\text {d}g(v,v_*), \end{aligned}$$

as well as

$$\begin{aligned} {\mathcal {I}}_7 = \frac{1}{2} \int _{R}^{\infty } \int _{-\infty }^{-R} \!\,\,\text {d}g(v,v_*), \quad {\mathcal {I}}_8 = \frac{1}{2} \int _{R}^{\infty } \int _{-R}^R \!\,\,\text {d}g(v,v_*), \quad {\mathcal {I}}_9 = \frac{1}{2} \int _{R}^{\infty } \int _{R}^\infty \!\,\,\text {d}g(v,v_*). \end{aligned}$$

It is immediately clear that ${\mathcal {I}}_1 = {\mathcal {I}}_9 = 0$, as ${{\widetilde{\nabla }}} \varphi _R$ vanishes in the respective ranges for $v,v_*$, whence $g(v,v_*)=0$. It is easy to verify that ${\mathcal {I}}_j \ge 0$, for $j\ne 5$. We expand on the argument for ${\mathcal {I}}_2$ and note that arguments along similar lines will allow us to treat the remaining terms. Indeed,

$$\begin{aligned} {\mathcal {I}}_2&= \int _{-\infty }^{-R}\int _{-R}^R |v - v_*| (v_* + R) (v_*-v) \!\,\,\text {d}{(f_t \otimes f_t)}(v,v_*) \ge 0, \end{aligned}$$

since $v_* \ge -R \ge v$ in the domain of integration. Substituting ${\mathcal {I}}_j\ge 0$, for $j\ne 5$, into (3.8), we get

$$\begin{aligned} \int _{{{\mathbb {R}}}} \varphi _R(v)\!\,\,\text {d}{f_t}(v) - \int _{{{\mathbb {R}}}} \varphi _R(v)\!\,\,\text {d}{f_s}(v) + \frac{1-e}{4} \int _s^t\int _{-R}^R\int _{-R}^R |v-v_*|^3 \!\,\,\text {d}{(f_t \otimes f_t)}(v,v_*) \le 0, \end{aligned}$$

having integrated in time. By the dominated convergence theorem and the finite initial kinetic energy, we obtain

$$\begin{aligned} \frac{1}{2} \int _{{{\mathbb {R}}}} |v|^2 \!\,\,\text {d}{f_t}(v) + \frac{1-e}{4} \int _0^t\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} |v-v_*|^3 \!\,\,\text {d}{(f_t \otimes f_t)}(v,v_*) \le \frac{1}{2} \int _{{{\mathbb {R}}}} |v|^2\!\,\,\text {d}{f_0}(v). \end{aligned}$$

$\square$

Remark 3.4

• (1) Let us highlight that the proof of the dissipation of the kinetic energy via the truncation argument using the test functions, $\varphi _R$, is absolutely independent of assumption (3.5). Indeed, it is not too surprising that we require the kinetic energy to be dissipated along the aggregation equation regardless of the metric setting. In particular, any weak solution from Definition 3.1 satisfies

  $$\begin{aligned} {{\mathcal {E}}}(f_T) + \int _0^T {{\mathcal {D}}}(f_t) \!\,\,\text {d}{t} \le {{\mathcal {E}}}(f_0). \end{aligned}$$

  (3.9)

• (2) Note that the statement of the theorem is true for any absolutely continuous curve, namely ${\{}f_t{\}}_{t\in [0,T]} \in {{\,\textrm{AC}\,}}([0,T];({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ with $f_0\in {{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}})$ and $\int _0^T {{\mathcal {D}}}(f_t)\!\,\,\text {d}t < \infty$. In this case the action is always bounded and implies the existence of an associated flux, using the characterisation of absolutely continuous curves stated in Proposition 2.20.

As direct consequence of the chain rule we have ${{\mathcal {D}}}^\frac{1}{2}$ is a strong upper gradient with respect to the distance $d_{{\mathcal {A}}}$ in the sense of [2, Definition 1.2.1]

Corollary 3.5

For any curve $f\in {{\,\textrm{AC}\,}}([0,T];({{\mathcal {P}}}({{\mathbb {R}}}),d_{{\mathcal {A}}}))$ with $f_0\in {{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}})$ it holds

$$\begin{aligned} |{{\mathcal {E}}}(f_t)-{{\mathcal {E}}}(f_s))|\le \int _s^t {{\mathcal {D}}}(f_r)^\frac{1}{2} |f'_r| \textrm{d} r, \qquad \forall \, 0\le s\le t\le T, \end{aligned}$$

that is ${{\mathcal {D}}}^\frac{1}{2}$ is a strong upper gradient for ${{\mathcal {E}}}$.

Proof

Without loss of generality, we can assume $\int _s^t {{\mathcal {D}}}(f_r)^\frac{1}{2} |f'_r| \!\,\,\text {d}r<\infty$, otherwise the claim is immediately true. The result follows from Lemma 3.3 by applying Cauchy-Schwartz inequality and using the characterisation of absolutely continuous curves stated in Proposition 2.20. $\square$

We are now able to characterise weak solutions as curves of maximal slope in the sense of Definition 3.2.

Theorem 3.6

(Weak solutions are curves of maximal slope) A curve $f\in {{\,\textrm{AC}\,}}{(}[0,T], ({{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}){)}$ is a weak solution to (1.14) in the sense of Definition 3.1 if and only if ${\mathcal {G}}_T(f)=0$.

Proof

Let f be a weak solution in the sense of Definition 3.1 with corresponding flux $U_t^{{\mathcal {E}}}(v,v_*)$ given by (3.1). It can be checked that ${{\mathcal {A}}}(f_t,U_t^{{\mathcal {E}}})={{\mathcal {D}}}(f_t)$ and by the energy dissipation (3.9) also follows that ${{\mathcal {E}}}(f_T) + \int _0^T {{\mathcal {D}}}(f_t)\!\,\,\text {d}t \le {{\mathcal {E}}}(f_0) < \infty$. In particular, ${{\mathcal {E}}}(f_T) - {{\mathcal {E}}}(f_0) + \frac{1}{2}\int _0^T ({{\mathcal {A}}}(f_t,U_t^{{\mathcal {E}}}) +{{\mathcal {D}}}(f_t)) \!\,\,\text {d}{t} \le 0$, whence ${{\mathcal {G}}}_T(f)\le 0$ and $f\in {{\,\textrm{AC}\,}}([0,T];({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$. Thus, by the chain rule Lemma 3.2, we have that ${{\mathcal {G}}}_T(f)\ge 0$. Hence, ${{\mathcal {G}}}_T(f)=0$.

Let us now assume that $f\in {{\,\textrm{AC}\,}}([0,T];({{\mathcal {P}}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ satisfies ${{\mathcal {G}}}_T(f)=0$. According to Proposition 2.20 there exists a unique family $\{\!\,\,\text {d}U_t = {{\hat{U}}}_t \!\,\,\text {d}(f_t \otimes f_t)\}_{t\in [0,T]}$ such that ${\{}(f_t, U_t){\}}_{t\in [0,T]}\in {\textrm{CE}}_T$ and $\int _0^T{{\mathcal {A}}}(f_t, U_t)\!\,\,\text {d}t<\infty$. By the chain rule Lemma 3.3, we obtain

$$\begin{aligned} 0&={{\mathcal {G}}}_T(f_t) = {{\mathcal {E}}}(f_T) - {{\mathcal {E}}}(f_0) + \frac{1}{2} \int _0^T {{\mathcal {A}}}(f_t, U_t) \!\,\,\text {d}t + \frac{1}{2} \int _0^T{{\mathcal {D}}}(f_t)\!\,\,\text {d}t\\&=\int _0^T\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {{\widetilde{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f} \sigma _e(|v_* - v|) {{\hat{U}}}_t(v,v_*) \sigma _e(|v_*-v|)\!\,\,\text {d}(f_t\otimes f_t)(v,v_*)\!\,\,\text {d}t \\&\qquad + \frac{1}{2} \int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \bigg ({ \left| {{\widetilde{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f}\right| ^2 + \left| {{\hat{U}}}_t\right| ^2}\bigg ) \sigma _e(|v_*-v|) \!\,\,\text {d}(f_t\otimes f_t)(v,v_*) \!\,\,\text {d}{t}\\&= \frac{1}{2} \int _0^T \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \left| {{\widetilde{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f} + {{\hat{U}}}_t\right| ^2\sigma _e(|v_*-v|) \!\,\,\text {d}(f_t\otimes f_t)(v,v_*) \!\,\,\text {d}{t} . \end{aligned}$$

Hence

$$\begin{aligned} {{\hat{U}}}_t(v,v_*) = - {{\widetilde{\nabla }}} \frac{\delta {{\mathcal {E}}}}{\delta f}(v,v_*) = v - v_*, \end{aligned}$$

which implies that $U_t = U_t^{{\mathcal {E}}}$, from (3.1). $\square$

To establish the existence of minimisers of the De Giorgi functional in (3.4), we have to prove lower semicontinuity of the dissipation.

Proposition 3.7

(Lower semicontinuity of the dissipation) Let ${\{}f^n{\}}_{n\in {{\mathbb {N}}}} \subset {{\mathcal {P}}}({{\mathbb {R}}})$ such that $f^n \rightarrow f \in {{\mathcal {P}}}({{\mathbb {R}}})$, then it holds

$$\begin{aligned} \liminf _{n\rightarrow \infty } {{\mathcal {D}}}(f^n) \ge {{\mathcal {D}}}(f). \end{aligned}$$

Proof

We consider a cut-off away from the diagonal. Let $\varphi _R(r)\in C_c^1({{\mathbb {R}}})$ be such that $\varphi _R(r) = 1$ for $r\in [-R,R]$ and $\varphi _R(r)= 0$ for $\left| r\right| \ge 2R$, then we have by positivity of the integrand in ${{\mathcal {D}}}(f^n)$ the estimate

$$\begin{aligned} {{\mathcal {D}}}(f^n) \ge \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} \varphi _R(|v-v_*|) |v-v_*|^2 \sigma _e(|v-v_*|) \!\,\,\text {d}(f^n \otimes f^n)(v,v_*). \end{aligned}$$

Hence, the proof is concluded by letting $n\rightarrow \infty$ first, and via monotone convergence for $R\rightarrow \infty$. $\square$

3.2 Stability and existence by particle approximation

To discuss the existence of curves of maximal slope, we proceed by a strategy similar to showing existence of solutions to the aggregation equation by finite-dimensional approximations, cf. [14].

Let us first summarise the given compactness and lower semicontinuity statements for the objects in the definition of the De Giorgi functional, cf. (3.4), which provide the stability of curves of maximal slope in our setting. By combining the lower semicontinuity of the action in Proposition 2.11 and the lower semicontinuity of the dissipation in Proposition 3.7, as well as noting that the kinetic energy (1.6) is lower semicontinuous with respect to narrow convergence due to the convexity of the integrand, we obtain the stability of curves of maximal slope.

Theorem 3.8

(Stability of curves of maximal slope) Let the sequence ${\{}f^n{\}}_{n\in {{\mathbb {N}}}} \subset {{\,\textrm{AC}\,}}([0,T], ({{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ be such that $\sup _{n\in {{\mathbb {N}}}}{{\mathcal {G}}}(f^n)< \infty$ and ${{\mathcal {E}}}(f^n_0)\rightarrow {{\mathcal {E}}}(f_0)$ with $f^n_0\rightarrow f_0$, then there exists some $f\in {{\,\textrm{AC}\,}}([0,T],({{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}))$ such that $f_t^n\rightarrow f_t$, for a.e. $t\in [0,T]$ and

$$\begin{aligned} \liminf _{n\rightarrow \infty } {{\mathcal {G}}}(f^n) \ge {{\mathcal {G}}}(f). \end{aligned}$$

Based on this stability statement for curves of maximal slope we may now construct solutions devising an approximation by particles. Let us stress that existence of minimisers for ${{\mathcal {G}}}_T$ can be shown by the direct method of calculus of variations. However, this does not provide that minima are zeros of ${{\mathcal {G}}}_T$.

Theorem 3.9

(Existence by particle approximation) For any $f_0 \in {{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}})$, that is ${{\mathcal {E}}}(f_0)<\infty$, there exists a curve of maximal slope.

Proof

The strategy is based on constructing a particle approximation of the initial measure, $f_0 \in {{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}})$, by arguing that there exists a sequence of empirical measures $\bigg ({f^n_0=\frac{1}{n}\sum _{i=1}^n \delta _{v_i^n(0)}}\bigg )_{n \in {{\mathbb {N}}}}$ such that

$$\begin{aligned} d_2(f_0, f^n_0) \rightarrow 0 , \qquad \text {as } n\rightarrow \infty . \end{aligned}$$

Taking the existence of $f^n_0$ for granted, we can then follow the atoms of the initial empirical measure $f^n_0$ along the solution of the associated system of ordinary differential equations

$$\begin{aligned} \frac{\!\,\,\text {d}{v}_i^n}{\!\,\,\text {d}{t}} = -\frac{2}{n} \sum _{j=1}^n \sigma _e\bigg ({\left| v_i^n(t)-v_j^n(t)\right| }{(}v_i^n(t)-v_j^n(t){)}\bigg ), \end{aligned}$$

whose existence is guaranteed by the classical Cauchy–Lipschitz theory. This gives rise to a family of curves $(f^n_t)_{t\in [0,T]}$ for each $n\in {{\mathbb {N}}}$, which are readily verified to be weak solutions to (1.14) and, by Theorem 3.6, also curves of maximal slope in the sense of Definition 3.2. In particular, this sequence of solution satisfies the a priori estimate (3.9), and they have uniformly bounded action, thus they are curves in ${{\,\textrm{AC}\,}}{(}[0,T],({{\mathcal {P}}}_2^{\textrm{cm}}({{\mathbb {R}}}),d_{{{\mathcal {A}}}}){)}$. Moreover, since convergence in $d_2$ implies $f^n_0 \rightarrow f_0$ and convergence of second order moments, we also obtain ${{\mathcal {E}}}(f^n_0) \rightarrow {{\mathcal {E}}}(f_0)$. Hence, we can conclude the proof by applying the stability statement from Theorem 3.8 in the limit $n\rightarrow \infty$ and conclude

$$\begin{aligned} 0 = \liminf _{n\rightarrow \infty } {{\mathcal {G}}}_T(f^n) \ge {{\mathcal {G}}}_T(f) \ge 0. \end{aligned}$$

Hence the limit f is also a curve of maximal slope.

Let us now turn to the construction of the approximation $f^n_0$ of the initial measure $f_0$, which consists of three steps: mollification, truncation, and approximation by particles. Let $\varepsilon > 0$ be arbitrary.

Step 1. In the mollification step, we find some $f_{\textrm{ac}}^\varepsilon \in {L}^1({{\mathbb {R}}}) \cap {{\mathcal {P}}}({{\mathbb {R}}})$ such that $d_2(f_0, f_{\textrm{ac}}^\varepsilon ) < \varepsilon /3$, which can be easily done by mollifying $f_0$ with a smooth bump function at a suitable scale $\delta =\delta (\varepsilon )>0$. Furthermore, we note that

$$\begin{aligned} \int _{{{\mathbb {R}}}}\left| v\right| ^2\!\,\,\text {d}f_{\textrm{ac}}^\varepsilon (v)&= \int _{{{\mathbb {R}}}}\int _{{{\mathbb {R}}}} \left| v\right| ^2 \varphi ^\delta (v-w) \!\,\,\text {d}{v} \!\,\,\text {d}{f_0}(w) \\&\le \int _{{{\mathbb {R}}}} \left( {2\left| w\right| ^2 + 2\delta ^2}\right) \!\,\,\text {d}{f_0}(w) = 4 {{\mathcal {E}}}(f_0) + 2\delta ^2 . \end{aligned}$$

Step 2. We will now use the fact that the second moment control on $f_{\textrm{ac}}^\varepsilon$, gives us uniform tightness which allows to cut off, in a quantitative fashion, its tails. The standard tightness estimate tells us that

$$\begin{aligned} \int _{[-R,R]^c} \!\,\,\text {d}{f_{\textrm{ac}}^\varepsilon } \le \frac{1}{R^2} \int _{[-R,R]^c} \left| v\right| ^2 \!\,\,\text {d}{f_{\textrm{ac}}^\varepsilon } \le \frac{4 {{\mathcal {E}}}(f_0) + 2\delta ^2}{R^2} \, . \end{aligned}$$

Consider now the cut off and renormalised measure $f_{\textrm{ac},R}^\varepsilon = f_{\textrm{ac}}^\varepsilon |_{[-R,R]}/\left\| f_{\textrm{ac}}^\varepsilon \right\| _{{L}^1([-R,R])}$. Using [25, Theorem 6.15], we have that

$$\begin{aligned} d_2(f_{\textrm{ac}}^\varepsilon ,f_{\textrm{ac},R}^\varepsilon ) \le&\left( {2\int _{{{\mathbb {R}}}}\left| v\right| ^2 \left| f_{\textrm{ac}}^\varepsilon -f_{\textrm{ac},R}^\varepsilon \right| \!\,\,\text {d}{v} }\right) ^{\frac{1}{2}} \\ \le&\left( {\frac{2\left( 1- \left\| f_{\textrm{ac}}^\varepsilon \right\| _{{L}^1([-R,R])}\right) }{\left\| f_{\textrm{ac}}^\varepsilon \right\| _{{L}^1([-R,R])}} }\right) ^{\frac{1}{2}} \left( {4 {{\mathcal {E}}}(f_0) + 2\delta ^2}\right) ^{\frac{1}{2}} +\sqrt{2} \int _{[-R,R]^c}\left| v\right| ^2\!\,\,\text {d}{f_{\textrm{ac}}^\varepsilon } \,. \end{aligned}$$

It is now clear that for a fixed $\varepsilon >0$, we can choose $R=R(\varepsilon )>0$ such that it holds that

$$\begin{aligned} d_2(f_{\textrm{ac}}^\varepsilon ,f_{\textrm{ac},R}^\varepsilon ) <\frac{\varepsilon }{3}. \end{aligned}$$

Step 3. Finally, we use a classical result from measure theory (for example cf. [9, Example 8.16 (i)]) that empirical measures are dense in probability measures in the narrow topology. However, since $f_{\textrm{ac},R}^\varepsilon$ has compact support, the sequence of empiricals we construct will necessarily converge in $d_2$. Thus, we can find a measure of the form $f^n_0:=\frac{1}{n}\sum _{i=1}^n\delta _{v_i}$ for some $n=n(\varepsilon )$ such that

$$\begin{aligned} d_2(f^n_0,f_{\textrm{ac},R}^\varepsilon ) < \frac{\varepsilon }{3}. \end{aligned}$$

This completes the proof of the existence of an approximating sequence of empirical measures and hence the proof. $\square$

Appendix

1.1 Formal derivation of the Boltzmann equation

We present a formal derivation of the Boltzmann equation from a gain-loss argument. For the subsequent argument, it is more useful to think of the collisions in terms of the matrix $T: {{\mathbb {R}}}^2 \rightarrow {{\mathbb {R}}}^2$ given by

$$\begin{aligned} T= \begin{pmatrix} \frac{1- e}{2} &{} \frac{1+ e}{2} \\ \frac{1+ e}{2} &{} \frac{1- e}{2} \end{pmatrix}, \end{aligned}$$

which maps the pre-collisional velocities to the post-collisional velocities, i.e.,

$$\begin{aligned} \begin{pmatrix} v'\\ v'_* \end{pmatrix}=T\begin{pmatrix} v\\ v_* \end{pmatrix}. \end{aligned}$$

Respectively, its inverse, given by

$$\begin{aligned} T^{-1}= \begin{pmatrix} \frac{1- e^{-1}}{2} &{} \frac{1+ e^{-1}}{2} \\ \frac{1+ e^{-1}}{2} &{} \frac{1- e^{-1}}{2} \end{pmatrix} \,, \end{aligned}$$

maps post-collisional velocities to pre-collisional velocities. Note that $\det T= -e$ and $\det (T^{-1})= - e^{-1}$.

A formal derivation for the inelastic Boltzmann equation can be obtained by describing the evolution of the velocity distribution, $f_t$, using a simple gain-loss balance argument. The density at a point v in velocity space is produced by all collisions of particles with ‘v’ as one of their post-collisional velocities and is destroyed by all collisions with ‘v’ as one of their pre-collisional velocities.

We thus split the derivation into two parts: gain and loss. We consider an $\varepsilon >0$ interval $\Omega _\varepsilon =[\nu -\varepsilon ,\nu +\varepsilon ]$ around some velocity $\nu$ and try to obtain the rate of production of density in this interval. Formally, we can integrate over the rate of production for those pre-collisional velocities $\alpha = T^{-1}_1(v,v_*)$ and $\beta = T^{-1}_2(v,v_*)$ that produce v after collision and arrive at

$$\begin{aligned} \bigg ({\int _{{\Omega _\varepsilon }} \partial _t {f_t}(v) \!\,\,\text {d}{v}}\bigg )_{\text {gain}} =\iint _{{{\mathbb {R}}}^2} {f_t}(\alpha ) {f_t}(\beta ) \sigma (\left| \alpha -\beta \right| ) \mathbbm {1}_{\Omega _\varepsilon }(v) \!\,\,\text {d}{\alpha } \!\,\,\text {d}{\beta }. \end{aligned}$$

The function $\sigma =\sigma (|v|)$ models the frequency of the collisions, depending on the strength of the relative velocities and referred to as the collision kernel. We now make the change of variables $(\alpha ,\beta ) \mapsto (v,v_*)$ to obtain

$$\begin{aligned} \bigg ({\int _{{\Omega _\varepsilon }} \partial _t {f_t}(v) \!\,\,\text {d}{v}}\bigg )_{\text {gain}} = e \iint _{{{\mathbb {R}}}^2} {f_t}(T^{-1}_1(v,v_*)) {f_t}(T^{-1}_2(v,v_*)) \sigma ( e^{-1}\left| v-v_*\right| ) \mathbbm {1}_{\Omega _\varepsilon } (v) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*}. \end{aligned}$$

The loss term is simpler as we obtain

$$\begin{aligned} \bigg ({\int _{{\Omega _\varepsilon }} \partial _t {f_t}(v) \!\,\,\text {d}{v}}\bigg )_{\text {loss}} = \iint _{{{\mathbb {R}}}^2} {f_t}(v) {f_t}(v_*) \sigma (\left| v-v_*\right| ) \mathbbm {1}_{\Omega _\varepsilon } (v) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*}, \end{aligned}$$

where we have integrated over the rate of destruction over all pre-collisional velocities with one of the particles having velocity v. Subtracting the two, dividing by $\varepsilon$, and passing to the limit we have the strong form as

$$\begin{aligned} \partial _t f_t(v) =&e \int _{{{\mathbb {R}}}} {f_t}(T^{-1}_1(v,v_*)) {f_t}(T^{-1}_2(v,v_*)) \sigma ( e^{-1}\left| v-v_*\right| ) \!\,\,\text {d}{v_*} \\&-\int _{{{\mathbb {R}}}} {f_t}(v) {f_t}(v_*) \sigma (\left| v-v_*\right| ) \!\,\,\text {d}{v_*}. \end{aligned}$$

The weak form can be obtained by testing against $\varphi \in C^\infty ({{\mathbb {R}}})$ as follows

$$\begin{aligned} \begin{aligned} {\langle }\varphi , \partial _t {f_t}{\rangle } =&e \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(T^{-1}_1(v,v_*)) {f_t}(T^{-1}_2(v,v_*)) \sigma ( e^{-1}\left| v-v_*\right| ) \varphi (v) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*}\\&-\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(v) {f_t}(v_*) \sigma (\left| v-v_*\right| ) \varphi (v) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*} \, . \end{aligned} \end{aligned}$$

We would now like to bring the collision operator into a more standard form. To this end, we relabel the gain term and change variables back to $(v,v_*)= T^{-1}(v',v'_*)$, to obtain

$$\begin{aligned} {\langle }\varphi , \partial _t f_t{\rangle }&=e \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(T^{-1}_1(v',v'_*)) {f_t}(T^{-1}_2(v',v'_*)) \sigma ( e^{-1}\left| v'-v'_*\right| ) \varphi (v') \!\,\,\text {d}{v'} \!\,\,\text {d}{v'_*}\\ {}&\quad -\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(v) {f_t}(v_*) \sigma (\left| v-v_*\right| ) \varphi (v) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*}\\&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(v) {f_t}(v_*) \sigma ( \left| v-v_*\right| ) \varphi (v') \!\,\,\text {d}{v} \!\,\,\text {d}{v_*}\\&\quad -\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(v) {f_t}(v_*) \sigma (\left| v-v_*\right| ) \varphi (v) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*} \\&= \iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(v) {f_t}(v_*) \sigma ( \left| v-v_*\right| ) (\varphi (v') -\varphi (v)) \!\,\,\text {d}{v} \!\,\,\text {d}{v_*} ={\langle }\varphi , Q({f_t},{f_t}_*){\rangle } \, . \end{aligned}$$

One can symmetrise once more by using the transformation $v \mapsto v_*$ which also induces the transformation $v' \mapsto v'_*$. Thus, one obtains

$$\begin{aligned} {\langle }\varphi ,Q({f_t},{f_t}_*){\rangle }&= \frac{1}{2}\iint _{{{\mathbb {R}}}^2_{\!\scriptscriptstyle \diagup }} {f_t}(v) {f_t}(v_*) \sigma ( \left| v-v_*\right| ) {(}\varphi (v')+ \varphi (v'_*) -\varphi (v) -\varphi (v_*){)} \!\,\,\text {d}{v} \!\,\,\text {d}{v_*} \, . \end{aligned}$$