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

TL;DR

This work establishes an $L^2$-gradient-flow framework for the one-dimensional pressureless Euler–alignment system, representing the dynamics as a gradient flow on the convex cone of nondecreasing mappings with an added indicator term. The authors prove existence and uniqueness of a Lagrangian $L^2$-gradient-flow solution, show that it induces a distributional solution of the Euler–alignment system and an entropy solution of the associated scalar balance law, and connect clustering to the projection onto the cone via the lower convex envelope $A^{**}$. By constructing particle approximations (sticky CS dynamics) and passing to the gradient-flow limit, they derive projection formulas and demonstrate globally sticky behavior, enabling a semigroup structure in the space of probability measures with finite quadratic moments. The analysis explains cluster formation and flocking in terms of convex-analytic objects (tangent/normal cones, $A^{**}$) and tail properties of the communication protocol $oldsymbol{\phi}$, providing a rigorous bridge between gradient-flow methods and entropy-conservation-law viewpoints in 1D. This framework broadens the toolkit for understanding alignment-driven aggregation in measure-valued settings and clarifies the role of convexity and stickiness in long-time dynamics.

Abstract

We show how the sticky dynamics for the one-dimensional pressureless Euler-alignment system can be obtained as an $L^2$-gradient flow of a convex functional. This is analogous to the Lagrangian evolution introduced by Natile and Savaré for the pressureless Euler system, and by Brenier et al. for the corresponding system with a self-interacting force field. Our Lagrangian evolution can be seen as the limit of sticky particle Cucker-Smale dynamics, similar to the solutions obtained by Leslie and Tan from a corresponding scalar balance law, and provides us with a uniquely determined distributional solution of the original system in the space of probability measures with quadratic moments and corresponding square-integrable velocities. Moreover, we show that the gradient flow also provides an entropy solution to the balance law of Leslie and Tan, and how their results on cluster formation follow naturally from (non-)monotonicity properties of the so-called natural velocity of the flow.

Figures (6)

• Figure 1: Illustration of the function $\Phi$ in the case of (a) bounded and (b) weakly singular communication protocols $\phi$. We will return to the convex function $W_{\Phi}$, the primitive of $\Phi$, in Section \ref{['s:Lag']}.
• Figure 2: Illustration of natural velocities $\bar{\psi}_i$ for six particles with different masses $m_i$, $i \in \{1,\dots,6\}$, where we have introduced the breakpoints $\theta_i = \sum_{k=1}^{i}m_k$.
• Figure 3: An $\Xi_{W} \in \mathscr{N}_X$ where $\Omega_{X} = (\omega_1^-,\omega_1^+) \sqcup (\omega_2^-,\omega_2^+)$.
• Figure 4: The various projections of $f(m) = -4\pi\cos\mleft(4\pi m\mright)$ and their primitives for $\Omega_X = (0,\frac{1}{8}) \sqcup (\frac{5}{16},\frac{1}{2}) \sqcup (\frac{5}{8},\frac{7}{8})$. We observe that the primitive of $\mathop{}\mathrm{P}_{\mathscr{K}}f$ coincides with the lower convex envelope of $\int_{0}^{m}f(\omega)\mathop{}\!\mathrm{d}\omega = -\sin(4\pi m)$ on $[0,1]$.
• Figure 5: Consider the six particles from Figure \ref{['fig:discflux']}, where the first collision happens between particles 3, 4 and 5 at time $t_1$. We have plotted the corresponding step functions $\bar{\Psi}^6$ and $\Psi^6_{t_1}$, as well as the primitive of their difference.

Theorems & Definitions (41)

• Lemma 1.1
• proof
• Definition 1.2: Distributional solutions of the Euler-alignment system
• Lemma 2.1
• Lemma 3.1: Characterization of the normal cone $N_X\mathscr{K}$
• Proposition 3.2: Projection on $\mathscr{K}_{(\alpha,\beta)}$
• Lemma 3.3
• Remark 3.4
• Lemma 4.1: Uniform continuity of $\Phi\ast\rho$
• proof