A Wasserstein approach to the one-dimensional sticky particle system

TL;DR

This paper introduces a Wasserstein space-based approach to analyze the one-dimensional pressureless Euler system, providing explicit estimates and a measure-theoretic evolution semigroup for sticky particle dynamics.

Contribution

It develops a novel measure-theoretic framework and explicit estimates for the sticky particle system using Wasserstein space and gradient flow techniques.

Findings

Explicit estimates of solutions based on initial mass and momentum

Construction of an evolution semigroup in measure space

Connection between the semigroup and gradient flow of Wasserstein distance

Abstract

We present a simple approach to study the one-dimensional pressureless Euler system via adhesion dynamics in the Wasserstein space of probability measures with finite quadratic moments. Starting from a discrete system of a finite number of "sticky" particles, we obtain new explicit estimates of the solution in terms of the initial mass and momentum and we are able to construct an evolution semigroup in a measure-theoretic phase space, allowing mass distributions with finite quadratic moment and corresponding L^2-velocity fields. We investigate various interesting properties of this semigroup, in particular its link with the gradient flow of the (opposite) squared Wasserstein distance. Our arguments rely on an equivalent formulation of the evolution as a gradient flow in the convex cone of nondecreasing functions in the Hilbert space L^2(0,1), which corresponds to the Lagrangian system of coordinates given by the canonical monotone rearrangement of the measures.