A Wasserstein Approach to the One-Dimensional Sticky Particle System

We present a simple approach for studying the one-dimensional pressureless Euler system via adhesion dynamics in the Wasserstein space $\mathcal{P}_2(\mathbb{R})$ 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 in $\mathcal{P}_2(\mathbb{R})$ and the 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.