Sticky particles and the pressureless Euler equations in one spatial dimension

TL;DR

This paper studies the dynamics of sticky particles on a line, showing that their behavior can be extended to solutions of the pressureless Euler equations, with a focus on semi-convex interaction potentials.

Contribution

It demonstrates that the sticky particle property persists in the continuum limit, leading to existence results for pressureless Euler solutions with entropy conditions.

Findings

Sticky particles remain stuck after collisions.

Existence of pressureless Euler solutions with entropy inequality.

Behavior preserved as number of particles tends to infinity.

Abstract

We consider the dynamics of finite systems of point masses which move along the real line. We suppose the particles interact pairwise and undergo perfectly inelastic collisions when they collide. In particular, once particles collide, they remain stuck together thereafter. Our main result is that if the interaction potential is semi-convex, this sticky particle property can quantified and is preserved upon letting the number of particles tend to infinity. This is used to show that solutions of the pressureless Euler equations exist for given initial conditions and satisfy an entropy inequality.