# A trajectory map for the pressureless Euler equations

## TL;DR

This paper develops a trajectory map approach to prove the existence of solutions for the pressureless Euler equations modeling inelastic particle collisions on a line.

## Contribution

It introduces a novel Lagrangian coordinate framework to establish solution existence for pressureless Euler equations with inelastic collisions.

## Key findings

- Solutions exist for given initial conditions.
- Reformulation in Lagrangian coordinates simplifies the analysis.
- Trajectory map approach effectively handles inelastic collisions.

## Abstract

We consider the dynamics of a collection of particles that interact pairwise and are restricted to move along the real line. Moreover, we focus on the situation in which particles undergo perfectly inelastic collisions when they collide. The equations of motion are a pair of partial differential equations for the particles' mass distribution and local velocity. We show that solutions of this system exist for given initial conditions by rephrasing these equations in Lagrangian coordinates and then by solving for the associated trajectory map.