Sticky particle dynamics with interactions

TL;DR

This paper studies one-dimensional pressureless fluid flows with self-interactions, using a differential inclusion framework and sticky particle dynamics, providing existence, stability, and explicit solution formulas especially for the Euler-Poisson system.

Contribution

It introduces a differential inclusion approach for self-interacting pressureless fluids and proves global existence and stability, with explicit solutions for the Euler-Poisson case.

Findings

Established a differential inclusion framework for the flow

Proved global existence and stability of solutions

Derived explicit formulas for Euler-Poisson solutions

Abstract

We consider compressible pressureless fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid itself. We explain how this flow can be described by a differential inclusion on the space of transport maps, in particular when a sticky particle dynamics is assumed. We study a discrete particle approximation and we prove global existence and stability results for solutions of this system. In the particular case of the Euler-Poisson system in the attractive regime our approach yields an explicit representation formula for the solutions.