Global regularity for 1D Eulerian dynamics with singular interaction forces

TL;DR

This paper proves that solutions to a one-dimensional Euler-Poisson-Alignment system with singular alignment forces and attraction/repulsion potentials remain globally regular, extending previous results to more general models.

Contribution

It generalizes prior work by incorporating attraction and repulsion potentials into the singular alignment framework, establishing global regularity for broader classes of initial data.

Findings

Singular alignment kernels regularize solutions effectively.

Global regularity persists with added attraction/repulsion potentials.

The results extend previous models to more general interaction forces.

Abstract

The Euler-Poisson-Alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in the paper by Do, Kiselev, Ryzhik and Tan. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.