Second order two-species systems with nonlocal interactions: existence and large damping limits

TL;DR

This paper develops a mathematical framework for second order two-species systems with nonlocal interactions in one dimension, proving existence, uniqueness, and convergence results, including large damping limits and long-time behavior, supported by numerical simulations.

Contribution

It introduces a novel Lagrangian approach to analyze second order two-species systems with nonlocal interactions, establishing existence, uniqueness, and asymptotic convergence results.

Findings

Existence and uniqueness of solutions for smooth nonlocal potentials.

Convergence to first order systems in large damping and large time limits.

Solutions tend to Dirac delta distributions over time.

Abstract

We study the mathematical theory of second order systems with two species, arising in the dynamics of interacting particles subject to linear damping, to nonlocal forces and to external ones, and resulting into a nonlocal version of the compressible Euler system with linear damping. Our results are limited to the $1$ space dimensional case but allow for initial data taken in a Wasserstein space of probability measures. We first consider the case of smooth nonlocal interaction potentials, not subject to any symmetry condition, and prove existence and uniqueness. The concept of solutions relies on a stickiness condition in case of collisions, in the spirit of previous works in the literature. The result uses concepts from classical Hilbert space theory of gradient flows (cf. Brezis [7]) and a trick used in [4]. We then consider a large-time and large-damping scaled version of our system and prove convergence to solutions to the corresponding first order system. Finally, we consider the case of Newtonian potentials -- subject to symmetry of the cross-interaction potentials -- and external convex potentials. After showing existence in the sticky particles framework in the spirit of [4], we prove convergence for large times towards Dirac delta solutions for the two densities. All the results share a common technical framework in that solutions are considered in a Lagrangian framework, which allows to estimate the behavior of solutions via $L^2$ estimates of the pseudo-inverse variables corresponding to the two densities. In particular, due to this technique, the large-damping result holds under a rather weak condition on the initial data, which does not require well-prepared initial velocities. We complement the results with numerical simulations.