Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

TL;DR

This paper establishes existence, uniqueness, and long-term behavior of measure solutions for a two-species nonlocal interaction system in one dimension, introducing a gradient-flow framework for initial data with atomic parts.

Contribution

It develops a gradient-flow solution concept for measure solutions, including atomic initial data, and demonstrates the necessity of this framework for uniqueness.

Findings

Proves global existence and uniqueness of measure solutions.

Shows solutions preserve certain norms and moments over time.

Provides examples of non-uniqueness without the gradient-flow framework.

Abstract

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space $L^2(0,1)^2$ according to the classical theory by Br\'ezis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the $L^m$-norms for all $m\in [1,+\infty]$ and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.