Geodesically convex energies and confinement of solutions for a multi-component system of nonlocal interaction equations

TL;DR

This paper establishes geodesic convexity of a multi-component nonlocal interaction energy, ensuring well-posedness and analyzing long-term behavior of solutions, including boundedness and blow-up prevention.

Contribution

It proves geodesic convexity for matrix-valued interaction energies under broad conditions, leading to existence, uniqueness, and qualitative analysis of solutions.

Findings

Existence and uniqueness of solutions are guaranteed.

Solutions have uniformly bounded support under certain convexity conditions.

Finite-time blow-up can be excluded with Lipschitz conditions.

Abstract

We consider a system of $n$ nonlocal interaction evolution equations on $\mathbb{R}^d$ with a differentiable matrix-valued interaction potential $W$. Under suitable conditions on convexity, symmetry and growth of $W$, we prove $\lambda$-geodesic convexity for some $\lambda\in\mathbb{R}$ of the associated interaction energy with respect to a weighted compound distance of Wasserstein type. In particular, this implies existence and uniqueness of solutions to the evolution system. In one spatial dimension, we further analyse the qualitative properties of this solution in the non-uniformly convex case. We obtain, if the interaction potential is sufficiently convex far away from the origin, that the support of the solution is uniformly bounded. Under a suitable Lipschitz condition for the potential, we can exclude finite-time blow-up and give a partial characterization of the long-time behaviour.