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Existence of periodic measure-valued solutions to the nonlocal continuity equation via optimal transport

Seung-Yeal Ha, Gyuyoung Hwang, Philippe Thieullen, Jaeyoung Yoon

Abstract

We investigate the existence of periodic solutions for a class of nonlocal continuity equations, which include mean-field equations derived from systems of coupled oscillators. While periodic solutions at the particle level have been studied through the construction of a Poincaré map on a section of an invariant set, extending this analysis to the level of continuity equations presents nontrivial challenges. In particular, setting an appropriate topology for the infinite-dimensional space to show invariance and apply the fixed point argument is not easy. To overcome this difficulty, we use fixed point theorem for geodesically convex spaces constructed by optimal transportation. Specifically, from the disintegration with respect to stationary variable, we define a metric using the Wasserstein-$2$ distance over one-dimensional space, which yields a $CAT(0)$ space. In this topology, we construct an invariance set of probability measures and prove the existence of the periodic measure-valued solution from Schauder's fixed point theorem on geodesically convex spaces. As a corollary, our method directly gives an existence of periodic graph measure solution.

Existence of periodic measure-valued solutions to the nonlocal continuity equation via optimal transport

Abstract

We investigate the existence of periodic solutions for a class of nonlocal continuity equations, which include mean-field equations derived from systems of coupled oscillators. While periodic solutions at the particle level have been studied through the construction of a Poincaré map on a section of an invariant set, extending this analysis to the level of continuity equations presents nontrivial challenges. In particular, setting an appropriate topology for the infinite-dimensional space to show invariance and apply the fixed point argument is not easy. To overcome this difficulty, we use fixed point theorem for geodesically convex spaces constructed by optimal transportation. Specifically, from the disintegration with respect to stationary variable, we define a metric using the Wasserstein- distance over one-dimensional space, which yields a space. In this topology, we construct an invariance set of probability measures and prove the existence of the periodic measure-valued solution from Schauder's fixed point theorem on geodesically convex spaces. As a corollary, our method directly gives an existence of periodic graph measure solution.
Paper Structure (21 sections, 18 theorems, 271 equations)

This paper contains 21 sections, 18 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Previous results
  4. A sufficient framework $\boldsymbol{({\mathcal{A}})}$
  5. Explicit examples for $\boldsymbol{\mathcal{F}^\nu(\kappa, \gamma)}$
  6. Kinetic Winfree flow
  7. Kinetic Kuramoto flow
  8. Collection of parameters and constants
  9. Preparatory lemmas
  10. Description of main results
  11. Topology of the metric space
  12. The metric space $\boldsymbol{(\mathcal{X}_\nu, d_2)}$
  13. Geodesic property
  14. Geodesic convexity and compactness
  15. Dispersion estimate of the characteristics
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $y = y(t)$ be a global unique solution to B-10 - B-11 whose explicit representation is given by B-11-0. Then, the following assertions hold.

Theorems & Definitions (41)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.1: Disintegration theorem
  • Theorem 2.2
  • ...and 31 more