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