McKean-Vlasov Stochastic Partial Differential Equations: Existence, Uniqueness and Propagation of Chaos

TL;DR

This paper develops a comprehensive framework for analyzing McKean-Vlasov stochastic PDEs, establishing existence, uniqueness, and propagation of chaos, with applications to fluid dynamics and phase transition models.

Contribution

It introduces new methods for proving existence and uniqueness of solutions and demonstrates propagation of chaos for complex stochastic PDE systems.

Findings

Existence of weak solutions via approximation and compactness methods.

Uniqueness of strong solutions under locally monotone conditions.

Propagation of chaos in Wasserstein distance for stochastic Navier-Stokes systems.

Abstract

In this paper, we provide a general framework for investigating McKean-Vlasov stochastic partial differential equations. We first show the existence of weak solutions by combining the localizing approximation, Faedo-Galerkin technique, compactness method and the Jakubowski version of the Skorokhod representation theorem. Then under certain locally monotone condition we further investigate the existence and uniqueness of (probabilistically) strong solutions. The applications of the main results include a large class of McKean-Vlasov stochastic partial differential equations such as stochastic 2D/3D Navier-Stokes equations, stochastic Cahn-Hilliard equations and stochastic Kuramoto-Sivashinsky equations. Finally, we show a propagation of chaos result in Wasserstein distance for weakly interacting stochastic 2D Navier-Stokes systems.