Dean-Kawasaki Equation with Singular Interactions and Applications to Dynamical Ising-Kac Model

TL;DR

This paper establishes the existence and uniqueness of solutions for the Dean-Kawasaki equation with singular interactions, and applies these results to the well-posedness of fluctuating Ising-Kac-Kawasaki dynamics, advancing understanding of nonlinear fluctuations in stochastic particle systems.

Contribution

It introduces a probabilistic weak renormalized kinetic solution framework for the Dean-Kawasaki equation with singular interactions and proves strong well-posedness under certain conditions.

Findings

Existence of weak solutions under Ladyzhenskaya-Prodi-Serrin condition

Pathwise uniqueness and strong well-posedness established

Application to fluctuating Ising-Kac-Kawasaki dynamics

Abstract

Inspired by [Fehrman, Gess; Invent. Math., 2023] and [Fehrman, Gess; Arch. Ration. Mech. Anal., 2024], we consider the Dean-Kawasaki equation with singular interactions and correlated noise which can be viewed as fluctuating mean-field limits. By imposing the Ladyzhenskaya-Prodi-Serrin condition on the interaction kernel, the existence of probabilistic weak renormalized kinetic solutions is established. Further, under an additional integrability assumption on the divergence of the interaction kernel, a kinetic formulation approach is applied to derive pathwise uniqueness, leading to the strong well-posedness of the equation. As an application, we obtain the well-posedness of a conservative stochastic partial differential equations known as fluctuating Ising-Kac-Kawasaki dynamics, which paves a step on the conjecture concerning nonlinear fluctuations of Kawasaki dynamics proposed by [Giacomin, Lebowitz, Presutti; Math. Surveys Monogr., 1999].