WELL-POSEDNESS OF DEAN--KAWASAKI EQUATION WITH SINGULAR INTERACTIONS

Inspired by [Fehrman and Gess, Invent. Math., 234 (2023), pp. 573--636] and [Fehrman and Gess, Arch. Ration. Mech. Anal., 248 (2024), 20], 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 equation known as the fluctuating Ising--Kac--Kawasaki dynamics.