  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].
