  Inspired by [Fehrman, Gess; Invent. Math., 2023], we provide a fine analysis
of the McKean-Vlasov PDE with singular interactions and drift terms of square
root form. As the corresponding skeleton equation of Dean-Kawasaki equation
with singular interactions (a stochastic, conservative PDE), it determines the
rate function of small noise large deviations. By imposing
Ladyzhenskaya-Prodi-Serrin type conditions on the interaction kernel, we
establish the large deviations in the framework of stochastic renormalized
kinetic solution, when the intensity and the correlation of the noise are
simultaneously sent to $0$ under a suitable scaling. This result contributes to
demonstrating the consistency between the macroscopic fluctuation theory
associated with singular interacting mean-field systems and fluctuating
hydrodynamics related to the Dean-Kawasaki equation. As an application, we also
obtain large deviations for the other stochastic conservative PDE called
fluctuating Ising-Kac-Kawasaki dynamics. It is of great importance in exploring
fluctuations of Kawasaki dynamical Ising-Kac model, since they formally
exhibits the same key features in terms of Gaussian fluctuations, large
deviations, and scaling limits near criticality.
