Dean-Kawasaki Equation with Biot-Savart and Keller-Segel Interactions: Existence and Large Deviations

TL;DR

This paper proves the existence of solutions and a large deviation principle for the Dean--Kawasaki equation with singular Biot--Savart and Keller--Segel interactions, addressing criticality challenges.

Contribution

It establishes probabilistically weak, renormalized solutions and a restricted large deviation principle for the Dean--Kawasaki equation with singular interactions, introducing a novel exponential tightness method.

Findings

Existence of weak, renormalized solutions for singular kernels.

A restricted large deviation principle under regularization.

Novel exponential tightness argument adapted to Dean--Kawasaki noise.

Abstract

We establish the existence of probabilistically weak, renormalized kinetic solutions to the Dean--Kawasaki equation with singular interaction kernels, including those of Biot--Savart and Keller--Segel type. Under a suitable regularization of the square-root noise coefficient, we further prove a restricted large deviation principle for probabilistically weak solutions to the regularized Dean--Kawasaki equation. The Biot--Savart and Keller--Segel type interactions introduce a scaling criticality within the $L^1$ framework of the Dean--Kawasaki equation and the associated skeleton equation, which gives rise to a significant new challenge. In contrast to [Fehrman, Gess; Invent. Math., 2023], our large deviation analysis relies on a novel exponential tightness argument specifically adapted to the Dean--Kawasaki noise. This approach, combined with a weak-strong uniqueness result for the associated skeleton equation, allows us to partially overcome the criticality induced by the singular interaction kernel.