Kinetic Theory with Fluctuations: Strong Well-Posedness of the Vlasov-Fokker-Planck-Dean-Kawasaki System

2025/11/13 by Zimo Hao, Zhengyan Wu, Hao, Zimo +3 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Mathematical Biology Tumor Growth #Navier-Stokes equation solutions #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2511.10194

openalex publication_date 2025/11/13 · openalex created_date 2025/11/15 · openalex updated_date 2026/07/28

Abstract

The strong well-posedness of the Vlasov-Fokker-Planck-Dean-Kawasaki (VFPDK) equation with correlated noise is established. This equation can be interpreted as the fluctuating mean-field limit of second-order Newtonian particle systems, combining kinetic theory with stochastic fluctuations. It includes bounded nonlocal interactions and a diffusion coefficient exhibiting a square-root structure. Key challenges stem from the complexity of the kinetic operator and the irregularity introduced by the conservative noise with square-root-type coefficients. The proof relies on a novel combination of kinetic semigroup estimates and the framework of renormalized kinetic solutions.

Cited by

• Ergodicity for the Dean--Kawasaki Equation with Dirichlet Boundary Conditions: Taming the Square-Root
• Fluctuating Hydrodynamics of the Ising-Kac-Kawasaki Model and Nonlinear Fluctuations Near Criticality

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