The Vlasov–Fokker–Planck–Dean–Kawasaki equation with correlated noise and bounded nonlocal interactions has a unique probabilistically strong renormalized kinetic solution for finite-mass, finite-entropy initial data.
Dean–Kawasaki equation with singular interac- tions and applications to dynamical Ising-Kac model
4 Pith papers cite this work. Polarity classification is still indexing.
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abstract
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 equation known as the fluctuating Ising-Kac-Kawasaki dynamics.
representative citing papers
Constructs weak solutions, proves anisotropic Besov regularity, and establishes uniqueness in the mass-preserving renormalized class for kinetic FP equations with nonlinear diffusion under mass-critical growth on Ψ.
Proves well-posedness of the obstacle problem for generalized Dean-Kawasaki equations via kinetic formulation of the reflection term, covering the full porous-medium regime under continuous obstacles.
Under relative scalings lim ε log(1/δ)=0 and stricter variants, the authors establish LLN, LDP, and CLT for an additive-noise model approximating fluctuating chemotactic particle hydrodynamics in distribution and function spaces via singular SPDE techniques.
citing papers explorer
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Kinetic Theory with Fluctuations: Strong Well-Posedness of the Vlasov-Fokker-Planck-Dean-Kawasaki System
The Vlasov–Fokker–Planck–Dean–Kawasaki equation with correlated noise and bounded nonlocal interactions has a unique probabilistically strong renormalized kinetic solution for finite-mass, finite-entropy initial data.
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Kinetic Fokker-Planck Equations with Nonlinear Diffusion
Constructs weak solutions, proves anisotropic Besov regularity, and establishes uniqueness in the mass-preserving renormalized class for kinetic FP equations with nonlinear diffusion under mass-critical growth on Ψ.
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Well-posedness of the obstacle problem for generalized Dean-Kawasaki equation
Proves well-posedness of the obstacle problem for generalized Dean-Kawasaki equations via kinetic formulation of the reflection term, covering the full porous-medium regime under continuous obstacles.
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An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Small-Noise Results
Under relative scalings lim ε log(1/δ)=0 and stricter variants, the authors establish LLN, LDP, and CLT for an additive-noise model approximating fluctuating chemotactic particle hydrodynamics in distribution and function spaces via singular SPDE techniques.