Well-posedness of Dean-Kawasaki Equation with Singular Interactions

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.

arXiv commentAccepted version

References in corpus (12)

• A theory of regularity structures
• Langevin Equation for the Density of a System of Interacting Langevin Processes
• Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1,\infty }$ kernels
• Scalar conservation laws with stochastic forcing
• Mean Field Limit for Coulomb-Type Flows
• A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations
• Propagation of chaos for the 2D viscous vortex model
• Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity
• On L_p- theory for stochastic parabolic integro-differential equations
• Modified Massive Arratia flow and Wasserstein diffusion
• Nonlinear diffusion equations with nonlinear gradient noise
• Global well-posedness of the viscous Camassa--Holm equation with gradient noise

Cited by in corpus (1)

• Well-Posedness of the generalised Dean-Kawasaki Equation with correlated noise on bounded domains