From particle systems to the stochastic compressible Navier-Stokes equations of a barotropic fluid
Jesus Correa, Christian Olivera
TL;DR
The authors derive the stochastic compressible Navier-Stokes equations for a barotropic fluid from a many-particle system with moderate long-range interactions and environmental noise. They formulate a rigorous mean-field-type limit using empirical measures $S_t^N$ and $V_t^N$, embed the analysis in Besov and Triebel-Lizorkin spaces, and employ the Ito-Kunita-Wentzell framework to pass to the limit, obtaining a stochastic PDE with pressure $p=\tfrac{1}{2}\varrho^2$. A central contribution is the quantitative convergence of the mollified empirical measures to the density and momentum fields, with explicit rates in Besov-type norms, under precise regularity and decay assumptions on the interaction and friction kernels. The work provides a mathematical bridge between particle-based SPH-like methods and macroscopic stochastic fluid models, offering both a first macroscopic derivation and a priori estimates that underpin stochastic hydrodynamic limits. This advances understanding of how random environmental effects propagate from microscopic particle dynamics to continuum stochastic fluid models, with potential implications for SPH-based simulations of stochastic flows.
Abstract
We propose a mathematical derivation of stochastic compressible Navier-Stokes equation. We consider many-particle systems with a Hamiltonian dynamics supplemented by a friction term and environmental noise. Both the interaction potential and the additional friction force are supposed to be long range in comparison with the typical distance between neighboring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of the stochastic compressible Navier-Stokes equations of a barotropic fluid. Moreover, we quantify the distance between particles and the limit in suitable in Besov and Triebel-Lizorkin spaces.