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From Hamiltonian Systems to Compressible Euler Equation driven by additive Hölder noise

Jesus Correa, Juan Londoño, Christian Olivera

TL;DR

The paper rigorously derives the stochastic compressible Euler equations driven by additive Hölder noise from a microscopic Hamiltonian particle system with moderate, long-range interactions. By employing the Itô–Wentzell–Kunita formula for Young integrals and a careful analysis in Besov and Triebel–Lizorkin spaces, it proves that the empirical measures of position and velocity converge to the macroscopic density and velocity fields as the particle number grows, with explicit convergence rates. The method bridges microscopic Hamiltonian dynamics and macroscopic stochastic fluid dynamics under Hölder noise, providing quantitative control via energy functionals and Grönwall-type arguments. This advances the theoretical understanding of how stochastic perturbations at the particle level propagate to stochastic fluid models, with potential implications for stochastic modeling in fluid dynamics and related transport systems.

Abstract

We derive stochastic compressible Euler Equation from a Hamiltonian microscopic dynamics. We consider systems of interacting particles with Hölder noise and potential whose range is large in comparison with the typical distance between neighbouring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of compressible Euler equations driven by additive Hölder path(noise), in the limit as the particle number tends to infinity, for a suitable scaling of the interactions. Furthermore, explicit rates for the convergence are obtained in Besov and Triebel-Lizorkin spaces. Our proof is based on the Itô-Wentzell-Kunita formula for Young integral.

From Hamiltonian Systems to Compressible Euler Equation driven by additive Hölder noise

TL;DR

The paper rigorously derives the stochastic compressible Euler equations driven by additive Hölder noise from a microscopic Hamiltonian particle system with moderate, long-range interactions. By employing the Itô–Wentzell–Kunita formula for Young integrals and a careful analysis in Besov and Triebel–Lizorkin spaces, it proves that the empirical measures of position and velocity converge to the macroscopic density and velocity fields as the particle number grows, with explicit convergence rates. The method bridges microscopic Hamiltonian dynamics and macroscopic stochastic fluid dynamics under Hölder noise, providing quantitative control via energy functionals and Grönwall-type arguments. This advances the theoretical understanding of how stochastic perturbations at the particle level propagate to stochastic fluid models, with potential implications for stochastic modeling in fluid dynamics and related transport systems.

Abstract

We derive stochastic compressible Euler Equation from a Hamiltonian microscopic dynamics. We consider systems of interacting particles with Hölder noise and potential whose range is large in comparison with the typical distance between neighbouring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of compressible Euler equations driven by additive Hölder path(noise), in the limit as the particle number tends to infinity, for a suitable scaling of the interactions. Furthermore, explicit rates for the convergence are obtained in Besov and Triebel-Lizorkin spaces. Our proof is based on the Itô-Wentzell-Kunita formula for Young integral.
Paper Structure (9 sections, 6 theorems, 65 equations)

This paper contains 9 sections, 6 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Related Works.
  3. Example of Hölder noise.
  4. Definitions, preliminaries and hypothesis.
  5. Young Integral.
  6. Space of functions
  7. Definition of solution
  8. Technical hypothesis
  9. Result

Key Result

Theorem 2.2

Let $Y\in C^{\alpha}([0,T];V)$ and $X\in C^{\beta}([0,T];L(V,W))$ for some $\alpha,\beta\in(0,1]$ with $\alpha+\beta>1$. Then the limit exists for every $t\in[0,T]$, where the limit is taken over any $\pi\in\mathcal{P}([0,t])$, and $\mathcal{P}([0,t])$ the set of all partitions $\pi$ of the interval $[0,t]$. This limit is called the Young integral of $X$ against $Y$, which moreover holds the foll

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 3.1
  • proof
  • ...and 2 more