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Quantitative particle approximations of stochastic 2D Navier-Stokes equation

Yufei Shao, Xianliang Zhao

Abstract

In this article, we investigate an interacting particle system featuring random intensities, individual noise, and environmental noise, commonly referred to as stochastic point vortex model. The model serves as an approximation for the stochastic 2-dimensional Navier-Stokes equation. We establish a quantitative mean-field convergence for the stochastic 2-dimensional Navier-Stokes equation in the form of relative entropy. To address challenges posed by environmental noise, random intensities, and singular kernel, we compare relative entropy for conditional distributions, employing technology of disintegration and the relative entropy method developed by Jabin and Wang in [JW18].

Quantitative particle approximations of stochastic 2D Navier-Stokes equation

Abstract

In this article, we investigate an interacting particle system featuring random intensities, individual noise, and environmental noise, commonly referred to as stochastic point vortex model. The model serves as an approximation for the stochastic 2-dimensional Navier-Stokes equation. We establish a quantitative mean-field convergence for the stochastic 2-dimensional Navier-Stokes equation in the form of relative entropy. To address challenges posed by environmental noise, random intensities, and singular kernel, we compare relative entropy for conditional distributions, employing technology of disintegration and the relative entropy method developed by Jabin and Wang in [JW18].
Paper Structure (21 sections, 28 theorems, 290 equations)

This paper contains 21 sections, 28 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Methodology and difficulties.
  4. Organization of the paper:
  5. Notations:
  6. Preliminary
  7. Assumptions
  8. Function spaces
  9. Conditional probability
  10. Relative entropy, entropy and Fisher information functional
  11. Definitions of solutions
  12. Conditional Mckean-Vlasov equation and Stochastic Fokker-Planck equation
  13. Well-posedness of SPDEs
  14. Conditional Mckean-Vlasov equation
  15. Quantitative convergence
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that It holds that In particular, where $H_N(\cdot|\cdot)$ denotes the nomalized relative entropy functional, $C_{\xi,0}$ is a positive deterministic constant depending on $\|v_0\|_{L^2(\mathbb{T}^2)},\underset{x\in \mathbb{T}^2 }{\inf}\bar{\rho}_0$ and $\mathcal{L}(\xi),$ and $( v_t)_{t\in [0,T]}$ is the probabilistically strong s

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 41 more