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A uniform point vortex approximation for the solution of the two-dimensional Navier Stokes equation with transport noise

Filippo Giovagnini, Dan Crisan

TL;DR

The paper proves a uniform-in-time mean-field limit for a stochastic point-vortex-like particle system approximating the 2d Navier–Stokes equations with transport noise. By mollifying the empirical measures and using a semigroup framework, the authors show convergence to the unique solution of the SPDE $\partial_t \omega + u\cdot \nabla \omega + \sum_k \sigma_k \circ dW^k_t \cdot \nabla \omega = \nu \Delta \omega$, with $u=K*\omega$. The analysis combines compactness arguments in fractional Sobolev/Besov spaces, limit-passage techniques for stochastic terms, and a truncation-lifting strategy to obtain global well-posedness. The results give a rigorous link between finite-particle stochastic dynamics and the continuum stochastic fluid model, with potential implications for stochastic turbulence modeling and mean-field approximations in 2D.

Abstract

We study a model of interacting particles represented by a system of N stochastic differential equations. We establish that the mollified empirical distribution of the system converges uniformly with respect to both time and spatial variables to the solution of the two dimensional Navier Stokes equation with transport noise. The proofs are based on a semigroup approach.

A uniform point vortex approximation for the solution of the two-dimensional Navier Stokes equation with transport noise

TL;DR

The paper proves a uniform-in-time mean-field limit for a stochastic point-vortex-like particle system approximating the 2d Navier–Stokes equations with transport noise. By mollifying the empirical measures and using a semigroup framework, the authors show convergence to the unique solution of the SPDE , with . The analysis combines compactness arguments in fractional Sobolev/Besov spaces, limit-passage techniques for stochastic terms, and a truncation-lifting strategy to obtain global well-posedness. The results give a rigorous link between finite-particle stochastic dynamics and the continuum stochastic fluid model, with potential implications for stochastic turbulence modeling and mean-field approximations in 2D.

Abstract

We study a model of interacting particles represented by a system of N stochastic differential equations. We establish that the mollified empirical distribution of the system converges uniformly with respect to both time and spatial variables to the solution of the two dimensional Navier Stokes equation with transport noise. The proofs are based on a semigroup approach.

Paper Structure

This paper contains 13 sections, 18 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The particle system and the main result
  4. Criterion of compactness
  5. Passing to the limit
  6. Lifting the truncation
  7. Existence and uniqueness of the 2d-SNS
  8. The limit of $(g^{N, +}, g^{N, -})$ coincides with the solution to the 2d-SNN
  9. Proof of the main result
  10. Technical estimates
  11. Appendix
  12. Properties of the Biot-Savart kernel
  13. The approximation of the identity $V^N$

Key Result

Theorem 1

Assume Assumption ass:assprinci. Then, $(g^{N, +}, g_t^{N, -})$ converges to the solution of the system eq:main_equation in the sense of definition def:solution_of_system, in the strong topology of $C([0, T]; \mathbb{H}^{\eta}_p(\mathbb{T}^2))^2$, for any $\eta \in \left(\frac{2}{p}, \alpha\right)$,

Theorems & Definitions (22)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 12 more