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Eddy viscosity by Lévy transport noises

Dejun Luo, Feifan Teng

TL;DR

The paper investigates stochastic 2D Euler equations on the torus with Lévy transport noise in the Marcus sense and demonstrates that, under a diffusive scaling of small jumps, weak solutions converge to the deterministic 2D Navier–Stokes equation with eddy viscosity $\kappa$, thereby extending diffusion-limit results from Brownian to discontinuous noise. The authors first analyze stochastic linear transport equations to establish the eddy-diffusion mechanism, showing the limit equation is $\partial_t \xi = \kappa \Delta \xi$ with $\kappa= C_d \int_{|z|\le1} z^2\,\nu(dz)$. They then prove weak existence for stochastic 2D Euler with Lévy noise via Galerkin approximations and demonstrate tightness and convergence to the Navier–Stokes dynamics in the limit, with the nonlinear transport term converging and the Lévy correction yielding the Laplacian diffusion in the limit. The results reveal that small-scale jump perturbations act as eddy viscosity, producing a Laplacian dissipation term even in discontinuous noise settings. This provides a rigorous bridge between stochastic transport with jump noise and classical dissipative fluid models, offering insight into turbulence modeling under non-Gaussian fluctuations and extending the diffusion-limit paradigm to Lévy-driven transport. The work has potential implications for stochastic fluid models where jump-like eddies are essential, and it highlights the robustness of eddy viscosity as a macroscopic dissipative mechanism across noise types.

Abstract

We consider stochastic 2D Euler equations with $L^2$-initial vorticity and driven by Lévy transport noise in the Marcus sense. Under a suitable scaling limit of the noises, we prove that the weak solutions converge weakly to the unique solution of the deterministic 2D Navier-Stokes equation. This shows that small scale jump noises generate eddy viscosity, extending the recent studies on Itô-Stratonovich diffusion limit to discontinuous setting.

Eddy viscosity by Lévy transport noises

TL;DR

The paper investigates stochastic 2D Euler equations on the torus with Lévy transport noise in the Marcus sense and demonstrates that, under a diffusive scaling of small jumps, weak solutions converge to the deterministic 2D Navier–Stokes equation with eddy viscosity , thereby extending diffusion-limit results from Brownian to discontinuous noise. The authors first analyze stochastic linear transport equations to establish the eddy-diffusion mechanism, showing the limit equation is with . They then prove weak existence for stochastic 2D Euler with Lévy noise via Galerkin approximations and demonstrate tightness and convergence to the Navier–Stokes dynamics in the limit, with the nonlinear transport term converging and the Lévy correction yielding the Laplacian diffusion in the limit. The results reveal that small-scale jump perturbations act as eddy viscosity, producing a Laplacian dissipation term even in discontinuous noise settings. This provides a rigorous bridge between stochastic transport with jump noise and classical dissipative fluid models, offering insight into turbulence modeling under non-Gaussian fluctuations and extending the diffusion-limit paradigm to Lévy-driven transport. The work has potential implications for stochastic fluid models where jump-like eddies are essential, and it highlights the robustness of eddy viscosity as a macroscopic dissipative mechanism across noise types.

Abstract

We consider stochastic 2D Euler equations with -initial vorticity and driven by Lévy transport noise in the Marcus sense. Under a suitable scaling limit of the noises, we prove that the weak solutions converge weakly to the unique solution of the deterministic 2D Navier-Stokes equation. This shows that small scale jump noises generate eddy viscosity, extending the recent studies on Itô-Stratonovich diffusion limit to discontinuous setting.
Paper Structure (18 sections, 25 theorems, 227 equations)

This paper contains 18 sections, 25 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Structure of noise
  4. Main results
  5. Stochastic linear transport equations
  6. Stochastic 2D Euler equations
  7. Preliminaries
  8. Properties of the flow $\varphi^{k,i}_t$ and Sobolev spaces
  9. Skorohod spaces $D([0,T];S)$
  10. Stochastic transport equations driven by Lévy noises
  11. Existence of solutions
  12. Scaling limit of Marcus stochastic transport equations
  13. Weak existence for stochastic 2D Euler equations with Lévy noises
  14. Galerkin approximation
  15. Tightness results
  16. ...and 3 more sections

Key Result

Theorem 1.2

As $n\to\infty$, the solutions $\xi^n$ to STE-Marcus-n converge in probability, in the weak-$\ast$ topology of $L^\infty([0,T];L^2(\mathbb{T}^d))$, to the unique solution of where $\kappa= C_d \int_{|z|\le 1} z^2 \,\nu(dz)$ for some dimension-dependent constant $C_d>0$.

Theorems & Definitions (45)

  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 35 more