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Scaling Limit and Large Deviation for 3D Globally Modified Stochastic Navier-Stokes Equations with Transport Noise

Chang Liu, Dejun Luo

TL;DR

This work analyzes the 3D globally modified stochastic Navier–Stokes equations with transport noise on the torus. It establishes well-posedness (existence and pathwise uniqueness) for Λ∈[1,2) with an appropriate δ, and proves a scaling limit in which stochastic solutions converge to a deterministic GMNSE with an enhanced dissipation term $(3ν/5)Δu$ under a structured noise scaling. The paper further proves a Large Deviation Principle for the hyperviscous regime Λ∈(1,2) with δ=0 via the weak convergence method, by studying skeleton equations and rate functions that quantify deviations from the deterministic limit. Together, the results quantify how transport noise regularizes the flow in the mean and provide rigorous asymptotics and probabilistic large-deviation behavior for high-dimensional stochastic fluid models. The combination of scaling limits and LDP offers insights into turbulence modeling and the dissipative role of transport noise in 3D GMNSE-type systems.

Abstract

We consider the globally modified stochastic (hyperviscous) Navier-Stokes equations with transport noise on 3D torus. We first establish the existence and pathwise uniqueness of the weak solutions, and then show their convergence to the solutions of the deterministic 3D globally modified (hyperviscous) Navier-Stokes equations in an appropriate scaling limit. Furthermore, we prove a large deviation principle for the stochastic globally modified hyperviscous system.

Scaling Limit and Large Deviation for 3D Globally Modified Stochastic Navier-Stokes Equations with Transport Noise

TL;DR

This work analyzes the 3D globally modified stochastic Navier–Stokes equations with transport noise on the torus. It establishes well-posedness (existence and pathwise uniqueness) for Λ∈[1,2) with an appropriate δ, and proves a scaling limit in which stochastic solutions converge to a deterministic GMNSE with an enhanced dissipation term under a structured noise scaling. The paper further proves a Large Deviation Principle for the hyperviscous regime Λ∈(1,2) with δ=0 via the weak convergence method, by studying skeleton equations and rate functions that quantify deviations from the deterministic limit. Together, the results quantify how transport noise regularizes the flow in the mean and provide rigorous asymptotics and probabilistic large-deviation behavior for high-dimensional stochastic fluid models. The combination of scaling limits and LDP offers insights into turbulence modeling and the dissipative role of transport noise in 3D GMNSE-type systems.

Abstract

We consider the globally modified stochastic (hyperviscous) Navier-Stokes equations with transport noise on 3D torus. We first establish the existence and pathwise uniqueness of the weak solutions, and then show their convergence to the solutions of the deterministic 3D globally modified (hyperviscous) Navier-Stokes equations in an appropriate scaling limit. Furthermore, we prove a large deviation principle for the stochastic globally modified hyperviscous system.
Paper Structure (15 sections, 22 theorems, 195 equations)

This paper contains 15 sections, 22 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of our paper
  4. Preliminaries
  5. Structure of the noise
  6. Analytical results
  7. Well-posedness
  8. Existence
  9. Pathwise uniqueness
  10. Convergence
  11. Scaling limit
  12. Quantitative convergence rate
  13. Large deviation principle
  14. Weak convergence method
  15. Proof of Theorem \ref{['LDP thm']}

Key Result

Theorem 1.2

Given any $u_0 \in L^2$ and $T>0$, for $\Lambda \in [1,2)$, there exists a pathwise unique weak solution $u$ to equation Ito form in the sense of Definition def of solutions.

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Example 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4: Interpolation inequality
  • ...and 34 more