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Gaussian Fluctuations for the Stochastic Landau-Lifshitz Navier-Stokes Equation in Dimension $D\geq2$

Sotiris Kotitsas, Marco Romito, Zhilin Yang, Xiangchan Zhu

TL;DR

This work analyzes Gaussian fluctuations for the stochastic LLNS equation on the torus across dimensions $d\ge2$ under diffusive scaling ($d\ge3$) and weak coupling ($d=2$). Building on a fluctuation-dissipation framework, the authors prove convergence to a renormalised stochastic heat equation with a dimension-dependent diffusion correction $D$, and they provide an explicit asymptotic expansion of $D$ for $d\ge3$, while also deriving a Replacement Lemma for $d=2$. A careful treatment of the Leray projection, the antisymmetric generator, and rotational symmetries enables decoupling of vector components and a rigorous description of the large-scale limit, clarifying renormalisation effects in stochastic fluid models. The results refute prior conjectures and offer precise control on the effective diffusivity, with potential impact on understanding hydrodynamic fluctuations in incompressible stochastic systems.

Abstract

We revisit the large-scale Gaussian fluctuations for the stochastic Landau-Lifshitz Navier-Stokes equation (LLNS) at and above criticality, using the method in \cite{CGT24}. With the classical diffusive scaling in $d\geq 3$ and weak coupling scaling in $d=2$, we obtain the convergence of the regularised LLNS to a stochastic heat equation with a non-trivially renormalized coefficient. Moreover, we obtain an asymptotic expansion of the effective coefficient when $d\geq3$, and show that the one in \cite[Conjecture 6.5]{JP24} is incorrect. The new ingredient in our proof is a case-by-case analysis to track the evolution of the vector under the action of the Leray projection, combined with the use of the anti-symmetric part of the generator and a rotational change of coordinates to derive the desired decoupled stochastic heat equation from the original coupled system.

Gaussian Fluctuations for the Stochastic Landau-Lifshitz Navier-Stokes Equation in Dimension $D\geq2$

TL;DR

This work analyzes Gaussian fluctuations for the stochastic LLNS equation on the torus across dimensions under diffusive scaling () and weak coupling (). Building on a fluctuation-dissipation framework, the authors prove convergence to a renormalised stochastic heat equation with a dimension-dependent diffusion correction , and they provide an explicit asymptotic expansion of for , while also deriving a Replacement Lemma for . A careful treatment of the Leray projection, the antisymmetric generator, and rotational symmetries enables decoupling of vector components and a rigorous description of the large-scale limit, clarifying renormalisation effects in stochastic fluid models. The results refute prior conjectures and offer precise control on the effective diffusivity, with potential impact on understanding hydrodynamic fluctuations in incompressible stochastic systems.

Abstract

We revisit the large-scale Gaussian fluctuations for the stochastic Landau-Lifshitz Navier-Stokes equation (LLNS) at and above criticality, using the method in \cite{CGT24}. With the classical diffusive scaling in and weak coupling scaling in , we obtain the convergence of the regularised LLNS to a stochastic heat equation with a non-trivially renormalized coefficient. Moreover, we obtain an asymptotic expansion of the effective coefficient when , and show that the one in \cite[Conjecture 6.5]{JP24} is incorrect. The new ingredient in our proof is a case-by-case analysis to track the evolution of the vector under the action of the Leray projection, combined with the use of the anti-symmetric part of the generator and a rotational change of coordinates to derive the desired decoupled stochastic heat equation from the original coupled system.

Paper Structure

This paper contains 14 sections, 13 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. The divergence-free vector field space
  4. Wiener space analysis
  5. The truncated equation and its generator
  6. The Fluctuation-Dissipation Theorem
  7. Proof of Theorem \ref{['thm:main']} when $d\geq3$
  8. Notations and main proposition
  9. Proof of Proposition \ref{['prop:limite']}
  10. Asymptotic expansion of the effective coefficient when $d\geq3$
  11. $d=2$: the Replacement Lemma
  12. The generator on the Fock space
  13. Technical Estimates in the Replacement Lemma
  14. Proof of Theorem \ref{['thm:F-D']} when $d\geq3$

Key Result

Theorem 1.1

Let $T>0$ and $\lambda$ be a fixed positive constant. For $N>0$, let Define $u^N$ be the stationary solution to eq:scaled-NS with the initial value $u^N(0,\cdot):=\mu,$ for $\mu$ a divergence-free and mean-free spatial white noise on $\mathbb{T}^d.$ Then there exists a strictly positive constant $D$, depending only on the dimension $d$ and $\lambda$, such that $u^N$ In the case $d=2$, $D=\sqrt{\

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4: Graded Sector Condition
  • Lemma 3.1: Itô trick
  • ...and 19 more