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Landau-Lifshitz-Navier-Stokes Equations: Large Deviations and Relationship to The Energy Equality

Benjamin Gess, Daniel Heydecker, Zhengyan Wu

Abstract

The dynamical large deviations principle for the three-dimensional incompressible Landau-Lifschitz-Navier-Stokes equations is shown, in the joint scaling regime of vanishing noise intensity and correlation length. This proves the consistency of the large deviations in lattice gas models \cite{QY}, with Landau-Lifschitz fluctuating hydrodynamics \cite{LL87}. Secondly, in the course of the proof, we unveil a novel relation between the validity of the deterministic energy equality for the deterministic forced Navier-Stokes equations and matching large deviations upper and lower bounds. In particular, we conclude that time-reversible uniqueness to the forced Navier-Stokes equations implies the validity of the energy equality, thus generalising the classical Lions-Ladyzhenskaya result. Thirdly, we prove that no non-trivial large deviations result can be true for local-in-time strong solutions.

Landau-Lifshitz-Navier-Stokes Equations: Large Deviations and Relationship to The Energy Equality

Abstract

The dynamical large deviations principle for the three-dimensional incompressible Landau-Lifschitz-Navier-Stokes equations is shown, in the joint scaling regime of vanishing noise intensity and correlation length. This proves the consistency of the large deviations in lattice gas models \cite{QY}, with Landau-Lifschitz fluctuating hydrodynamics \cite{LL87}. Secondly, in the course of the proof, we unveil a novel relation between the validity of the deterministic energy equality for the deterministic forced Navier-Stokes equations and matching large deviations upper and lower bounds. In particular, we conclude that time-reversible uniqueness to the forced Navier-Stokes equations implies the validity of the energy equality, thus generalising the classical Lions-Ladyzhenskaya result. Thirdly, we prove that no non-trivial large deviations result can be true for local-in-time strong solutions.
Paper Structure (34 sections, 30 theorems, 185 equations)

This paper contains 34 sections, 30 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Definitions
  4. Function Spaces
  5. Path Space and its topologies
  6. Divergence-Free White Noise and Regularisation
  7. A Solution Class to (\ref{['SNS-1']})
  8. Main Results
  9. Parameter Tuning
  10. Large Deviations of Initial Data
  11. The Rate Function
  12. Structure of the paper
  13. Comments on the literature
  14. Stochastic Navier Stokes and Large Deviations
  15. Independence of Uniqueness Theory
  16. ...and 19 more sections

Key Result

Proposition 2.2

For any $\epsilon, \delta>0$, there exists a stochastic basis $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0}, \mathbb{P})$ on which are defined a divergence-free space-time white noise $W$ and a stochastic Leray solution ${U}^\epsilon_\delta$ to (SNS-1) for the white noise $W$. Under integrability

Theorems & Definitions (57)

  • Definition 2.1
  • Proposition 2.2: Proposition \ref{['existence']}
  • Lemma 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9: Triviality of large deviations of the strong solution
  • Lemma 3.1
  • ...and 47 more