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

TL;DR

This paper establishes a large deviations principle for the 3D incompressible Landau-Lifschitz-Navier-Stokes equations, linking it to lattice gas models and exploring the connection between energy equality and large deviations.

Contribution

It introduces a novel large deviations principle for these equations and uncovers a new relation between energy equality and large deviations bounds, extending classical results.

Findings

Large deviations principle proven for the equations.

Relation between energy equality and large deviations established.

No non-trivial large deviations for local-in-time strong solutions.

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.