Invariant Gibbs measures of the energy for shell models of turbulence; the inviscid and viscous cases

TL;DR

This paper constructs and analyzes invariant Gibbs measures for shell models of turbulence, proving the existence of global flows in both viscous and inviscid cases, with invariance of these measures.

Contribution

It introduces Gibbs measures based on energy for shell models and proves the existence of unique global flows that preserve these measures in both viscous and inviscid scenarios.

Findings

Existence of a unique global flow for viscous shell models.

Existence of a global flow for inviscid shell models.

Invariance of Gibbs measures under these flows.

Abstract

Gaussian measures of Gibbsian type are associated with some shell models of 3D turbulence; they are constructed by means of the energy, a conserved quantity for the 3D inviscid and unforced shell model. We prove the existence of a unique global flow for a stochastic viscous shell model and a global flow for the deterministic inviscid shell model, with the property that these Gibbs measures are invariant for these flows.