Ergodicity for a Constantin-Lax-Majda-DeGregorio model of turbulent flow

TL;DR

This paper analyzes a stochastic one-dimensional turbulence model, proving the existence and uniqueness of an invariant measure and exponential mixing, advancing understanding of anomalous cascades in turbulent flows.

Contribution

It establishes the existence, uniqueness, and exponential mixing of an invariant measure for a stochastic gCLMG turbulence model, linking dynamical systems theory to turbulence phenomena.

Findings

Existence of an invariant measure proved

Uniqueness and exponential mixing established under large viscosity

First theoretical construction connecting turbulence cascades and dynamical systems

Abstract

This paper presents a mathematical analysis of a one-dimensional model of turbulence based on a stochastic generalized Constantin-Lax-Majda-DeGregorio (gCLMG) equation. We focus on the specific case where the nonlinearity in the equation allows the existence of the anomalous enstrophy cascade, which is an inviscid conserved quantity, and some effective energy estimates for mathematical analysis. The existence of an invariant measure in the attractor is proved via the classical Krylov-Bogoliubov argument. The uniqueness of the measure and exponential mixing are proved under a sufficiently large viscosity condition, in which the nonlocal structure of the nonlinear term plays a prominent role. The construction of this invariant measure is the first step towards a theoretical understanding of turbulent phenomena that cause anomalous cascades in the zero viscous limit, viewed from the dynamical systems theory.