Ergodicity of Burgers' system

TL;DR

This paper investigates the ergodic properties of a stochastic coupled Burgers' system, extending previous work on single equations to a more complex fluid dynamics model.

Contribution

It establishes the existence of a unique invariant measure for the stochastic Burgers' system, generalizing prior results from single-equation models to coupled systems.

Findings

Existence of a unique invariant measure proven.

Extension of ergodic results to coupled Burgers' equations.

Generalization of prior single-equation turbulence models.

Abstract

We consider a stochastic version of a system of coupled two equations formulated by Burgers with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness of the solution as well as the irreducibility property of such system were given by Twardowska and Zabczyk. In the paper the existence of a unique invariant measure is investigated. The paper generalizes the results of Da Prato, Debussche and Temam, and Da Prato and Gatarek, dealing with one equation describing the turbulent motion only.