On Stochastic Shell Models of Turbulence

TL;DR

This paper establishes the existence, uniqueness, and invariant measures for a stochastic shell model of turbulence, analyzing energy balances and asymptotic behavior of structure functions.

Contribution

It provides a rigorous mathematical analysis of a stochastic shell model, including solutions, invariant measures, and energy flux relations, advancing understanding of turbulence modeling.

Findings

Existence and uniqueness of solutions proven.

Invariant measures for the stochastic model established.

Balance relation among energy injection, dissipation, and flux derived.

Abstract

We prove existence of weak and strong solutions and uniqueness for a viscous dyadic model driven by additive white noise in time using a path-wise approach. Existence of invariant measures also established and a simple balance relation among the mean rates of energy injection, dissipation and flux is derived and we investigate the asymptotic exponents $\zeta_{p}$ of the $p$-order structure functions.