Time Averaged Statistics of the 3D Stochastic Ladyzenskaya-Smagorinsky Equations

TL;DR

This paper studies the time-averaged statistics of the $3$D stochastic Ladyzhenskaya–Smagorinsky equations on a periodic domain driven by space-time Gaussian noise. It derives a rigorous upper bound on the first moment of the energy dissipation rate $\varepsilon$ that remains finite as the viscosity $\nu$ tends to zero, in alignment with Kolmogorov's $K41$ phenomenology. The bound is expressed in terms of scales $U$ and $L$ and dimensionless groups $\mathrm{Re}_{\nu}=\frac{U L}{\nu}$, $\mathrm{Re}_{\bar{\nu}}=\frac{L^{r-1}}{\bar{\nu} U^{r-3}}$, and a stochastic-forcing parameter $\rho_{\infty}$, with a key condition $\rho_{\infty}<\nu\lambda_1$ (or a finite additive-noise case) ensuring uniform-in-time control. The results extend known bounds for deterministic NSE and Smagorinsky models to the stochastic LS framework and demonstrate the absence of artificial over-dissipation in the boundaryless setting, reinforcing the dissipative anomaly in a non-Newtonian stochastic context.

Abstract

Due to the chaotic nature of turbulence, statistical quantities are often more informative than pointwise characterizations. In this work, we consider the stochastic Ladyzhenskaya-Smagorinsky equation driven by space-time Gaussian noise on a three-dimensional periodic domain. We derive a rigorous upper bound on the first moment of the energy dissipation rate and show that it remains finite in the vanishing viscosity limit, consistent with Kolmogorov's phenomenological theory. This estimate also agrees with classical results obtained for the Navier-Stokes equations and demonstrates that, in the absence of boundary layers, as considered here, the model does not over-dissipate.

Theorems & Definitions (12)

• Definition 2.1
• Theorem 2.1
• Remark 2.3
• Definition 3.1: Energy dissipation rate
• Remark 3.1
• Definition 3.2
• Remark 3.2
• Theorem 3.3
• Lemma 3.4
• Remark 3.5