Statistical Estimates for 2D stochastic Navier-Stokes Equations
Anuj Kumar, Ali Pakzad
TL;DR
This work studies the stochastic two-dimensional Navier–Stokes equations on the periodic torus to bound the long-time mean energy and enstrophy dissipation, $\mathbb{E}[\varepsilon]$ and $\mathbb{E}[\chi]$, under broad stochastic forcing. By employing martingale solutions and Itô calculus, the authors derive rigorous upper bounds that relate $\mathbb{E}[\varepsilon]$ and $\mathbb{E}[\chi]$ to large-scale quantities $U$, $L$, the Reynolds number $Re$, and noise intensities $F$, $G$, and $\tilde{G}$, while incorporating a forcing-time scale $\tau$. The main results show $\mathbb{E}[\chi] \le (\tau+1+Re^{-1}) \frac{U^3}{L^3} + \tilde{G}^2$ and $\mathbb{E}[\varepsilon] \lesssim Re^{-1/2} \big[(\tau+1+Re^{-1})+(\tilde{G}^2 L^3 U^{-3})\big]^{1/2} \frac{U^3}{L}$, which yield $\mathbb{E}[\varepsilon] \to 0$ and $\mathbb{E}[\chi] \lesssim \mathcal{O}(1)$ as $Re \to \infty$, aligning with the 2D dual-cascade picture. The analysis extends deterministic dissipation bounds to stochastic forcing and clarifies how noise contributes to sustaining enstrophy transfer while dissipating energy at high $Re$. Future work may address extending to multiplicative noise and more precise dependence on the stochastic forcing term $\tilde{G}$.
Abstract
The statistical features of homogeneous, isotropic, two-dimensional stochastic turbulence are discussed. We derive some rigorous bounds for the mean value of the bulk energy dissipation rate $\mathbb{E} [\varepsilon ]$ and enstrophy dissipation rates $\mathbb{E} [χ] $ for 2D flows sustained by a variety of stochastic driving forces. We show that $$\mathbb{E} [ \varepsilon ] \rightarrow 0 \hspace{0.5cm}\mbox{and} \hspace{0.5cm} \mathbb{E} [ χ] \lesssim \mathcal{O}(1)$$ in the inviscid limit, consistent with the dual-cascade in 2D turbulence.