[Paper Review] Averaging principle for two dimensional stochastic Navier-Stokes equations

Shihu Li, Xiaobin Sun+2 more|arXiv (Cornell University)|Oct 4, 2018

Stochastic processes and financial applications26 references4 citations

TL;DR

This paper establishes the strong averaging principle for a two-time-scale system where the slow component is a 2D stochastic Navier-Stokes equation and the fast component is a stochastic reaction-diffusion equation. Using the Khasminskii time-discretization approach, the authors prove that the slow component converges strongly to the solution of the averaged equation under suitable conditions, overcoming challenges from nonlinear terms and multiplicative noise via stopping time techniques and Sobolev regularity assumptions on initial data.

ABSTRACT

The averaging principle is established for the slow component and the fast component being two dimensional stochastic Navier-Stokes equations and stochastic reaction-diffusion equations, respectively. The classical Khasminskii approach based on time discretization is used for the proof of the slow component strong convergence to the solution of the corresponding averaged equation under some suitable conditions. Meanwhile, some powerful techniques are used to overcome the difficulties caused by the nonlinear term and to release the regularity of the initial value.

Motivation & Objective

• To establish the strong averaging principle for a multiscale system with a 2D stochastic Navier-Stokes equation as the slow component and a stochastic reaction-diffusion equation as the fast component.
• To address the challenge of strong convergence in the presence of highly nonlinear terms and multiplicative noise in the Navier-Stokes equation.
• To relax the regularity requirement on the initial data by avoiding the need for initial values in $ H^\theta $ with $ \theta > 0 $, using refined estimates.
• To extend the classical Khasminskii approach to SPDEs with nonlinearities and multiplicative noise, ensuring convergence in the strong sense.

Proposed method

• Employing the Khasminskii time-discretization method to split the time interval $[0,T]$ into subintervals of size $ \delta $, dependent on $ \varepsilon $.
• Introducing an auxiliary process $ (\hat{X}_t^\varepsilon, \hat{Y}_t^\varepsilon) $ on each subinterval to decouple the slow and fast components.
• Using stopping time techniques to control the difference $ |X_t^\varepsilon - \bar{X}_t| $ by bounding $ |X_t^\varepsilon - \hat{X}_t^\varepsilon| $ and $ |\hat{X}_t^\varepsilon - \bar{X}_t| $ before the stopping time.
• Applying priori estimates to control the difference after the stopping time, ensuring uniform convergence in probability.
• Establishing coercivity and local monotonicity for the SPDE system in the Gelfand triple $ \mathcal{V} \subset \mathcal{H} \subset \mathcal{V}' $, using Sobolev and $ L^4 $-norm estimates.
• Using finite-dimensional Galerkin approximations and weak convergence arguments, with verification of the necessary conditions for the coefficients in the SPDE formulation.

Experimental results

Research questions

• RQ1 Does the slow component of a two-time-scale 2D stochastic Navier-Stokes system converge strongly to the solution of the averaged equation as $ \varepsilon \to 0 $?
• RQ2 Can the classical Khasminskii approach be adapted to SPDEs with nonlinear Navier-Stokes terms and multiplicative noise?
• RQ3 How can the regularity requirement on the initial data be relaxed in the strong averaging principle for such systems?
• RQ4 What conditions on the nonlinearities $ f $, $ g $, and noise coefficients $ \sigma_1 $, $ \sigma_2 $ ensure strong convergence?
• RQ5 How do stopping time techniques and time discretization help control the nonlinear and multiplicative noise terms in the convergence proof?

Key findings

• The slow component $ X_t^\varepsilon $ converges strongly to the averaged solution $ \bar{X}_t $ in the sense that $ \lim_{\varepsilon \to 0} \mathbb{E}\left( \sup_{t \in [0,T]} |X_t^\varepsilon - \bar{X}_t|^{2p} \right) = 0 $ for all $ p \geq 1 $.
• The proof overcomes the difficulty of nonlinear terms in the Navier-Stokes equation by using stopping time techniques and refined $ L^4 $-norm estimates on velocity differences.
• The coercivity condition is verified via $ \langle \tilde{A}w + F(w), w \rangle + |\sigma(w)|^2_{\mathcal{L}_Q} \leq -C_\varepsilon \|w\|^2_{\mathcal{V}} + C(1 + |w|^2_{\mathcal{H}}) $, ensuring stability.
• Local monotonicity is established with $ \langle \tilde{A}w_1 + F(w_1) - \tilde{A}w_2 - F(w_2), w_1 - w_2 \rangle + |\sigma(w_1) - \sigma(w_2)|^2_{\mathcal{L}_Q} \leq C(1 + |u_2|_{L^4}^4) |w_1 - w_2|^2_{\mathcal{H}} $.
• The method avoids requiring initial data in $ H^\theta $ for $ \theta > 0 $, allowing for less regular initial conditions through careful pathwise control.
• The convergence result holds under standard assumptions on $ f $, $ g $, $ \sigma_1 $, and $ \sigma_2 $, including Lipschitz continuity and boundedness of derivatives.