Stochastic tamed 3D Navier-Stokes equations with locally weak monotonicity coefficients: existence, uniqueness and averaging principle
Shuaishuai Lu, Xue Yang, Yong Li
TL;DR
This work analyzes stochastic tamed 3D Navier–Stokes equations with locally weak monotonicity coefficients on $\mathbb{R}^3$ and $\mathbb{T}^3$, addressing non-Lipschitz drift and diffusion that preclude Gronwall-based uniqueness. A two-step approach using Galerkin approximations and a tailored control function yields pathwise uniqueness, with the Yamada–Watanabe theorem delivering a unique strong solution; a martingale solution is constructed via a limiting argument. The paper also proves an averaging principle for highly oscillatory systems, showing convergence to an averaged equation under the first Bogolyubov framework using Khasminskii discretization. These results extend well-posedness and homogenization theory for stochastic NS with irregular coefficients, providing a rigorous basis for turbulence analysis in inhomogeneous or rapidly oscillating environments.
Abstract
This paper investigates the stochastic tamed 3D Navier-Stokes equations with locally weak monotonicity coefficients in the whole space as well as in the three-dimensional torus, which play a crucial role in turbulent flows analysis. A significant issue is addressed in this work, specifically, the reduced regularity of the coefficients and the inapplicability of Gronwall's lemma complicates the establishment of pathwise uniqueness for weak solutions. Initially, the existence of a martingale solution for the system is established via Galerkin approximation; thereafter, the pathwise uniqueness of this martingale solution is confirmed by constructing a specialized control function. Ultimately, the Yamada-Watanabe theorem is employed to establish the existence and uniqueness of the strong solution to the system. Furthermore, an averaging principle, referred to as the first Bogolyubov theorem, is established for stochastic tamed 3D Navier-Stokes equations with highly oscillating components, where the coefficients satisfy the assumptions of linear growth and locally weak monotonicity. This result is achieved using classical Khasminskii time discretization, which illustrates the convergence of the solution from the original Cauchy problem to the averaged equation over a finite interval [0, T].