Continuous data assimilation for 2D stochastic Navier-Stokes equations

TL;DR

This paper extends continuous data assimilation methods to stochastic 2D Navier-Stokes equations, establishing conditions for convergence of approximate solutions to true stochastic flows under noise influence.

Contribution

It introduces a stochastic generalization of the AOT nudging framework, providing convergence guarantees for stochastic Navier-Stokes equations with additive and multiplicative noise.

Findings

Convergence in expectation for multiplicative noise with specific conditions.

Exponential convergence in both expectation and pathwise for additive noise.

Conditions on nudging parameters and observation scale ensure synchronization.

Abstract

Continuous data assimilation methods, such as the nudging algorithm introduced by Azouani, Olson, and Titi (AOT) [2], are known to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with observed dynamics. In this work, we extend this framework to a stochastic regime by considering the two-dimensional incompressible Navier-Stokes equations subject to either additive or multiplicative noise. We establish sufficient conditions on the nudging parameter and the spatial observation scale that guarantee convergence of the nudged solution to the true stochastic flow. In the case of multiplicative noise, convergence holds in expectation, with exponential or polynomial rates depending on the growth of the noise covariance. For additive noise, we obtain the exponential convergence both in expectation and pathwise. These results yield a stochastic generalization of the AOT theory, demonstrating how the interplay between random forcing, viscous dissipation and feedback control governs synchronization in stochastic fluid systems.