Continuous data assimilation for 2D stochastic Navier-Stokes equations

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.

💡 Research Summary

This paper extends the continuous data assimilation (CDA) framework, originally introduced by Azouani, Olson, and Titi (AOT), to the stochastic two‑dimensional incompressible Navier‑Stokes equations. The authors consider both additive and multiplicative noise, modelled by a cylindrical Wiener process, and study the convergence of a nudged solution (U(t)) toward the true stochastic flow (u(t)) when only coarse spatial observations are available.

The deterministic AOT algorithm augments the original dynamics with a feedback term (\mu R_h(u-U)), where (\mu>0) is the nudging parameter and (R_h) is an interpolant operator representing observations at resolution (h). In the stochastic setting the modified system reads

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