A note on continuous data assimilation for stochastic convective Brinkman-Forchheimer equations in 2D and 3D

TL;DR

This paper develops and analyzes a continuous data assimilation algorithm for stochastic convective Brinkman-Forchheimer equations in 2D and 3D, demonstrating convergence under certain conditions and highlighting the benefits of nonlinear damping.

Contribution

It introduces a CDA algorithm for stochastic CBFEs, providing convergence analysis and showing improved stability and convergence properties due to nonlinear damping.

Findings

Convergence in mean-square sense for the assimilated solution.

Pathwise convergence established with additive noise.

Nonlinear damping enhances stability and enables CDA in 3D.

Abstract

Continuous data assimilation (CDA) methods, such as the nudging algorithm introduced by Azouani, Olson, and Titi (AOT), have proven to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with observed dynamics. In this note, we introduce and analyze an algorithm for CDA for the two- and three-dimensional stochastic convective Brinkman-Forchheimer equations (CBFEs) driven by either additive or multiplicative Gaussian noise. The model is believed to provide an accurate description when the flow velocity exceeds the regime of validity of Darcy's law and the porosity remains moderately large. We derive sufficient conditions on the nudging parameter and the spatial resolution of observations that ensure convergence of the assimilated solution to the true stochastic flow. We demonstrate convergence in the mean-square sense, and additionally establish pathwise convergence in the presence of additive noise. The CBFEs, also known as Navier-Stokes equations with damping, exhibit enhanced stability properties due to the presence of nonlinear damping term. In particular, we show that nonlinear damping not only enables the implementation of CDA in three dimensions but also yields improved convergence results in two dimensions when compared to the classical Navier-Stokes equations.