On Continuous Data Assimilation for a class of 2D and 3D stochastic non-Newtonian fluids of differential type

TL;DR

This paper develops a continuous data assimilation scheme for stochastic non-Newtonian fluids, proving convergence under certain conditions in both 2D and 3D settings.

Contribution

It introduces a CDA method for stochastic third-grade fluids and establishes convergence criteria, including mean-square and almost sure convergence.

Findings

Convergence of the assimilated state is guaranteed under specific nudging gain and mesh size.

Proves mean-square convergence for all stochastic forcing types.

Establishes almost sure convergence for additive noise cases.

Abstract

Continuous data assimilation (CDA) techniques, most notably the nudging approach proposed by Azouani, Olson, and Titi (AOT), have been shown to be very successful in deterministic frameworks for achieving long-time synchronization between an approximate state and true state. In this note, we develop and study a CDA scheme for a class of stochastic non-Newtonian fluids, namely third-grade fluids, subject to either additive or multiplicative Gaussian stochastic forcing in both two- and three-dimensional settings. We establish sufficient criteria on the nudging gain and the observational mesh size that guarantee convergence of the assimilated state toward the underlying stochastic solution. Convergence is proved in the mean-square sense, and, in the case of additive noise, we further obtain almost sure (pathwise) convergence.