Continuous Data Assimilation for Semilinear Parabolic Equations with Multiplicative Observation Noise

TL;DR

This paper develops a theoretical framework for continuous data assimilation of semilinear parabolic PDEs with multiplicative noise, proving convergence and applying it to various fluid and reaction models.

Contribution

It introduces a general Gelfand triple-based theory for nudging equations with multiplicative noise, covering weak and strong formulations, and establishes convergence results.

Findings

Proved mean square convergence of the assimilation error.

Established almost sure convergence under additional noise conditions.

Applied the framework to models like 2D Navier-Stokes and Allen-Cahn equations.

Abstract

The problem of continuous data assimilation for semilinear parabolic equations based on partial observations corrupted by noise is investigated. The noise is allowed to be multiplicative, with additive noise arising as a special case. In a general Gelfand triple framework, an abstract theory for the nudging equation is developed that covers both weak and strong formulations. Mean square convergence of the assimilation error is proved under suitable assumptions, and, under additional integrability conditions on the noise, a uniform almost sure convergence result is established. Finally, the framework is applied to several PDE models, including the 2D Navier-Stokes, 2D magnetohydrodynamics, 2D quasi-geostrophic, and 1D Allen-Cahn equations.