Continuous Data Assimilation for Semilinear Parabolic Equations: A General Approach by Evolution Equations

TL;DR

This paper presents a unified framework for continuous data assimilation in semilinear parabolic equations, demonstrating exponential convergence of approximate solutions to true states across various complex systems.

Contribution

It introduces a general nudged model approach for data assimilation in semilinear parabolic equations, establishing well-posedness and exponential convergence under broad conditions.

Findings

Proves global well-posedness of the nudged and reference systems.

Shows exponential convergence of the approximate solution to the true solution.

Applies the framework to systems like Allen-Cahn, Cahn-Hilliard, and bidomain models.

Abstract

This article develops a general framework for continuous deterministic data assimilation for semilinear parabolic equations by means of evolution equations. Introducing a nudged model driven by partial observations, the global well-posedness of the reference and the approximating systems is established under natural assumptions. In addition, it is shown that the approximating solution converges exponentially to the solution of the reference system, provided the observational resolution and the nudging parameter are suitably chosen. The approach allows us to consider many systems, such as the Allen-Cahn, Cahn-Hilliard, Sellers-type energy balance, and bidomain systems, for the first time.