Continuous Data Assimilation For the 3D Ladyzhenskaya Model: Analysis and Computations

TL;DR

This paper analyzes and demonstrates the effectiveness of continuous data assimilation via nudging for the 3D Ladyzhenskaya equations, establishing conditions for synchronization with coarse observational data through theoretical analysis and numerical experiments.

Contribution

It provides the first rigorous analysis of data assimilation for the 3D Ladyzhenskaya model, including conditions for synchronization and numerical validation.

Findings

Synchronization achieved under certain spatial resolution conditions

Algorithm effective in both 2D and 3D simulations

Theoretical guarantees for convergence despite potential non-uniqueness

Abstract

We analyze continuous data assimilation by nudging for the 3D Ladyzhenskaya equations. The analysis provides conditions on the spatial resolution of the observed data that guarantee synchronization to the reference solution associated with the observed, spatially coarse data. This synchronization holds even though it is not known whether the reference solution, with initial data in $L^2$, is unique; a particular reference solution is determined by the observed, coarse data. The efficacy of the algorithm in both 2D and 3D is demonstrated by numerical computations.