Determining Map, Data Assimilation and an Observable Regularity Criterion for the Three-Dimensional Boussinesq System

TL;DR

This paper establishes velocity-based conditions ensuring the global well-posedness and regularity of solutions to the 3D Boussinesq system using the AOT data assimilation algorithm, introducing a new observable regularity criterion.

Contribution

It provides the first rigorous analysis of the AOT algorithm for the 3D Boussinesq system, deriving conditions based solely on observed velocity data.

Findings

Conditions guarantee convergence of the AOT algorithm.

Constructs a determining map for the system.

Introduces a new observable regularity criterion.

Abstract

In this paper, we provide conditions, \emph{based solely on the observed velocity data}, for the global well-posedness, regularity and convergence of the Azouni-Olson-Titi data assimilation algorithm (AOT algorithm) for a Leray-Hopf weak solutions of the three dimensional Boussinesq system. This condition also guarantees the construction of the {\it determining map}. The aforementioned conditions on the (finite-dimensional) velocity observations, which in this case comprise either of a finite-dimensional \emph{modal} projection or finitely many \emph{volume element observations}, are automatically satisfied for solutions that are globally regular and are uniformly bounded in the $H^1$-norm. However, neither regularity nor uniqueness is {\it a priori} assumed on the solutions. To the best of our knowledge, this is the first such rigorous analysis of the AOT data assimilation algorithm for the three-dimensional Boussinesq system. As a corollary, we obtain that the condition that we imposed is in fact {\it a new observable regularity criterion on the weak global attractor.} The proof of this fact proceeds through the construction of the determining map.