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

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.