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

Abstract

In this paper, we consider the analysis of a nudging based algorithm for data assimilation for the three-dimensional Boussinesq system, which we call the AOT system. A rigorous analysis of this algorithm for well-posed dissipative partial differential equations was first provided by Azouani, Olson and Titi (J Nonlinear Sci 24:277–304, 2014); thus justifying our terminology for the associated nudging system. We provide a sufficient condition, based solely on the observed velocity data obtained from a Leray-Hopf weak solution, for the global well-posedness, regularity and most crucially, the asymptotic tracking property of solutions of the associated (three-dimensional) nudging system. It is to be noted that neither regularity nor any knowledge of a uniform ℍ¹-norm bound is a priori assumed on the solution of the original three-dimensional Boussinesq system from which the observations are obtained. As a corollary of our result, we obtain a novel observable regularity criterion based on finitely many observational data. Our condition also guarantees the construction of the Lipschitz continuous determining map, which is known to play a crucial role in the construction of the so-called determining form and in statistical data assimilation.