Infinite Dimensional Dynamical Systems In Fluid Dynamics And Fluid-Structure Interaction

Abstract

In Part 1 of this thesis, three results are presented : (1) A sufficient condition, \emph{based solely on the observed velocity data}, for the global well-posedness, regularity and the asymptotic tracking property of a data assimilation algorithm for the three-dimensional Boussinesq system employing nudging, (2) a data assimilation algorithm for the 3D Navier-Stokes equation (3D NSE) using \emph{nodal observations}, and, as a consequence (3) a novel regularity criterion for the 3D NSE \emph{based on finitely many observations} of the velocity. The observations are drawn from a Leray-Hopf weak solution of the of the underlying system. For the data assimilated 3D Boussinesq system the observations are comprised either of a finite-dimensional \emph{modal} projection or finitely many \emph{volume element observations}, whereas for the data assimilated 3D NSE, the observations could be a finite dimensional \emph{modal} projection, finitely many \emph{volume element observations} or finitely many \emph{nodal observations}. The proposed conditions on the data in each case are automatically satisfied for solutions that are globally regular and are uniformly bounded in the $\h^1$-norm. However, neither regularity nor any knowledge of a uniform $\h^1$-norm bound is {\it a priori} assumed on the solutions. To the best of our knowledge, this is the first such rigorous analysis of \emph{any} data assimilation algorithm for the \emph{three-dimensional} Boussinesq system for which global regularity and well-posedness is unknown. Our condition also guarantees the construction of the {\it determining map} for the 3D Boussinesq system, thus extending prior work on its existence for the two-dimensional NSE. Additionally, the regularity criterion for the 3D NSE is \emph{fundamentally different} from any preexisting regularity criterion as it is based on \emph{finitely many pointwise observations} and \emph{does not require knowing the solution almost everywhere in space}. Lastly, we show that the regularity criterion we propose is both a necessary and sufficient condition for regularity. In Part 2 of this thesis the {strong asymptotic stabilization} of 3D hyperbolic dynamics is achieved by a damped 2D elastic structure evolving on a bounded subset. The model is a Neumann wave-type equation with low regularity coupling conditions given in terms of a nonlinear von Karman plate. This problem is motivated by the elimination of aeroelastic instability (sustained oscillations of bridges, airfoils, etc.) in engineering applications. Empirical observations indicate that the subsonic wave-plate system to equilibria. Classical approaches which decouple the plate and wave dynamics have fallen short. Here, we operate on the model as it appears in the engineering literature with {no regularization} and achieve stabilization by microlocalizing the Neumann boundary data for the wave equation (given by the plate). We observe {a compensation} by the plate dynamics { precisely where the regularity of the 3D wave is compromised} (in the characteristic sector).