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Doctoral Dissertation Defense: Abhishek Balakrishna

Advisors: Animikh Biswas and Justin Webster


Mathematics/Psychology : 412

Date & Time

April 28, 2023, 10:00 am12:00 pm


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

Advised By:
Animikh Biswas and Justin Webster (UMBC)

Addional Members of the Ph.D. Committee:
Andrei Draganescu (UMBC, reader)
Irena Lasiecka (University of Memphis, reader)
Rouben Rostamian (UMBC)
Muruhan Rathinam (UMBC)

For those joining remotely, you can use this Zoom information:
Meeting ID: 871 9717 7516 Passcode: k7RzUa

In Part 1 of this thesis, three results are presented : (1) A sufficient condition, 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 sys- tem employing nudging, (2) a data assimilation algorithm for the 3D Navier-Stokes equation (3D NSE) using nodal observations, and, as a consequence (3) a novel regularity criterion for the 3D NSE 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 modal projection or finitely many volume element observations, whereas for the data assimilated 3D NSE, the observations could be a finite dimensional modal projection, finitely many vol- ume element observations or finitely many 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 H1 -norm. However, neither regularity nor any knowledge of a uni- form H1 -norm bound is a priori assumed on the solutions. To the best of our knowledge, this is the first such rigorous analysis of any data assimilation algorithm for the three-dimensional Boussinesq system for which global regularity and well-posedness is unknown. Our condition also guarantees the construction of the determining map for the 3D Boussinesq system, thus extending prior work on its existence for the two-dimensional NSE. Additionally, the reg- ularity criterion for the 3D NSE is fundamentally different from any preexisting regularity criterion as it is based on finitely many pointwise observations and 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 aeroelas- tic 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).