# Doctoral Dissertation Defense: Abhishek Balakrishna

### Advisors: Animikh Biswas and Justin Webster

Location

Mathematics/Psychology : 412

Doctoral Dissertation Defense: Abhishek Balakrishna – Online Event

Date & Time

April 28, 2023, 10:00 am – 12:00 pm

Description

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

Abstract:

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).

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