Electronic Theses and Dissertations

Date

2019

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Irena Lasiecka

Committee Member

Roberto Triggiani

Committee Member

Irena Lasiecka

Committee Member

Roberto Triggiani

Committee Member

Hongqiu Chen

Committee Member

Bentuo Zheng

Abstract

We consider 2- or 3-dimensional Navier-Stokes equations defined on a bounded domain $\Omega$ subject to an external force, assumed to cause instability. We then seek to uniformly stabilize such N-S system, in the vicinity of an unstable equilibrium solution, in $L^q$-based Sobolev and Besov spaces, by finite dimensional feedback controls. This work is divided in to two parts. In Part I, the finite dimensional feedback controls are localized on an arbitrarily small open interior subdomain $\omega$ of $\Omega$. Instead, in Part II seeks tangential boundary feedback stabilizing controls. It provides a solution to the following recognized open problem in the theory of uniform stabilization of d-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of tangential boundary localized feedback controls: can these stabilizing controls be asserted to be finite dimensional also in the physical dimension $d=3$? To achieve the desired finite dimensionality result of the feedback tangential boundary controls, it was then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with an appropriate Lq-based/Besov setting with tight parameters related to the physical dimension $d$, where the compatibility conditions are not recognized. This result is also a new contribution to the area of maximal regularity with inhomogeneous boundary feedback.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest

Notes

embargoed

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