Dissertation Defense Announcement

College of Arts and Sciences announces the Final Dissertation Defense of

Buddhika Priyasad Sembukutti Liyanage

for the Degree of Doctor of Philosophy

April 3, 2019 at 10:30 AM in Dunn Hall, Room 249

Advisor: Roberto Triggiani

Uniform Stabilization of Navier-Stokes Equations in Lq-based Sobolev and Besov Spaces

ABSTRACT: The present dissertation studies the problem of uniform stabilization of the Navier-Stokes equations, defined on a bounded domain of dimension d =2,3, in the vicinity of an unstable equilibrium solution, by means of a finite dimensional feedback controller. The setting is in the L^q-based Sobolev and Besov spaces, for specific, tight indices. A justification of this functional setting is given below. The treatment is divided in two parts. Part I considers the case where the stabilizing, finite dimensional, feedback control is localized in the interior; that is, it acts on an arbitrarily small subdomain. It intends to test the L^q-Besov techniques. Part II addresses the case of primary interest, with a boundary tangential feedback control localized on an arbitrarily small part of the boundary of the bounded domain. It then solves in the affirmative a long standing open problem: whether such stabilizing tangential boundary control can be asserted to be finite dimensional also in dimension d=3 in full generality. It is the aim of solving this open problem that forces the L^q-based, Besov setting for explicit values of the parameters. It replaces the Hilbert-Sobolev setting of the literature where finite dimensionality of the stabilizing controller for d=3 was achieved only with compactly supported data, because of the issue of compatibility conditions. To counter this, the Besov-setting is chosen as to not recognize boundary conditions. Critical technical ingredients include: (i) establishing maximal regularity for the boundary feedback linearized problem; and (ii) the role of unique continuation properties for over-determined Oseen eigen-problems.