Tangential boundary uniform stabilization of 3-D Navier-Stoke equations by localized finite dimensional feedback controls

Abstract:  The present dissertation 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? The result is known for d=2 and also for d=3, however only for compactly supported initial conditions. For physical dimension d=3, the N-S nonlinearity forces a topological level sufficiently high as to dictate compatibility conditions. 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. Eventually, well-posedness of the nonlinear N-S problem as well as its uniform stabilization are obtained in an explicit Besov space with tight parameters related to the physical dimension d, where the compatibility conditions are not recognized. The proof is constructive and is "optimal" also regarding the "minimal" amount of tangential boundary control action needed.

The new setting requires the solution of new technical and conceptual issues. These include establishing maximal regularity in the required Besov setting for the overall closed-loop linearized problem with feedback control applied on the boundary. This result is also a new contribution to the area of maximal regularity. The minimal amount of tangential boundary action is linked to the issue of unique continuation properties of over-determined Oseen eigenproblems.