Uniform Stabilization of 3D Navier–Stokes Equations in Low Regularity Besov Spaces with Finite Dimensional, Tangential-Like Boundary, Localized Feedback Controllers


The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier–Stokes equations in the vicinity of an unstable equilibrium solution, by means of a ‘minimal’ and ‘least’ invasive feedback strategy which consists of a control pair { v, u} (Lasiecka and Triggiani in Nonlinear Anal 121:424–446, 2015). Here v is a tangential boundary feedback control, acting on an arbitrary small part Γ~ of the boundary Γ; u is a localized, interior feedback control, acting tangentially on an arbitrarily small subset ω of the interior supported by Γ~. The ideal strategy of taking u= 0 on ω is not sufficient. A question left open in the literature is: can such feedback control v of the pair { v, u} be asserted to be finite dimensional also in dimension d= 3 ? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control v, it is here then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with a “right" Besov space setting of lower regularity. These spaces are ‘close’ to L3(Ω) for d= 3. This functional setting is significant. It is in line with recent well-posedness results in the full space of the non-controlled N–S equations (Escauriaza et al. in Math Subj Classif 35K:76D, 1991; Rusin and Sverak in Minimal initial data for potential Navier–Stokes singularities. arXiv:0911.0500; Jia and Šverák in SIAM J Math Anal 45(3):1448–1459, 2013; Gallagher et al. in Math Ann 355(4):1527–1559, 2013). A double key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions while having a sufficiently high topological level to handle the 3d-nonlinearity in the analysis of well-posedness and uniform stabilization. The proof is constructive and is “optimal” also regarding the “minimal” number of tangential boundary feedback controllers needed. The new setting requires the solution of novel technical and conceptual issues. These include establishing maximal regularity up to T= ∞ in the required suitably identified “right" Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. This result is also a new contribution to the area of maximal regularity as the operator to which it applies incorporates a boundary feedback control term rather than homogeneous boundary conditions. It escapes direct use of perturbation theory. Finally, the very ability to stabilize even the finite dimensional unstable projected system is linked to a Unique Continuation Property of a suitably over-determined (adjoint) Oseen eigenproblem, which requires the presence of the interior tangential-like control u acting on ω.

Publication Title

Archive for Rational Mechanics and Analysis