Global Stabilization of the Navier-Stokes Equations Around an Unstable Steady State with Mixed Boundary Kinetic Energy Controller

The paper, benefiting from techniques developed in Ngom et al. (Evol Equ Control Theory. 2015;4:89–106), presents a mixed (Dirichlet-Neumann) boundary feedback controller for stabilizing the Navier-Stokes equations around a prescribed steady state, in a bounded domain \({\Omega }\). The Neumann part of the boundary controller is designed to be zero when the inflow vanishes, and to have the magnitude of the kinetic energy. Like in Ngom et al. (Evol Equ Control Theory. 2015;4:89–106), the present paper proves exponential decrease of the perturbation in \(L^{2}\), without blowup. In addition, it goes further than (Ngom et al., Evol Equ Control Theory. 2015;4:89–106) by proving, on the one hand, that the exponential convergence towards zero holds in \(H^{1}\), on the other hand, that the weak solution is unique when the computational domainis two-dimensional.