Communications of the
Korean Mathematical Society CKMS

We study the stabilization of 2D $g$-Navier-Stokes equations in bounded domains with no-slip boundary conditions. First, we stabilize an unstable stationary solution by using finite-dimensional feedback controls, where the designed feedback control scheme is based on the finite number of determining parameters such as determining Fourier modes or volume elements. Second, we stabilize the long-time behavior of solutions to 2D $g$-Navier-Stokes equations under action of fast oscillating-in-time external forces by showing that in this case there exists a unique time-periodic solution and every solution tends to this periodic solution as time goes to infinity.

Keywords: 2D $g$-Navier-Stokes equations, stabilization, stationary solution, time-periodic solution, finite-dimensional feedback controls, oscillating-in-time forces

MSC numbers: 35Q35, 35B35, 93D15