Acta Mathematica Vietnamica On the stationary solutions to 2D $g$-Navier-Stokes equations

We consider the $g$-Navier-Stokes equations in a two-dimensional bounded domain
$\Omega$. First, we study the existence and exponential stability of a stationary
solution under some certain conditions. Second, we prove that any unstable steady
state can be stabilized by proportional controller with support in an open subset
$\omega \subset\Omega$ such that $\Omega\backslash\omega$ is sufficiently "small".