Stability and instability of the 3D incompressible viscous flow in a bounded domain

In this paper, we investigate the stability and instability of the steady state (0, p(s)) (p(s) is a constant) for the 3D homogeneous incompressible viscous flow in a bounded simply connected domain with a smooth boundary where the velocity satisfies the Navier boundary conditions. It is shown that there exists a critical slip length -C-r mu, where C-r > 0 is an explicit generic constant depending only on the domain (given in (1.7)) and mu > 0 is the viscosity coefficient, such that when the slip length zeta is less than -C-r mu, the steady state (0, p(s)) is linearly and nonlinearly unstable; and conversely, the steady state (0, p(s)) is linearly and nonlinearly stable when zeta > -C-r mu.