Long-time behavior of Navier-Stokes flow on a two-dimensional torus excited by an external sinusoidal force

In this paper we study the Navier Stokes flow on the two-dimensional torus S1 x S1 excited by the external force (k2 sin ky, 0) and find the long-time behavior for the flow starting from some states, where S1 = [ 0, 2π](mod 2π). Especially for the case k = 2, it follows from an analysis and computation that the Navier-Stokes flow with the initial state cos(mx + ny) or sin(mx + ny) will likely evolve through at most one step bifurcation to either a steady-state solution or a time-dependent periodic solution for any Reynolds number and integers m and n