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 S-1 x S-1 excited by the external force (k(2) sin ky, 0) and find the long-time behavior for the now starting from some states, where S-1 = [0, 2 pi](mod 2 pi). 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.