A Nine-Modes Truncation of the Plane Incompressible Navier-Stokes Equations

Abstract

In this paper a nine-modes truncation of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained. The stationary solutions, the existence of attractor and the global stability of the equations are firmly proved. What is more, that the force $f$ acts on the mode $k_3$ and $k_7$ respectively produces two systems, which lead to a much richer and varied phenomenon. Numerical simulation is given at last, which shows a stochastic behavior approached through an involved sequence of bifurcations.