A Nine-Modes Truncation of the Plane Incompressible Navier-Stokes Equations (Communications in Mathematical Research, 2021)
In this paper a nine-modes truncation of Navier-Stokes equations for atwo-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 twosystems, which lead to a much richer and varied phenomenon. Numerical simulationis given at last, which shows a stochastic behavior approached through an involvedsequence of bifurcations.
Keywords: the Navier-Stokes equation; the strange attractor; Lyapunov function; bifurcation; chaos.