Mathematical proof of spatial symmetry invariance of two-dimensional Kolmogorov flow

We prove a mathematical theorem that the exact solution of the two-dimensional (2D) Kolmogorov flow governed by the Navier–Stokes (NS) equations with the periodic boundary condition maintains the same spatial symmetry as its initial condition globally in time, i.e., for all t≥ 0 . This mathematical theorem can be used as an absolute scale to check the correctness and reliability of numerical simulations of NS turbulence. It is found that all reported CNS (clean numerical simulation) results of the NS turbulence maintain the same spatial symmetry as the corresponding initial condition in the whole time interval of simulation, but all of the corresponding DNS (direct numerical simulation) results lose their spatial symmetry quickly. In other words, these DNS results violate this mathematical theorem. Therefore, this mathematical theorem rigorously confirms that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. It also illustrates that CNS can provide helpful enlightenment to deepen our understanding of turbulence, since it can reveal mathematical truths and realities of NS turbulence.