Time-asymptotics and the self-organization hypothesis for 2D Navier-Stokes equations

Abstract

In this paper we study the long-time behaviour of solutions with a two-dimensional structure of the Navier-Stokes equations on a periodic grid. From a rigorous investigation of the decrease of the energy E and the enstrophy W, it follows that the Rayleigh quotient Q= W E is monotonically decreasing on solutions. This is shown to imply that the spatial structure of any solution tends to a critical point of Q, which is the structure of some planar vortex, and that all these structures are unstable except for the fundamental one which has the longest wavelength. The time dependence of the approach towards this self-organized state is investigated in some detail. In the spectral plane, the mean-squared wave-numbers of both the spectral energy- and enstrophy density are shown to decrease as a function of time. For the enstrophy this implies that the normal cascade does not match the discriminating effect of the viscous dissipation.