Relations Between Energy and Enstrophy on the Global Attractor of the 2-D Navier-Stokes Equations

Abstract

We examine how the global attractor $\mathcal {A}$ of the 2-D periodic Navier–Stokes equations projects in the normalized, dimensionless energy–enstrophy plane (e, E). We treat time independent forces, with the view of understanding how the attractor depends on the nature of the force. First we show that for any force, $\mathcal {A}$ is bounded by the parabola E = e^1/2 and the line E=e. We then show that for $\mathcal {A}$ to have points near enough to the parabola, the force must be close to an eigenvector of the Stokes operator A; it can intersect the parabola only when the force is precisely such an eigenvector, and does so at a steady state parallel to this force. We construct a thin region along the parabola, pinched at such steady states, that the attractor can never enter. We show that 0 cannot be on the attractor unless the force is in H^m for all m. Different lower bound estimates on the energy and enstrophy on $\mathcal {A}$ are derived for both smooth and nonsmooth forces, as are bounds on invariant sets away from 0 and near the line E = e. Motivation for the particular attention to the regions near the parabola and near 0 comes from turbulence theory, as explained in the introduction.