Universal bounds on the attractor of the Navier-Stokes equation in the energy, enstrophy plane

We continue our examination of the projection of the global attractor of the two-dimensional Navier-Stokes equations in a normalized, dimensionless energy, enstrophy plane. In this plane the attractor must lie above the line of slope 1 through the origin (an immediate consequence of the Poincare inequality). We showed in Dascaliuc [J. Dyn. Differ. Equ. 17, 629-649 (2005)] that the attractor must also lie below a parabola (the square root curve). Here we optimize this parabola to obtain a sharper bound. These bounds are universal; they are independent of viscosity, domain size, and strength of the force. The effectiveness of the bounds is demonstrated on Galerkin approximations of modest size (up to 1372 modes). (c) 2007 American Institute of Physics.