Effect of vorticity coherence on energy-enstrophy bounds for the 3D Navier-Stokes equations

Bounding curves in the energy,enstrophy-plane are derived for the 3D Navier-Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is H"older continuous with exponent $r=1\mathrm{/}2$ results in a curve with extraordinarily large maximal enstrophy (exponential in Grashof), in marked contrast to the subcritical case, $r>1\mathrm{/}2$ (algebraic in Grashof).