I read that one of the main differences between the establishing global regularity for the Navier-Stokes equations for viscous and incompressible flow in 2 spatial dimensions versus in 3 is that in 2D there is something known as the enstrophy miracle which is given as (getting this straight from arxiv),
$$\int_{\Omega_{\epsilon}}\left(f\left(u\cdot\nabla f\right)\right)\,dV = 0$$
What does this mean? How does this come about? Please respond as if you're responding to somebody whose only familiar with undergraduate vector calculus, linear algebra, differential equations, and a bit of fluid dynamics and real analysis.