Maximum Production of Enstrophy in Swirling Viscous Flows

We study a family of axisymmetric vector fields that maximize the instantaneous production of enstrophy in 3-dimensional (3D) incompressible viscous flows. These vector fields are parametrized by their energy $\mathcal{K}$, enstrophy $\mathcal{E}$ and helicity $\mathcal{H}$, and are obtained as the solution of suitable constrained optimization problems. The imposed symmetry is justified by the results reported in the seminal work of Doering \& Lu (2008), recently confirmed independently by Ayala \& Protas (2016), where highly-localized pairs of colliding vortex rings are found to be optimal for enstrophy production. The connection between these optimal axisymmetric fields and the ``blow-up'' problem in the 3D Navier-Stokes equation is discussed.