rtex states already found in §\[sec:3D\_InstOpt\_E0to0\] and §\[sec:3D\_InstOpt\_E\], and to the Poincaré limit. Another interesting possibility is to replace the energy $\K({\mathbf{u}})$ with the helicity [$\H({\mathbf{u}}) :=
  \int_{\Omega} {\mathbf{u}}\cdot(\bnabla\times{\mathbf{u}})\,d\Omega$]{} in the multiobjective formulation , as this might allow one to obtain extreme vortex states with a more complicated topology (i.e., a certain degree of “knottedness”). We note that all the extreme vortex states found in the present study were “unknotted”, i.e., were characterized by $\H({\widetilde{\mathbf{u}}_{\E_0}}) = 0$, as the vortex rings were in all cases disjoint (cf. figure \[fig:ring\]).

Finally, another promising possibility to find initial data producing a larger growth of enstrophy is to solve a [*finite-time*]{} optimization problem of the type already studied by [@ap11a] in the context of the 1D Burgers equation, namely

\[pb:maxdE\] $$\tilde{{\mathbf{u}}}_{0;\E_0,T}  = \mathop{\arg\max}_{{\mathbf{u}}_0\in{\mathcal{S}_{\E_0}}} \, \E(T),$$

where $T>0$ is the length of the time interval of interest and ${\mathbf{u}}_0$ the initial data for the Navier-Stokes system . In contrast to problems \[pb:maxdEdt\_E\] and \[pb:maxdEdt\_KE\], solution of problem \[pb:maxdE\] is more complicated as it involves flow evolution. It represents therefore a formidable computational task for the 3D Navier-Stokes system. However, it does appear within reach given the currently available computational resources and will be studied in the near future.

Acknowledgements {#acknowledgements .unnumbered}
================

The authors are indebted to Charles Doering for many enlightening discussions concerning the research problems studied in this work. The authors are grateful to Nicholas Kevlahan for making his parallel Navier-Stokes solver available, which was used to obtain the results reported in §\[sec:timeEvolution\]. Anonymous referees provided many insightful comments which helped us improve this work. This