Charles R. Doering, Professor of Mathematics & Physics, University of Michigan

It is still not known whether solutions to the 3D Navier-Stokes equations for incompressible flows in a finite periodic box can become singular in finite time. (Indeed, this question is the subject of a $1M Clay Prize problem.) It is known that a solution remains smooth as long as the enstrophy, i.e., the mean-square vorticity, of the solution is finite. The generation rate of enstrophy is given by a functional that can be bounded using elementary functional estimates. Those estimates establish short-time regularity but do not rule out finite-time singularities in the solutions. In this work we formulate and solve the variational problem for the maximal growth rate of enstrophy and display flows that generate enstrophy at the greatest possible rate. Implications for questions of regularity or singularity in solutions of the 3D Navier-Stokes equations are discussed. This is joint work with Lu Lu (Wachovia Investments).