It's still an open question whether solutions to the 3D Navier-Stokes equations for incompressible flows in a periodic box can become singular in finite time. It is known, however, that as long as the enstrophy (the mean-square vorticity) of a solution remains finite, the solution remains smooth. 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. 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. The results are discussed in the context of the search for either regularity or singularity in solutions of the 3D Navier-Stokes equations. This is joint work with Lu Lu.
|