Maximal vorticity amplification and enstrophy production in the 3D Navier-Stokes equations

It is not known if solutions to the 3D Navier-Stokes equations for incompressible Newtonian fluids flowing in a finite periodic box can become singular in finite time. Indeed, this open issue is the subject of one of the $1M Clay Prize problems. It is known that solutions remains smooth as long as the enstrophy, i.e., the mean-square vorticity in the flow filed, is finite. The instantaneous generation rate of enstrophy is given by a functional that can be bounded in terms of the enstrophy itself using elementary functional estimates. That classical analysis establishes short-time regularity of solutions starting with arbitrarily large (but finite) enstrophy but does not rule out the possibility of singularities subsequently appearing in the flow in finite time. In this work we formulate and solve the variational problem for the maximal growth rate of enstrophy and display flows that saturate the functional analytic estimates and amplify vorticity at the greatest possible rate. Implications for questions of regularity or singularity in solutions of the 3D Navier-Stokes equations are discussed.

This joint work with Lu Lu was published in Indiana University Mathematics Journal Vol. 57, pp. 2693-2727 (2008).