Twist & Shout: Maximal Enstrophy Production in the 3D Navier-Stokes 
Equations

Charles R. Doering
Professor of Mathematics & Physics
University of Michigan, Ann Arbor, MI 49109-1043

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 one of the $1M Clay 
Prize problems.)  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).