Departmental Colloquium
Charlie Doering
University of Michigan, Ann Arbor
Twist \& Shout: Maximal enstrophy production in the 3D Navier-Stokes equations
Abstract: 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, Indiana University Mathematics
Journal Vol. 57, pp. 2693-2727 (2008).
Friday March 6, 2009 at 3:00 PM in SEO 636