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Geometric aspects of hydrodynamic blowup

Friday, February 26, 2010 - 11:00am - 11:45am
EE/CS 3-180
Stephen Preston (University of Colorado)
The geometric approach to hydrodynamics was developed by Arnold
to study Lagrangian stability of ideal fluids. It identifies a
Lagrangian fluid flow with a geodesic on the Riemannian
manifold of volume-preserving diffeomorphisms. The curvature of
this manifold is typically negative but sometimes positive, and
positivity leads to conjugate points (where initially close
geodesics spread apart and come together again).

In this talk we suppose a fluid in src=http://www.ima.umn.edu/springer/greek/bbbT-small.png>3 satisfies a
pointwise version of the Beale-Kato-Majda criterion for blowup
at a finite time T. I will describe a theorem which states
that either the geodesic experiences an infinite sequence of
consecutive conjugate pairs approaching the blowup time, or the
deformation tensor has a fairly special form at the blowup
time. The first possibility suggests that one could see
blowup geometrically in a weak space, such as the space of
L2 measure-preserving transformations.
MSC Code: 
85A30
Keywords: