# 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>

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

L

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 apointwise 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

L

^{2}measure-preserving transformations.MSC Code:

85A30

Keywords: