Steve Shkoller, UC Davis
Recent developments in the geometry of hydrodynamics
Fluid dynamics is an extremely interesting and challenging field
of mathematics,
which relies upon PDE theory, numerical analysis, and geometry to reveal its
many secrets. Traditionally, only the first two areas, PDE and numerics,
benefited greatly from one another's successes, but recent advances in the
geometry of the volume-preserving diffeomorphism group has served to unify
all three areas.
We shall describe these geometric developments and explain how they have been
used to show the equivalence between seemingly unrelated fluids models, as
well as provide well-posedness results previously unavailable by traditional
techniques. In a surprising twist, we shall explain how the mathematical
equations of non-Newtonian fluids which model polymer flow are, in fact, the
famous vortex numerical method for integrating the Euler equations of inviscid
incompressible fluids. Both of these systems are geodesics of new subgroups
of the volume-preserving diffeomorphism group with respect to a recently
discovered Riemannian metric. Other hydrodynamical systems will be discussed
in this context as well.