Jun 2
Geometry and analysis of the Euler and average Euler equations
J. Marsden (Caltech)
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This talk will review some of the background on
geometric fluid dynamics appropriate for the Euler and
averaged Euler equations. In particular, we shall recall
how one regards the equations as geodesics on the
diffeomorphism group. Specifically, as Arnold showed,
these are geodesics with respect to the $L^2$ metric for
the Euler equations. We discuss how this approach gives
insight into a variety of things, ranging from existence
and uniqueness theorems, results on limits of zero
viscosity and the variational and Hamiltonian structure of
the equations and of vortex methods. Using
Euler-Poincar\'e theory, we show how the equations may be
regarded as geodesic equations for the
$H^1$ metric on the volume preserving diffeomorphism
group. Then we will present some of the analytical
theorems including the convergence as viscosity tends to
zero, even in the presense of boundaries. We will also
briefly indicate some of the interesting computational
aspects of the equations. The geometric mechanics approach
to hydrodynamics also gives insight into the derivation of
new classes of equations, namely the averaged Euler
equations, both the isotropic and the nonisotropic forms;
this derivation and related issues is the subject of the
talk of Steve Shkoller at this meeting.
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