Schedule Jun 2
Geometry and analysis of the Euler and average Euler equations
J. Marsden (Caltech)
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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