On the local structure of the set of steady-state solutions to the 2D Euler equations.

Abstract

The main result of this thesis is based on the interpretation of Euler's flow of an incompressible fuid as a geodesic flow on the infinite-dimensional Lie group of volume-preserving diffeomorphisms of the region occupied by the fluid equipped with a one-sided invariant metric. In finite dimensions, the dynamics on the cotangent bundle of a Lie group equipped with a one-sided invariant metric can be reduced to a family of Hamiltonian systems on the co-adjoint orbits in the dual Lie algebra. Thus, non-degenerate stationary points are in a (local) one-to-one correspondence with the co-adjoint orbits. We prove that this holds for the most part for two-dimensional Euler's equations of hydrodynamics. Here, the co-adjoint orbits are the sets of isovorticed flows, i.e. sets of vorticity functions obtained by composition with volume-preserving diffeomorphisms, and these are invariant under the vorticity equation. (The latter statement is equivalent to Kelvin-Helmholtz' theorem on conservation of vorticity.) This result is valid for annulus domains in two dimensions, in the category of smooth functions, and in a neighborhood of fairly general steady-states. The co-adjoint orbits are not smooth manifolds if one works in the usual Banach spaces and therefore the proof is based on an application of the Nash-Moser inverse function theorem.