Journal
GEOMETRIC AND FUNCTIONAL ANALYSIS
Volume 22, Issue 1, Pages 136-201Publisher
BIRKHAUSER VERLAG AG
DOI: 10.1007/s00039-012-0149-8
Keywords
Incompressible Euler; stationary flows; groups of diffeomorphisms; Lie-Poisson reduction; Nash-Moser inverse function theorem
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Funding
- NSF [DMS 0800908]
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It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler's equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.
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