## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 136-201 |

Number of pages | 66 |

Journal | Geometric and Functional Analysis |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2012 |

### Bibliographical note

Funding Information:Poisson reduction, Nash–Moser inverse function theorem 2010 Mathematics Subject Classification: 35Q35, 37C25, 37K65, 46T05, 58C15, 58D05 V.Sˇ. supported in part by NSF grant DMS 0800908.

## Keywords

- Incompressible Euler
- Lie-Poisson reduction
- Nash-Moser inverse function theorem
- groups of diffeomorphisms
- stationary flows