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Local Structure of The Set of Steady-State Solutions to The 2d Incompressible Euler Equations

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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.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Leipzig, PF 10.09.20, D-04009, Leipzig, Germany

    Antoine Choffrut

  2. School of Mathematics, University of Minnesota, Vincent Hall 236, 206 Church St. SE, 55455, Minneapolis, USA

    Vladimír Šverák

Authors
  1. Antoine Choffrut
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  2. Vladimír Šverák
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Correspondence to Antoine Choffrut.

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V.Š. supported in part by NSF grant DMS 0800908.

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Choffrut, A., Šverák, V. Local Structure of The Set of Steady-State Solutions to The 2d Incompressible Euler Equations. Geom. Funct. Anal. 22, 136–201 (2012). https://doi.org/10.1007/s00039-012-0149-8

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  • DOI: https://doi.org/10.1007/s00039-012-0149-8

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