TY - JOUR

T1 - Local Structure of The Set of Steady-State Solutions to The 2d Incompressible Euler Equations

AU - Choffrut, Antoine

AU - Šverák, Vladimír

PY - 2012/2

Y1 - 2012/2

N2 - 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.

AB - 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.

KW - Incompressible Euler

KW - Lie-Poisson reduction

KW - Nash-Moser inverse function theorem

KW - groups of diffeomorphisms

KW - stationary flows

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U2 - 10.1007/s00039-012-0149-8

DO - 10.1007/s00039-012-0149-8

M3 - Article

AN - SCOPUS:84858862424

VL - 22

SP - 136

EP - 201

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 1

ER -