Local structure of the set of steady-state solutions to the 2D incompressible Euler equations

Details Local structure of the set of steady-state solutions to the 2D incompressible Euler equations Local structure of the set of steady-state solutions to the 2D incompressible Euler equations

2010-12-13

arxiv.org/abs/1012.2736v1

Mathematics

Analysis of PDEs

81 pages

Scientific paper

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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