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Local structure of the set of steady-state solutions to the 2D incompressible Euler equations. (English) Zbl 1256.35076
Author’s 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.”

35Q35 PDEs in connection with fluid mechanics
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
46T05 Infinite-dimensional manifolds
58C15 Implicit function theorems; global Newton methods on manifolds
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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