KAM theory and the 3D Euler equation |
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Authors: | Boris Khesin Sergei Kuksin Daniel Peralta-Salas |
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Institution: | 1. Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada;2. Université Paris-Diderot (Paris 7), UFR de Mathématiques – Batiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris Cedex 13, France;3. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain |
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Abstract: | We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold M is not mixing in the Ck topology (k>4 and non-integer) for any prescribed value of helicity and sufficiently large values of energy. This can be regarded as a 3D version of Nadirashvili's and Shnirelman's theorems showing the existence of wandering solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the mixing under the Euler flow of Ck-neighborhoods of divergence-free vectorfields on M . On the way we construct a family of functionals on the space of divergence-free C1 vectorfields on the manifold, which are integrals of motion of the 3D Euler equation. Given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types. We use the KAM theory to establish some continuity properties of these functionals in the Ck-topology. This allows one to get a lower bound for the Ck-distance between a divergence-free vectorfield (in particular, a steady solution) and a trajectory of the Euler flow. |
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Keywords: | KAM theory Euler equation Integrals of motion Ergodicity |
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