Chaos in the incompressible Euler equation on manifolds of high dimension

• 成果类型: Article
• 署名作者: de Lizaur, Francisco Torres
• 署名单位: University of Toronto
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01089-3
• 发表日期: 2022
• 页码: 687-715
• 关键词: billiards

摘要

We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of any dimension. The main theorem of the paper, from which these families of solutions are obtained, states that for any given vector field X on a closed manifold N, there is a Riemannian manifold M on which the following holds: N is diffeomorphic to a finite dimensional manifold in the phase space of fluid velocities (the space of divergence-free vector fields on M) that is invariant under the Euler evolution, and on which the Euler equation reduces to a finite dimensional ODE that is given by an arbitrarily small smooth perturbation of the vector field X on N.