Isolated steady solutions of the 3D Euler equations

• 成果类型: Article
• 署名作者: Enciso, Alberto; Kepplinger, Willi; Peralta-Salas, Daniel
• 署名单位: Consejo Superior de Investigaciones Cientificas (CSIC); CSIC - Instituto de Ciencias Matematicas (ICMAT); University of Vienna
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-14930
• DOI: 10.1073/pnas.2414730122
• 发表日期: 2024-03-25
• 关键词: beltrami fields; FLOWS; THEOREM

摘要

We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the 1-topology. The proof of this fact combines ideas from dynamical systems, which appear naturally because these isolated states have strongly chaotic dynamics, with techniques from spectral geometry and contact topology, which can be effectively used to analyze the steady Euler equations on carefully chosen Riemannian manifolds. Interestingly, much of this strategy carries over to the Euler equations in Euclidean space, leading to the weaker result that there exist analytic steady solutions on T3 such that the only analytic steady Euler flows in a C1-neighborhood must belong to a certain linear space of dimension six. For comparison, note that in any Ck-neighborhood of shear flow, there are infinitely many linearly independent analytic shears.