论文标题
稳定的欧拉在$ \ mathbb {r}^3 $上流动使用野生和通用动力学
Steady Euler flows on $\mathbb{R}^3$ with wild and universal dynamics
论文作者
论文摘要
了解流体力学的复杂性是一个主要问题，在过去的几十年中吸引了物理学家和数学家的注意。使用动力学重新归一化的概念，我们显示了在$ \ mathbb r^3 $中向Euler方程的固定解决方案的局部密集设置的存在，因此每个矢量字段$ x \ in \ mathscr g $都可以通过$ x的diffeemorphisk（均可）（均可限制$ x.一些横向部分。我们指出，这种普遍性是近似的，但在所有尺度上都发生。特别是，我们的结果表明，稳定的Euler流可以表现出任何保守的有限的编核动力学现象。这包括存在椭圆岛积累的马蹄铁，增加了Hausdorff Dimension的马蹄铁$ 3 $或任意高的多样性的同层次切线。我们构建的稳定解决方案是无穷大的Beltrami领域，具有锐利的衰变。为了证明这些结果，我们在Beltrami领域的背景下介绍了新的扰动方法，这些方法使我们能够从分叉理论中导入深度技术：Gonchenko-Shilnikov-Turaev普遍性理论以及Newhouse和Newhouse和Duarte Theorems在野生多余的体内的几何形状上。这些扰动方法依赖于线性PDE的两个工具：全局近似和Cauchy-Kovalevskaya定理。这些结果暗示了V.I.阿诺德（Arnold）对欧几里得空间中Beltrami田地复杂性的愿景。
Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincaré map of $X$ at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.
