$\mathbb{R}^3$上的定常欧拉流及其普遍动力学
Steady Euler flows on $\mathbb{R}^3$ with wild and universal dynamics
摘要：
理解流体力学中的复杂性是一个主要问题，在过去的几十年里吸引了物理学家和数学家的注意力。利用动力学中重整化的概念，证明了欧拉方程定常解的局部稠密集$\mathscr G$的存在性，使得每个向量场$X\in \mathscr G$在保持圆盘的非对称性的任何区域都可以用$X$在某个横截面上的Poincaré映射来近似（具有任意精度）的意义下是普适的.我们注意到，这种普遍性是近似的，但发生在所有尺度。特别是，我们的研究结果建立了一个稳定的欧拉流可能会表现出任何保守的有限余维动力学现象，这包括存在的马蹄积累椭圆岛，增加工会的豪斯多夫维数$3$或同宿切线的任意高的多重性的马蹄。我们构造的定常解是在无穷远处有急剧衰减的贝尔特拉米场。为了证明这些结果，我们引入了新的扰动法的贝尔特拉米领域的背景下，使我们能够进口深技术从分歧理论：Gonchenko-Shilnikov-Turaev普遍性理论和纽豪斯和Duarte定理的几何野生双曲集。这些扰动法依赖于线性偏微分方程的两个工具：全局逼近和Cauchy-Kovalevskaya定理。这些结果暗示了V.I. Arnold对欧几里得空间中Beltrami场的复杂性的看法。
英文摘要：
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.
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