二维欧拉方程尺度不变旋转对称解的稠密轨道
Dense orbits for scale-invariant rotationally symmetric solutions of the 2D Euler equations
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中文总结 AI 辅助
本文证明二维欧拉方程在尺度不变m重对称涡量类中存在稠密前向轨道，通过线性叠加与稳定性估计迭代构造，表明受控涡量松弛假设不可放宽。
中文摘要 AI 辅助
设 $m \ge 4$。我们证明，对于尺度不变的 $m$ 重对称涡量，二维欧拉系统存在一个前向轨道，即 \begin{equation*} \partial_tg + 2G\partial_\theta g = 0, \quad 4G + \partial_{\theta\theta}G = g, \end{equation*} 该轨道在 $S:=\{g \in L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1): g\sin(m\theta) \ge 0 \text{ 且 }\\|g\\|_{L^\infty} \le 1\}$ 中稠密，其中 $S$ 装备了 $L^\infty(\mathbb{S}^1)$ 上的弱$^*$收敛拓扑，而 $L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1)$ 表示 $L^\infty(\mathbb{S}^1)$ 中奇且 $m$ 重对称函数的空间。上述系统最早由 Elgindi 和 Jeong 推导，他们证明了在 $m$ 重对称类中的适定性。在随后的工作中，Elgindi、Murray 和 Said 证明了具有受控涡量的解松弛到具有有限跳跃的分片常数稳态；我们的构造表明这一正则性假设不能被放宽。证明从一个简单的线性机制开始：通过从选定时间向后演化规定的阶跃函数并叠加所得数据，可以安排单个线性轨道逼近一个可数的稠密族。我们证明该机制对非线性欧拉耦合仍然成立。关键的稳定性估计表明，当下一近似时间取得足够远时，每个阶段引入的误差变得任意小，从而使构造可以无限迭代。
英文摘要
Let $m \ge 4$. We prove that there is a forward orbit of the 2D Euler system for scale-invariant $m$-fold symmetric vorticities, namely \begin{equation*} \partial_tg + 2G\partial_θg = 0, \quad 4G + \partial_{θθ}G = g, \end{equation*} which is dense in $S :=\{g \in L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1): g\sin(mθ) \ge 0 \text{ and }\|g\|_{L^\infty} \le 1\}$ equipped with the topology of weak$^*$ convergence on $L^\infty(\mathbb{S}^1)$, where $L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1)$ denotes the space of odd and $m$-fold symmetric functions in $L^\infty(\mathbb{S}^1)$. The system above was first derived by Elgindi and Jeong, who showed well-posedness in the $m$-fold symmetric class. In subsequent work by Elgindi, Murray and Said, it was shown that solutions with regulated vorticity relax to piece-wise constant steady states with finitely many jumps; our construction shows that this regularity assumption cannot be relaxed. The proof starts from a simple linear mechanism: by evolving prescribed step functions backwards from chosen times and superimposing the resulting data, one can arrange for a single linear orbit to approximate a countable dense family. We show that this mechanism persists for the nonlinear Euler coupling. The key stability estimates imply that the error introduced at each stage becomes arbitrarily small when the next approximation time is taken sufficiently far in the future, allowing the construction to be iterated indefinitely.
发表机构
- Courant Institute School of Mathematics, Computing and Data Science, New York University（纽约大学库朗数学科学研究所）
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