3倍对称尺度不变欧拉流的梯度增长与向跃变剖面的松弛
Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- University of Chinese Academy of Sciences（中国科学院大学）
- Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文针对Said等人m≥4松弛理论未解决的3倍对称尺度不变欧拉流，证明其解的梯度增长性，给出极限剖面性质及预紧轨道生成结论。
AI中文摘要：
我们研究二维欧拉方程的零齐次约化，针对3倍对称情形m=3，该情形在Said、Elgindi和Murray的m≥4松弛理论中未解决。我们证明，对1<p≤∞，每个非常数W^{1,p}解满足∥g_θ(t)∥_{L^p}在t→±∞时趋于无穷；对p=1，总变差守恒，但当初始有限时，LlogL模趋于无穷。若D_θg_0是可和的非原子同号分量与原子之和，两个ω极限集中的每个剖面均为跃变剖面，且每个半轨道在W^{α,r}中趋近其ω极限集（αr<1）；D_θg_0的结构假设对C^1数据自动成立。此外，每个弱L^2无穷时间极限生成一个完全L^2预紧轨道。
英文摘要：
We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. Éc. Norm. Supér. (4) \textbf{58} (2025), no.~4, 943--970], which treats $m$-fold symmetry solutions with $m\geq4$. We prove that every nonconstant $W^{1,p}$ solution satisfies $\|g_θ(t)\|_{L^p}\to\infty$ as $t\to\pm\infty$ for $1<p\leq\infty$. For $p=1$, the total variation is conserved, but the $L\log L$ modular tends to infinity whenever it is initially finite. If $D_θg_0$ is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in $W^{α,r}$ for $αr<1$. For such data, every weak $L^2$ infinite-time limit generates a complete $L^2$-precompact orbit. This structural assumption on $D_θg_0$ is automatic for $C^1$ data. In particular, these results answer the question concerning small-scale creation and compact orbit raised by Drivas and Elgindi [EMS Surv. Math. Sci. \textbf{10} (2023), no.~1, 1--100, Problem~5] for all nonconstant smooth 3-fold symmetric scale-invariant flows. Together with the known theory for $m\geq4$, they cover the full well-posed scale-invariant range $m\geq3$.
