发表机构
The Hong Kong Polytechnic University; Beijing Institute of Technology; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences（香港理工大学; 北京理工大学; 中国科学院数学与系统科学研究院; 中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过在三维环面构造双线性算子及在表面准地转方程构造解，证明精确二次能量抵消不强制空间解析半径增长，精确能量身份无法决定解析半径增长的频率几何。
AI 中文摘要
我们证明仅精确二次能量抵消并不强制空间解析半径的增长。在三维环面$\boldsymbol{\top}^3$上，我们构造了一个显式的对称、平移不变的一阶双线性算子$Q$，该算子保持无散类，且对任意光滑实值无散向量场$v$满足$\boldsymbol{\top}^3$上的积分$\boldsymbol{\top}^3 Q(v,v)\boldsymbol{\top} v \boldsymbol{\top} dx=0$。对任意给定的足够小的时间$T>0$，方程$\boldsymbol{\top}_t u-\boldsymbol{\top} u=Q(u,u)$存在全局光滑、实值、零均值、无散解$u$，使得$\boldsymbol{\top}(u(0))=\boldsymbol{\top}(u(T))=1$。该构造将不变循环剪切类上的动力学约化为粘性伯格斯方程，并调节Cole-Hopf热剖面，使其最近的复零点回到其与实环面的初始距离。对任意$1<\boldsymbol{\top}<2$和任意给定的足够小的$T>0$，我们还构造了一个对称稀疏频率集、其相关的傅里叶投影，以及投影耗散表面准地转方程的三角多项式初始数据。所得的唯一全局光滑解$\boldsymbol{\theta}$初始时具有无限解析半径，但满足$0<\boldsymbol{\top}(\boldsymbol{\theta}(T))\boldsymbol{\top}1$。加性分离的傅里叶级联产生系数的指数下界，而均匀比较估计控制反馈相互作用。因此，即使从三角多项式数据出发，整体解析性也不必持续。两个构造共同表明，仅精确能量身份并不能决定支配解析半径增长的频率几何。
英文摘要
We prove that exact quadratic energy cancellation alone does not force growth of the spatial analyticity radius. On $\mathbb{T}^3$, we construct an explicit symmetric, translation-invariant, first-order bilinear operator $Q$ that preserves the divergence-free class and, for every smooth real-valued divergence-free vector field $v$, satisfies $ \int_{\mathbb{T}^3} Q(v,v)\cdot v\,dx=0.$ For every prescribed sufficiently small time $T>0$, the equation $\partial_t u-Δu=Q(u,u)$ admits a global smooth, real-valued, mean-zero, divergence-free solution $u$ such that $\operatorname{rad}(u(0))=\operatorname{rad}(u(T))=1$. The construction reduces the dynamics on an invariant cyclic-shear class to viscous Burgers and tunes a Cole-Hopf heat profile so that its nearest complex zero returns to its initial distance from the real torus. For every $1<α<2$ and every prescribed sufficiently small $T>0$, we also construct a symmetric sparse frequency set, its associated Fourier projection, and trigonometric-polynomial initial data for the projected dissipative surface quasi-geostrophic equation. The resulting unique global smooth solution $θ$ has infinite analyticity radius initially but satisfies $0<\operatorname{rad}(θ(T))\leq1$. An additively separated Fourier cascade yields coefficientwise exponential lower bounds, while uniform comparison estimates control the feedback interactions. Thus entire analyticity need not persist even from trigonometric-polynomial data. Together, the two constructions show that an exact energy identity alone does not determine the frequency geometry governing analyticity-radius growth.
