二维欧拉流在 $W^{3,1}(\mathbb R^2)$ 中的紧支撑强不适定性
Compactly Supported Strong Ill-Posedness of the Two-Dimensional Euler Flow in $W^{3,1}(\mathbb R^2)$
AI总结：
本文证明二维不可压缩欧拉方程在端点空间 $W^{3,1}$ 中的强不适定性：对任意紧支撑光滑背景，可构造任意接近的紧支撑数据，其解在任意正时间内 $W^{3,1}$ 范数本质无界，方法结合局部双曲插入、非乘子机制与多包稀释。
AI中文摘要：
我们证明了二维不可压缩欧拉方程在端点速度空间 $W^{3,1}_\sigma(\mathbb R^2)$ 中的强不适定性。给定任意紧支撑光滑无散度背景场以及任意 $\delta>0$，我们构造一个紧支撑无散度数据，使其在 $W^{3,1}$ 范数下与背景场的距离小于 $\delta$。该扰动可被局部化在背景支撑的任意指定非空开邻域内，同时保持与该支撑不相交。相应的唯一全局有限能量解 $u$ 满足 ${\rm curl} u\in C([0,\infty);W^{2,1}(\mathbb R^2))$，但 \\[ \operatorname*{ess\\,sup}_{0<t<T}\\|u(t)\\|_{W^{3,1}}=\infty \qquad\text{对于任意 }T>0. \\] 证明结合了局部化双曲插入、Bonami--Poornima 非乘子机制以及多包稀释。矩消去和正物质分离控制非局部 Biot--Savart 相互作用，从而允许在一个固定紧区域内进行嵌套构造。若不对极限数据施加紧支撑要求，一个互补的远场论证可得到稠密 $G_\delta$ 集的数据，其解在每一个正有理数时刻以及一个剩余时间集上离开 $W^{3,1}$，并且在每个非空开时间区间上具有无界本质 supremum。
英文摘要：
We prove strong ill-posedness for the two-dimensional incompressible Euler equation in the endpoint velocity space $W^{3,1}_σ(\mathbb R^2)$. Given any compactly supported smooth divergence-free background and any $δ>0$, we construct a single compactly supported divergence-free datum within $δ$ of the background in $W^{3,1}$. The perturbation can be localized inside any prescribed nonempty open neighborhood of the background support while remaining disjoint from that support. The associated unique global finite-energy solution $u$ satisfies ${\rm curl} u\in C([0,\infty);W^{2,1}(\mathbb R^2))$, but \[ \operatorname*{ess\,sup}_{0<t<T}\|u(t)\|_{W^{3,1}}=\infty \qquad\text{for every }T>0. \] The proof combines localized hyperbolic insertion, the Bonami--Poornima nonmultiplier mechanism, and many-packet dilution. Moment cancellation and positive material separation control the nonlocal Biot--Savart interactions, allowing a nested construction inside one fixed compact region. Without the compact-support requirement on the limiting datum, a complementary far-field argument yields a dense $G_δ$ set of data whose solutions leave $W^{3,1}$ at every positive rational time and on a residual set of times, with unbounded essential supremum on every nonempty open time interval.
