Compactly Supported Strong Ill-Posedness of the Two-Dimensional Euler Flow in $W^{3,1}({\mathbb{R}}^2)$

We prove strong ill-posedness for the two-dimensional incompressible Euler equation in the endpoint velocity space $W_{\sigma }^{3,1}({\mathbb{R}}^2)$. Given any compactly supported smooth divergence-free background and any $\delta >0$, we construct a single compactly supported divergence-free datum within $\delta$ 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 $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}u\in C([0,\mathrm{\infty });W^{2,1}({\mathbb{R}}^2))$, but $$\underset{0<t<T}{\mathrm{ess\hspace{0.17em}sup}}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{W^{3,1}}=\mathrm{\infty }\text{for every }T>0.$$ 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 $G_{\delta }$ 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.