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

Abstract

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 $\mathrm{curl}u\in C([0,\infty );W^{2,1}({\mathbb{R}}^2))$, 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 $G_{\delta }$ 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.