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 $W^{3,1}_σ(\mathbb R^2)$. 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 \[ \operatorname*{ess\,sup}_{0 0. \] 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.