Compactly Supported Strong Ill-Posedness of the Two-Dimensional Euler Flow in W3,1(ℝ2)
Abstract
We prove strong ill-posedness for the two-dimensional incompressible Euler equation in the endpoint velocity space W3,1σ(ℝ2). 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 W3,1. 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 curl u∈ C([0,∞);W2,1(ℝ2)), but *ess sup0<t<T|u(t)|W3,1=∞ 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 Gδ set of data whose solutions leave W3,1 at every positive rational time and on a residual set of times, with unbounded essential supremum on every nonempty open time interval.
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BibTeX
@misc{ban2026compactly,
title = {{Compactly Supported Strong Ill-Posedness of the Two-Dimensional Euler Flow in $W^3,1(\mathbb R^2)$}},
author = {Yingzhe Ban and Maotuo Guo and Qi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04872},
eprint = {2610.04872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04872},
}
RIS
TY - UNPB
TI - Compactly Supported Strong Ill-Posedness of the Two-Dimensional Euler Flow in W^3,1(ℝ^2)
AU - Yingzhe Ban
AU - Maotuo Guo
AU - Qi Zhang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04872
UR - https://arxiv.org/abs/2610.04872
AB - We prove strong ill-posedness for the two-dimensional incompressible Euler equation in the endpoint velocity space W^3,1_σ(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 curl u∈ C([0,∞);W^2,1(ℝ^2)), but *ess sup_0<t<T|u(t)|_W^3,1=∞ 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.
ER -