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A Sharp Uniqueness-Nonuniqueness Dichotomy for Neutral Vortex Arrays in $2$D Euler Equations
Abstract
We establish a sharp uniqueness--nonuniqueness criterion for the two-dimensional incompressible Euler equations within a class of stationary arrays of disjoint neutral vortices. Let Θ be any finite nondecreasing envelope for the vorticity Lp norms, and set IΘ=∫p0∞ Θ(p)−1 dp. If IΘ=∞, every array in the corresponding class admits a unique Euler evolution, even among bounded energy-admissible weak velocity solutions. If IΘ<∞, there exists a compactly supported array in the same class admitting two distinct unforced Lagrangian Euler evolutions from the same initial datum. The criterion holds for arbitrary nondecreasing envelopes Θ.
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BibTeX
@misc{hou2026sharp,
title = {{A Sharp Uniqueness-Nonuniqueness Dichotomy for Neutral Vortex Arrays in $2$D Euler Equations}},
author = {Thomas Y. Hou and Xiang Qin and Peicong Song and Wenhan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40552},
eprint = {2610.40552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40552},
}
RIS
TY - UNPB
TI - A Sharp Uniqueness-Nonuniqueness Dichotomy for Neutral Vortex Arrays in $2$D Euler Equations
AU - Thomas Y. Hou
AU - Xiang Qin
AU - Peicong Song
AU - Wenhan Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40552
UR - https://arxiv.org/abs/2610.40552
AB - We establish a sharp uniqueness--nonuniqueness criterion for the two-dimensional incompressible Euler equations within a class of stationary arrays of disjoint neutral vortices. Let Θ be any finite nondecreasing envelope for the vorticity L^p norms, and set I_Θ=∫_p_0^∞ Θ(p)^−1 dp. If I_Θ=∞, every array in the corresponding class admits a unique Euler evolution, even among bounded energy-admissible weak velocity solutions. If I_Θ<∞, there exists a compactly supported array in the same class admitting two distinct unforced Lagrangian Euler evolutions from the same initial datum. The criterion holds for arbitrary nondecreasing envelopes Θ.
ER -