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A double-logarithmic lifespan bound for axisymmetric Euler equations with a small initial swirl
Abstract
We study the axisymmetric incompressible Euler equations in ℝ3 with a small initial swirl. Fix s>7/2, and let v0∈ Hs(ℝ3) be axisymmetric and divergence-free. If ε=|∇×(v0,θeθ)|L∞ is sufficiently small, then the lifespan T* of the corresponding Hs solution satisfies T*≥Cs/(1+|v0|Hs) [ loglog (ε−1) ]3/7 . This removes the assumption by Danchin (Proc. Amer. Math. Soc. 141:1979-1993, 2013) that the domain must be distant from the axis of symmetry. Moreover, even for v0∈ H3, our result improves the fourfold iterated-logarithmic lower bound for the lifespan obtained by Li and Zhou (Z. Angew. Math. Phys. 75:219, 2024) to a threefold iterated-logarithmic bound.
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BibTeX
@misc{li2026doublelogarithmic,
title = {{A double-logarithmic lifespan bound for axisymmetric Euler equations with a small initial swirl}},
author = {Zijin Li and Xu Xiao and Linbin Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25888},
eprint = {2610.25888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25888},
}
RIS
TY - UNPB
TI - A double-logarithmic lifespan bound for axisymmetric Euler equations with a small initial swirl
AU - Zijin Li
AU - Xu Xiao
AU - Linbin Yang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.25888
UR - https://arxiv.org/abs/2610.25888
AB - We study the axisymmetric incompressible Euler equations in ℝ^3 with a small initial swirl. Fix s>7/2, and let v_0∈ H^s(ℝ^3) be axisymmetric and divergence-free. If ε=|∇×(v_0,θe_θ)|_L^∞ is sufficiently small, then the lifespan T_* of the corresponding H^s solution satisfies T_*≥C_s/(1+|v_0|_H^s) [ loglog (ε^−1) ]^3/7 . This removes the assumption by Danchin (Proc. Amer. Math. Soc. 141:1979-1993, 2013) that the domain must be distant from the axis of symmetry. Moreover, even for v_0∈ H^3, our result improves the fourfold iterated-logarithmic lower bound for the lifespan obtained by Li and Zhou (Z. Angew. Math. Phys. 75:219, 2024) to a threefold iterated-logarithmic bound.
ER -