Vorticity growth beyond bounded relative vorticity for axisymmetric Euler flows without swirl

Abstract

We establish upper bounds for vorticity and related quantities in axisymmetric Euler flows without swirl in ℝ³, assuming that the initial relative vorticity ω₀/r belongs to L¹∩ L^p with 3<p≤ ∞. For each fixed p ∈ (3,∞], we obtain polynomial-in-time upper bounds for the radial moments, the radial velocity, and the L^p norm of vorticity. We then consider initial data with unbounded relative vorticity whose growth near the axis is controlled by Φ(log 1/r) for a subexponential profile Φ, so that ω₀/r belongs to L^p for every p<∞ but not to L^∞. For such data, we derive profile dependent upper bounds for the vorticity maximum. In particular, for any polynomial function Φ(s)=s^k with k≥ 0, we show that |ω(·,t)|_L^∞ is bounded by (1+t)^4/3+k (log (2+t) )^2k/3. Moreover, for initial data satisfying ω₀/r ∈ L¹∩ L^p with a fixed p∈ (3,∞), under additional symmetry and vertical confinement assumptions, we obtain a subexponential-in-time upper bound for the vorticity maximum, thereby improving the classical exponential bound for such data.

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BibTeX RIS

BibTeX

@misc{egamberganov2026vorticity,
  title = {{Vorticity growth beyond bounded relative vorticity for axisymmetric Euler flows without swirl}},
  author = {Khakim Egamberganov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26559},
  eprint = {2610.26559},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26559},
}

RIS

TY  - UNPB
TI  - Vorticity growth beyond bounded relative vorticity for axisymmetric Euler flows without swirl
AU  - Khakim Egamberganov
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26559
UR  - https://arxiv.org/abs/2610.26559
AB  - We establish upper bounds for vorticity and related quantities in axisymmetric Euler flows without swirl in ℝ^3, assuming that the initial relative vorticity ω_0/r belongs to L^1∩ L^p with 3<p≤ ∞. For each fixed p ∈ (3,∞], we obtain polynomial-in-time upper bounds for the radial moments, the radial velocity, and the L^p norm of vorticity. We then consider initial data with unbounded relative vorticity whose growth near the axis is controlled by Φ(log 1/r) for a subexponential profile Φ, so that ω_0/r belongs to L^p for every p<∞ but not to L^∞. For such data, we derive profile dependent upper bounds for the vorticity maximum. In particular, for any polynomial function Φ(s)=s^k with k≥ 0, we show that |ω(·,t)|_L^∞ is bounded by (1+t)^4/3+k (log (2+t) )^2k/3. Moreover, for initial data satisfying ω_0/r ∈ L^1∩ L^p with a fixed p∈ (3,∞), under additional symmetry and vertical confinement assumptions, we obtain a subexponential-in-time upper bound for the vorticity maximum, thereby improving the classical exponential bound for such data.
ER  -

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