Axisymmetric Navier--Stokes with Swirl: Regularity and Blow-Up Constraints
Abstract
For smooth finite-energy axisymmetric Navier--Stokes solutions with swirl, we give a continuation criterion in terms of the signed cumulative work of the actual post-cut source-generated meridional current, with quartic viscous dissipation counted once. In the stated strong Sobolev class the criterion is necessary and sufficient and requires neither a global circulation bound nor spatial moments. We also prove cumulative inward-compression criteria, collect quantitative necessary conditions at a first singular time, and give a complete-history mixed-residual test for noncontinuation. Auxiliary results quantify quartic dissipation, energy-funded source bounds, and causal source--receiver identities. An explicit smooth whole-space datum has positive newborn source production.
Related
- Critical Structure of Axisymmetric Navier--Stokes with Swirl
- Regularity of asymptotically axisymmetric Navier-Stokes flows with rotationally invariant curl of the force
- A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations
- Regularity for axisymmetric Navier-Stokes with an Euler length
- Regularity of asymptotically axisymmetric solutions to the 3D Navier-Stokes equations with analytic forcing
BibTeX
@misc{shahmurov2026axisymmetric,
title = {{Axisymmetric Navier--Stokes with Swirl: Regularity and Blow-Up Constraints}},
author = {Rishad Shahmurov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24142},
eprint = {2610.24142},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24142},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Axisymmetric Navier--Stokes with Swirl: Regularity and Blow-Up Constraints
AU - Rishad Shahmurov
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24142
UR - https://arxiv.org/abs/2610.24142
AB - For smooth finite-energy axisymmetric Navier--Stokes solutions with swirl, we give a continuation criterion in terms of the signed cumulative work of the actual post-cut source-generated meridional current, with quartic viscous dissipation counted once. In the stated strong Sobolev class the criterion is necessary and sufficient and requires neither a global circulation bound nor spatial moments. We also prove cumulative inward-compression criteria, collect quantitative necessary conditions at a first singular time, and give a complete-history mixed-residual test for noncontinuation. Auxiliary results quantify quartic dissipation, energy-funded source bounds, and causal source--receiver identities. An explicit smooth whole-space datum has positive newborn source production.
ER -