Global regularity for axisymmetric, swirl-free solutions of the Euler equation in four dimensions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911589056118784 |
|---|---|
| author | Miller, Evan |
| author_facet | Miller, Evan |
| contents | In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions. Previous works establishing global regularity for certain axisymmetric, swirl-free solutions of the Euler equation in four dimensions required the additional assumption that $\frac{ω^0}{r^2}\in L^\infty$, which can fail even for Schwartz class initial data. The key advance is a new bound on the vortex stretching term that only requires $\frac{ω^0}{r^2}\in L^{2,1}(\mathbb{R}^4)$, which is generically true for any axisymmetric, swirl-free initial data $u^0\in H^s\left(\mathbb{R}^4\right), s>4$, with reasonable decay at infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13963 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global regularity for axisymmetric, swirl-free solutions of the Euler equation in four dimensions Miller, Evan Analysis of PDEs 35Q31, 76B03 In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions. Previous works establishing global regularity for certain axisymmetric, swirl-free solutions of the Euler equation in four dimensions required the additional assumption that $\frac{ω^0}{r^2}\in L^\infty$, which can fail even for Schwartz class initial data. The key advance is a new bound on the vortex stretching term that only requires $\frac{ω^0}{r^2}\in L^{2,1}(\mathbb{R}^4)$, which is generically true for any axisymmetric, swirl-free initial data $u^0\in H^s\left(\mathbb{R}^4\right), s>4$, with reasonable decay at infinity. |
| title | Global regularity for axisymmetric, swirl-free solutions of the Euler equation in four dimensions |
| topic | Analysis of PDEs 35Q31, 76B03 |
| url | https://arxiv.org/abs/2602.13963 |