On partial type I solutions to the Axially symmetric Navier-Stokes equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915926563094528 |
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| author | Zhang, Qi S |
| author_facet | Zhang, Qi S |
| contents | Let $v= v_{r}e_{r} + v_þe_þ + v_{3}e_{3}$ be a Leray-Hopf solution to the axially symmetric Navier-Stokes equations (ASNS). We call it a partial type I solution if $v_r(x, t) \ge -C/\sqrt{T-t}$ for some constant $C>0$ and $(x, t) \in \mathbf{R}^3 \times [0, T)$. In this paper, it is proven that such solution does not blow up at time $T$ under the extra mild assumption that $|v_θ(x, 0)| |x'|$ is bounded. This extends a well known result by two groups of people who proved the no blowup conclusion under the full type I condition: $|v(x, t)| \le C/\sqrt{T-t}$. The result also confirms the physical intuition that potential blow ups for ASNS are caused by super-critical inward radial velocity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07785 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On partial type I solutions to the Axially symmetric Navier-Stokes equations Zhang, Qi S Analysis of PDEs 35Q30, 76N10 Let $v= v_{r}e_{r} + v_þe_þ + v_{3}e_{3}$ be a Leray-Hopf solution to the axially symmetric Navier-Stokes equations (ASNS). We call it a partial type I solution if $v_r(x, t) \ge -C/\sqrt{T-t}$ for some constant $C>0$ and $(x, t) \in \mathbf{R}^3 \times [0, T)$. In this paper, it is proven that such solution does not blow up at time $T$ under the extra mild assumption that $|v_θ(x, 0)| |x'|$ is bounded. This extends a well known result by two groups of people who proved the no blowup conclusion under the full type I condition: $|v(x, t)| \le C/\sqrt{T-t}$. The result also confirms the physical intuition that potential blow ups for ASNS are caused by super-critical inward radial velocity. |
| title | On partial type I solutions to the Axially symmetric Navier-Stokes equations |
| topic | Analysis of PDEs 35Q30, 76N10 |
| url | https://arxiv.org/abs/2604.07785 |