A limsup fast dynamo on $\mathbb{T}^3$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914175751553024 |
|---|---|
| author | Sorella, Massimo Villringer, David |
| author_facet | Sorella, Massimo Villringer, David |
| contents | We construct a time-dependent, incompressible, and uniformly-in-time Lipschitz continuous velocity field on $\mathbb{T}^3$ that produces exponential growth of the magnetic energy along a subsequence of times, for every positive value of the magnetic diffusivity. Because this growth is not uniform in time but occurs only along a diverging sequence of times, we refer to the resulting mechanism as a limsup fast dynamo. Our construction is based on suitably rescaled Arnold-Beltrami-Childress (ABC) flows, each supported on long time intervals. The analysis employs perturbation theory to establish continuity of the exponential growth rate with respect to both the initial data and the diffusivity parameter. This proves the weak form of the fast dynamo conjecture formulated by Childress and Gilbert on $\mathbb{T}^3$, but the considerably more challenging version proposed by Arnold on $\mathbb{T}^3$ remains an open problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23024 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A limsup fast dynamo on $\mathbb{T}^3$ Sorella, Massimo Villringer, David Analysis of PDEs 35Q35, 34L05, 47A55, 76E25, 47B12 We construct a time-dependent, incompressible, and uniformly-in-time Lipschitz continuous velocity field on $\mathbb{T}^3$ that produces exponential growth of the magnetic energy along a subsequence of times, for every positive value of the magnetic diffusivity. Because this growth is not uniform in time but occurs only along a diverging sequence of times, we refer to the resulting mechanism as a limsup fast dynamo. Our construction is based on suitably rescaled Arnold-Beltrami-Childress (ABC) flows, each supported on long time intervals. The analysis employs perturbation theory to establish continuity of the exponential growth rate with respect to both the initial data and the diffusivity parameter. This proves the weak form of the fast dynamo conjecture formulated by Childress and Gilbert on $\mathbb{T}^3$, but the considerably more challenging version proposed by Arnold on $\mathbb{T}^3$ remains an open problem. |
| title | A limsup fast dynamo on $\mathbb{T}^3$ |
| topic | Analysis of PDEs 35Q35, 34L05, 47A55, 76E25, 47B12 |
| url | https://arxiv.org/abs/2511.23024 |