Turbulent Dynamos on Bounded Domains and Their Generalization to the Geometric Transport Equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916030424547328 |
|---|---|
| author | Del Nin, Giacomo Faraco, Daniel Lindberg, Sauli Mengual, Francisco |
| author_facet | Del Nin, Giacomo Faraco, Daniel Lindberg, Sauli Mengual, Francisco |
| contents | For any smooth bounded domain $Ω\subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(Ω)$ for all $p < \infty$, and magnetic fields $B^ε\in L_t^p C^{m}(Ω)$ for all $p < \infty$ and $m\in \mathbb{N}$, that solve the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, uniformly in the vanishing-diffusivity limit $ε\to 0$. The construction is based on the convex integration scheme of Modena-Székelyhidi and Cheskidov-Luo. The main novelty lies in the introduction of explicit potentials, which allow the solutions to be localized and avoid the need to work with the anti-curl operator. In addition, we present a unified scheme for the geometric transport equation (GTE), which encompasses both the transport and Maxwell equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20451 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Turbulent Dynamos on Bounded Domains and Their Generalization to the Geometric Transport Equation Del Nin, Giacomo Faraco, Daniel Lindberg, Sauli Mengual, Francisco Analysis of PDEs For any smooth bounded domain $Ω\subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(Ω)$ for all $p < \infty$, and magnetic fields $B^ε\in L_t^p C^{m}(Ω)$ for all $p < \infty$ and $m\in \mathbb{N}$, that solve the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, uniformly in the vanishing-diffusivity limit $ε\to 0$. The construction is based on the convex integration scheme of Modena-Székelyhidi and Cheskidov-Luo. The main novelty lies in the introduction of explicit potentials, which allow the solutions to be localized and avoid the need to work with the anti-curl operator. In addition, we present a unified scheme for the geometric transport equation (GTE), which encompasses both the transport and Maxwell equations. |
| title | Turbulent Dynamos on Bounded Domains and Their Generalization to the Geometric Transport Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.20451 |