Existence of Vortex Patch Equilibria for Active Scalars Equations
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913591974690816 |
|---|---|
| author | Cuba, Edison |
| author_facet | Cuba, Edison |
| contents | In this paper, we investigate the existence of a finite number of vortex patches for the generalized surface quasi-geostrophic (gSQG) equations with $α\in [1,2)$, focusing on configurations that may rotate uniformly, translate, or remain stationary. Using a desingularization technique, we reformulate the problem to resolve singularities arising in the point vortex limit. Assuming a nondegenerate equilibrium of the point vortices, we apply the implicit function theorem to construct time-periodic solutions to the gSQG equations, offering asymptotic descriptions of the vortex patch boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00236 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of Vortex Patch Equilibria for Active Scalars Equations Cuba, Edison Analysis of PDEs 35Q35, 76B03, 76B47 In this paper, we investigate the existence of a finite number of vortex patches for the generalized surface quasi-geostrophic (gSQG) equations with $α\in [1,2)$, focusing on configurations that may rotate uniformly, translate, or remain stationary. Using a desingularization technique, we reformulate the problem to resolve singularities arising in the point vortex limit. Assuming a nondegenerate equilibrium of the point vortices, we apply the implicit function theorem to construct time-periodic solutions to the gSQG equations, offering asymptotic descriptions of the vortex patch boundaries. |
| title | Existence of Vortex Patch Equilibria for Active Scalars Equations |
| topic | Analysis of PDEs 35Q35, 76B03, 76B47 |
| url | https://arxiv.org/abs/2412.00236 |