Hölder regularity for collapses of point vortices
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arXiv
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| Format: | Preprint |
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2021
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| author | Donati, Martin Godard-Cadillac, Ludovic |
| author_facet | Donati, Martin Godard-Cadillac, Ludovic |
| contents | The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of $α$ models. In these models the kernel of the Biot-Savart law is a power function of exponent $-α$. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to, and including, the time of collapse. The Hölder exponent obtained is $1/(α+1)$ and this exponent is proved to be optimal for all $α$ by exhibiting an example of a $3$-vortex collapse.
The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as $t\to T$, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established : we prove that their distance to the boundary is Hölder regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_14230 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hölder regularity for collapses of point vortices Donati, Martin Godard-Cadillac, Ludovic Analysis of PDEs Dynamical Systems 76B47 The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of $α$ models. In these models the kernel of the Biot-Savart law is a power function of exponent $-α$. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to, and including, the time of collapse. The Hölder exponent obtained is $1/(α+1)$ and this exponent is proved to be optimal for all $α$ by exhibiting an example of a $3$-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as $t\to T$, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established : we prove that their distance to the boundary is Hölder regular. |
| title | Hölder regularity for collapses of point vortices |
| topic | Analysis of PDEs Dynamical Systems 76B47 |
| url | https://arxiv.org/abs/2111.14230 |