Construction of unstable concentrated solutions of the Euler and gSQG equations
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929317813944320 |
|---|---|
| author | Donati, Martin |
| author_facet | Donati, Martin |
| contents | In this paper we construct solutions to the Euler and gSQG equations that are concentrated near unstable stationary configurations of point-vortices. Those solutions are themselves unstable, in the sense that their localization radius grows from order $\varepsilon$ to order $\varepsilon^β$ (with $β< 1$) in a time of order $|\ln\varepsilon|$. This proves in particular that the logarithmic lower-bound obtained in previous papers (in particular [P. Buttà and C. Marchioro, Long time evolution of concentrated Euler flows with planar symmetry, SIAM J. Math. Anal., 50(1):735-760, 2018]) about vorticity localization in Euler and gSQG equations is optimal. In addition we construct unstable solutions of the Euler equations in bounded domains concentrated around a single unstable stationary point. To achieve this we construct a domain whose Robin's function has a saddle point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Construction of unstable concentrated solutions of the Euler and gSQG equations Donati, Martin Analysis of PDEs Dynamical Systems 76B47, 34A34 In this paper we construct solutions to the Euler and gSQG equations that are concentrated near unstable stationary configurations of point-vortices. Those solutions are themselves unstable, in the sense that their localization radius grows from order $\varepsilon$ to order $\varepsilon^β$ (with $β< 1$) in a time of order $|\ln\varepsilon|$. This proves in particular that the logarithmic lower-bound obtained in previous papers (in particular [P. Buttà and C. Marchioro, Long time evolution of concentrated Euler flows with planar symmetry, SIAM J. Math. Anal., 50(1):735-760, 2018]) about vorticity localization in Euler and gSQG equations is optimal. In addition we construct unstable solutions of the Euler equations in bounded domains concentrated around a single unstable stationary point. To achieve this we construct a domain whose Robin's function has a saddle point. |
| title | Construction of unstable concentrated solutions of the Euler and gSQG equations |
| topic | Analysis of PDEs Dynamical Systems 76B47, 34A34 |
| url | https://arxiv.org/abs/2303.14657 |