Stability of the Inviscid Power-Law Vortex
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911418949828608 |
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| author | Binz, Tim Coiculescu, Matei P. |
| author_facet | Binz, Tim Coiculescu, Matei P. |
| contents | We prove that the power-law vortex $\overlineω(x) = β|x|^{-α}$, which explicitly solves the stationary unforced incompressible Euler equations in $\mathbb{R}^2$ in both physical and self-similar coordinates, is exponentially linearly stable in self-similar coordinates with the natural scaling. This result, which is valid for functions in a weighted $L^2$ space and in the un-weighted $L^2$ space with a mild symmetry condition, answers a question from the monograph by Albritton et al. Moreover, we prove that in physical coordinates the linearization around the power law vortex cannot generate an unstable $C_0$-semigroup. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_13397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of the Inviscid Power-Law Vortex Binz, Tim Coiculescu, Matei P. Analysis of PDEs Fluid Dynamics We prove that the power-law vortex $\overlineω(x) = β|x|^{-α}$, which explicitly solves the stationary unforced incompressible Euler equations in $\mathbb{R}^2$ in both physical and self-similar coordinates, is exponentially linearly stable in self-similar coordinates with the natural scaling. This result, which is valid for functions in a weighted $L^2$ space and in the un-weighted $L^2$ space with a mild symmetry condition, answers a question from the monograph by Albritton et al. Moreover, we prove that in physical coordinates the linearization around the power law vortex cannot generate an unstable $C_0$-semigroup. |
| title | Stability of the Inviscid Power-Law Vortex |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2411.13397 |