Long-time behavior of an Arc-shaped Vortex Filament and its Application to the Stability of a Circular Vortex Filament
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914721529069568 |
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| author | Aiki, Masashi |
| author_facet | Aiki, Masashi |
| contents | We consider a nonlinear model equation, known as the Localized Induction Equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. We show stability estimates for an arc-shaped vortex filament, which is an exact solution to an initial-boundary value problem for the Localized Induction Equation. An arc-shaped filament travels along an axis at a constant speed without changing its shape, and is oriented in such a way that the arc stays in a plane that is perpendicular to the axis. We prove that an arc-shaped filament is stable in the Lyapunov sense for general perturbations except in the axis-direction, for which the perturbation can grow linearly in time. We also show that this estimate is optimal. We then apply the obtained stability estimates to study the stability of a circular vortex filament under some symmetry assumptions on the initial perturbation. We do this by dividing the circular filament into arcs, apply the stability estimate to each arc-shaped filament, and combine the estimates to obtain estimates for the whole circle. The optimality of the stability estimates for an arc-shaped filament also shows that a circular filament is not stable in the Lyapunov sense, namely, certain perturbations can grow linearly in time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time behavior of an Arc-shaped Vortex Filament and its Application to the Stability of a Circular Vortex Filament Aiki, Masashi Analysis of PDEs 35Q35 (Primary) 76B47, 35Q31 (Secondary) We consider a nonlinear model equation, known as the Localized Induction Equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. We show stability estimates for an arc-shaped vortex filament, which is an exact solution to an initial-boundary value problem for the Localized Induction Equation. An arc-shaped filament travels along an axis at a constant speed without changing its shape, and is oriented in such a way that the arc stays in a plane that is perpendicular to the axis. We prove that an arc-shaped filament is stable in the Lyapunov sense for general perturbations except in the axis-direction, for which the perturbation can grow linearly in time. We also show that this estimate is optimal. We then apply the obtained stability estimates to study the stability of a circular vortex filament under some symmetry assumptions on the initial perturbation. We do this by dividing the circular filament into arcs, apply the stability estimate to each arc-shaped filament, and combine the estimates to obtain estimates for the whole circle. The optimality of the stability estimates for an arc-shaped filament also shows that a circular filament is not stable in the Lyapunov sense, namely, certain perturbations can grow linearly in time. |
| title | Long-time behavior of an Arc-shaped Vortex Filament and its Application to the Stability of a Circular Vortex Filament |
| topic | Analysis of PDEs 35Q35 (Primary) 76B47, 35Q31 (Secondary) |
| url | https://arxiv.org/abs/2403.13546 |