The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V=1αψ(x)καV=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α<0\alpha \lt 0 or α>1\alpha \gt 1, where x∈[0,2mπ]x\in \left[0,2m\pi ] is the tangential angl...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2022-08-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2022-0245 |
| _version_ | 1828108156843065344 |
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| author | Wang Yaping Wang Xiaoliu |
| author_facet | Wang Yaping Wang Xiaoliu |
| author_sort | Wang Yaping |
| collection | DOAJ |
| description | In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V=1αψ(x)καV=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α<0\alpha \lt 0 or α>1\alpha \gt 1, where x∈[0,2mπ]x\in \left[0,2m\pi ] is the tangential angle at the point on evolving curves. For −1≤α<0-1\le \alpha \lt 0, we show the flow exists globally and the rescaled flow has a full-time convergence. For α<−1\alpha \lt -1 or α>1\alpha \gt 1, we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ\psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence. |
| first_indexed | 2024-04-11T10:43:22Z |
| format | Article |
| id | doaj.art-77eddf36c4274255bd6a583713d9b381 |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-04-11T10:43:22Z |
| publishDate | 2022-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-77eddf36c4274255bd6a583713d9b3812022-12-22T04:29:08ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2022-08-0112111713110.1515/anona-2022-0245The evolution of immersed locally convex plane curves driven by anisotropic curvature flowWang Yaping0Wang Xiaoliu1School of Mathematics, Southeast University, Nanjing 210096, PR ChinaSchool of Mathematics, Southeast University, Nanjing 210096, PR ChinaIn this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V=1αψ(x)καV=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α<0\alpha \lt 0 or α>1\alpha \gt 1, where x∈[0,2mπ]x\in \left[0,2m\pi ] is the tangential angle at the point on evolving curves. For −1≤α<0-1\le \alpha \lt 0, we show the flow exists globally and the rescaled flow has a full-time convergence. For α<−1\alpha \lt -1 or α>1\alpha \gt 1, we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ\psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.https://doi.org/10.1515/anona-2022-0245curvature flowanisotropylong-time behavioursingularity35b4035k1535k5553e10– nonlinear analysis: perspectives and synergies |
| spellingShingle | Wang Yaping Wang Xiaoliu The evolution of immersed locally convex plane curves driven by anisotropic curvature flow Advances in Nonlinear Analysis curvature flow anisotropy long-time behaviour singularity 35b40 35k15 35k55 53e10 – nonlinear analysis: perspectives and synergies |
| title | The evolution of immersed locally convex plane curves driven by anisotropic curvature flow |
| title_full | The evolution of immersed locally convex plane curves driven by anisotropic curvature flow |
| title_fullStr | The evolution of immersed locally convex plane curves driven by anisotropic curvature flow |
| title_full_unstemmed | The evolution of immersed locally convex plane curves driven by anisotropic curvature flow |
| title_short | The evolution of immersed locally convex plane curves driven by anisotropic curvature flow |
| title_sort | evolution of immersed locally convex plane curves driven by anisotropic curvature flow |
| topic | curvature flow anisotropy long-time behaviour singularity 35b40 35k15 35k55 53e10 – nonlinear analysis: perspectives and synergies |
| url | https://doi.org/10.1515/anona-2022-0245 |
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