Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number
Several non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of these flows for plane curves with the rotation number one. In p...
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doaj-3b550bfe51474498a67ed59d43824cfb2021-03-24T01:32:01ZengAIMS PressMathematics in Engineering2640-35012021-03-013612610.3934/mine.2021047Asymptotic analysis for non-local curvature flows for plane curves with a general rotation numberTakeyuki Nagasawa0Kohei Nakamura1Graduate School of Science and Engineering, Saitama University, JapanGraduate School of Science and Engineering, Saitama University, JapanSeveral non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of these flows for plane curves with the rotation number one. In particular, when the initial curve is strictly convex, the flow exists globally in time, and converges to a circle as time tends to infinity. Even if the initial curve is not strictly convex, a global solution, if it exists, converges to a circle. Here, we deal with curves with a general rotation number, and show, not only a similar result for global solutions, but also a blow-up criterion, upper estimates of the blow-up time, and blow-up rate from below. For this purpose, we use a geometric quantity which has never been considered before.http://www.aimspress.com/article/doi/10.3934/mine.2021047?viewType=HTMLnon-local curvature flowrotation numberblow-upasymptotic behaviorthe isoperimetric inequalitythe isoperimetric deficit |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Takeyuki Nagasawa Kohei Nakamura |
spellingShingle |
Takeyuki Nagasawa Kohei Nakamura Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number Mathematics in Engineering non-local curvature flow rotation number blow-up asymptotic behavior the isoperimetric inequality the isoperimetric deficit |
author_facet |
Takeyuki Nagasawa Kohei Nakamura |
author_sort |
Takeyuki Nagasawa |
title |
Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number |
title_short |
Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number |
title_full |
Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number |
title_fullStr |
Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number |
title_full_unstemmed |
Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number |
title_sort |
asymptotic analysis for non-local curvature flows for plane curves with a general rotation number |
publisher |
AIMS Press |
series |
Mathematics in Engineering |
issn |
2640-3501 |
publishDate |
2021-03-01 |
description |
Several non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of these flows for plane curves with the rotation number one. In particular, when the initial curve is strictly convex, the flow exists globally in time, and converges to a circle as time tends to infinity. Even if the initial curve is not strictly convex, a global solution, if it exists, converges to a circle. Here, we deal with curves with a general rotation number, and show, not only a similar result for global solutions, but also a blow-up criterion, upper estimates of the blow-up time, and blow-up rate from below. For this purpose, we use a geometric quantity which has never been considered before. |
topic |
non-local curvature flow rotation number blow-up asymptotic behavior the isoperimetric inequality the isoperimetric deficit |
url |
http://www.aimspress.com/article/doi/10.3934/mine.2021047?viewType=HTML |
work_keys_str_mv |
AT takeyukinagasawa asymptoticanalysisfornonlocalcurvatureflowsforplanecurveswithageneralrotationnumber AT koheinakamura asymptoticanalysisfornonlocalcurvatureflowsforplanecurveswithageneralrotationnumber |
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1724205234505383936 |