Nakamura, Kohei An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. (English) Zbl 1466.53005 Discrete Contin. Dyn. Syst., Ser. S 14, No. 3, 1093-1102 (2021). Summary: In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption. Cited in 2 Documents MSC: 53A04 Curves in Euclidean and related spaces 53E10 Flows related to mean curvature 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations Keywords:length-preserving flow; isoperimetric ratio; curvature; interpolation inequalities PDFBibTeX XMLCite \textit{K. Nakamura}, Discrete Contin. Dyn. Syst., Ser. S 14, No. 3, 1093--1102 (2021; Zbl 1466.53005) Full Text: DOI arXiv References: [1] G. Dziuk; E. Kuwert; R. Schätzle, Evolution of elastic curves in \(\Bbb R^n\): Existence and computation, SIAM J. Math. Anal., 33, 1228-1245 (2002) · Zbl 1031.53092 [2] J. Escher; G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126, 2789-2796 (1998) · Zbl 0909.53043 [3] M. Gage, On an area-preserving evolution equation for plane curves, Nonlinear Problems in Geometry, Contemp. Math., Amer. Math. Soc., Providence, RI, 51, (1986), 51-62. [4] L. S. Jiang; S. L. Pan, On a non-local curve evolution problem in the plane, Comm. Anal. Geom., 16, 1-26 (2008) · Zbl 1151.35392 [5] L. Ma; A. Q. Zhu, On a length preserving curve flow, Monatsh. Math., 165, 57-78 (2012) · Zbl 1235.35175 [6] U. F. Mayer, A singular example for the averaged mean curvature flow, Experiment. Math., 10, 103-107 (2001) · Zbl 0982.53061 [7] T. Nagasawa; K. Nakamura, Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows, Adv. Differential Equations, 24, 581-608 (2019) · Zbl 1442.53004 [8] D. Ševčovič; S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35, 1784-1798 (2012) · Zbl 1255.35148 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.