# American Institute of Mathematical Sciences

## An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow

 Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama 338-8570, Japan

Received  January 2019 Revised  February 2020 Published  June 2020

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.

Citation: Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020385
##### References:
 [1] G. Dziuk, E. Kuwert and R. Schätzle, Evolution of elastic curves in $\Bbb R^n$: Existence and computation, SIAM J. Math. Anal., 33 (2002), 1228-1245.  doi: 10.1137/S0036141001383709.  Google Scholar [2] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3.  Google Scholar [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. doi: 10.1090/conm/051/848933.  Google Scholar [4] L. S. Jiang and S. L. Pan, On a non-local curve evolution problem in the plane, Comm. Anal. Geom., 16 (2008), 1-26.  doi: 10.4310/CAG.2008.v16.n1.a1.  Google Scholar [5] L. Ma and A. Q. Zhu, On a length preserving curve flow, Monatsh. Math., 165 (2012), 57-78.  doi: 10.1007/s00605-011-0302-8.  Google Scholar [6] U. F. Mayer, A singular example for the averaged mean curvature flow, Experiment. Math., 10 (2001), 103-107.  doi: 10.1080/10586458.2001.10504432.  Google Scholar [7] T. Nagasawa and K. Nakamura, Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows, Adv. Differential Equations, 24 (2019), 581-608.   Google Scholar [8] D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35 (2012), 1784-1798.  doi: 10.1002/mma.2554.  Google Scholar

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##### References:
 [1] G. Dziuk, E. Kuwert and R. Schätzle, Evolution of elastic curves in $\Bbb R^n$: Existence and computation, SIAM J. Math. Anal., 33 (2002), 1228-1245.  doi: 10.1137/S0036141001383709.  Google Scholar [2] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3.  Google Scholar [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. doi: 10.1090/conm/051/848933.  Google Scholar [4] L. S. Jiang and S. L. Pan, On a non-local curve evolution problem in the plane, Comm. Anal. Geom., 16 (2008), 1-26.  doi: 10.4310/CAG.2008.v16.n1.a1.  Google Scholar [5] L. Ma and A. Q. Zhu, On a length preserving curve flow, Monatsh. Math., 165 (2012), 57-78.  doi: 10.1007/s00605-011-0302-8.  Google Scholar [6] U. F. Mayer, A singular example for the averaged mean curvature flow, Experiment. Math., 10 (2001), 103-107.  doi: 10.1080/10586458.2001.10504432.  Google Scholar [7] T. Nagasawa and K. Nakamura, Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows, Adv. Differential Equations, 24 (2019), 581-608.   Google Scholar [8] D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35 (2012), 1784-1798.  doi: 10.1002/mma.2554.  Google Scholar
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