Comparing motion of curves and hypersurfaces in $ \mathbb{R}^m $

Jiří Minarčík; Masato Kimura; Michal Beneš

doi:10.3934/dcdsb.2019032

Article Contents

2019, Volume 24, Issue 9: 4815-4826. Doi: 10.3934/dcdsb.2019032

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Comparing motion of curves and hypersurfaces in $ \mathbb{R}^m $

1.
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic
2.
Faculty of Mathematics and Physics, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan

^* Corresponding author: Jiří Minarčík
^* Corresponding author: Jiří Minarčík

Received: December 31, 2017

Revised: May 31, 2018

Early access: February 2019

Published: July 2019

The first and third author were partly supported by the project No. 15-27178A of Ministry of Health of the Czech Republic, the first author was partly supported by the Foundation Nadání Josefa, Marie a Zdeňky Hlávkových, by the Study at KU Scholarship from Kanazawa University and by the project of the Student Grant Agency of the Czech Technical University in Prague No. SGS17/194/OHK4/3T/14, and the second author was partly supported by JSPS KAKENHI Grant Numbers 15H03632 and 16H03953

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Abstract

This article aims to contribute to the understanding of the curvature flow of curves in a higher-dimensional space. Evolution of curves in $ \mathbb{R}^m $ by their curvature is compared to the motion of hypersurfaces with constrained normal velocity. The special case of shrinking hyperspheres is further analyzed both theoretically and numerically by means of a semi-discrete scheme with discretization based on osculating circles. Computational examples of evolving spherical curves are provided along with the measurement of the experimental order of convergence.

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Mathematics Subject Classification: Primary: 53C44, 14Q05, 14J70; Secondary: 34K12, 35B51.

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Figure 1. Visualization for the definition of $ \mathbf{y}(u,t) $ and $ \bar{u}(t) $

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Figure 2. Visualization of the geometric quantities defined in the osculating plane

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Figure 3. Results of the numerical simulation from Example 1. The discretized curve is visualized at four different time levels along with the corresponding sphere

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Figure 4. Results of the numerical simulation from Example 2. The discretized curve is visualized at four different time levels along with the corresponding sphere

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Table 1. Results of the numerical computation from Example 1. The error measurements were taken during time interval $ [0,0.45] $

$N$	$\mathcal{E}_{\infty}(N)$	EOC$_{\infty}$	$\mathcal{E}_{1}(N)$	EOC₁	$\mathcal{E}_{2}(N)$	EOC₂
100	$1.0205 \cdot 10^{-2}$	2.0283	$9.3770 \cdot 10^{-3}$	2.0540	$6.4629 \cdot 10^{-3}$	2.0794
200	$2.5018 \cdot 10^{-3}$	1.9372	$2.2581 \cdot 10^{-3}$	1.9449	$1.5292 \cdot 10^{-3}$	1.9758
400	$6.5329 \cdot 10^{-4}$	1.9746	$5.8649 \cdot 10^{-4}$	2.0234	$3.8876 \cdot 10^{-4}$	1.9924
800	$1.6623 \cdot 10^{-4}$	1.7062	$1.4426 \cdot 10^{-4}$	2.0071	$9.7702 \cdot 10^{-5}$	2.0183
1600	$5.0943 \cdot 10^{-5}$		$3.5888 \cdot 10^{-5}$		$2.4118 \cdot 10^{-5}$

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Table 2. Results of the numerical computation from Example 2. The error measurements were taken during time interval $ [0,0.45] $

$N$	$\mathcal{E}_{\infty}(N)$	EOC$_{\infty}$	$\mathcal{E}_{1}(N)$	EOC$_1$	$\mathcal{E}_{2}(N)$	EOC$_2$
400	$1.1538 \cdot 10^{-1}$	1.9639	$1.1538 \cdot 10^{-1}$	1.9640	$1.0340 \cdot 10^{-1}$	2.2232
800	$2.9575 \cdot 10^{-2}$	2.0276	$2.9574 \cdot 10^{-2}$	2.0280	$2.2146 \cdot 10^{-2}$	2.0975
1200	$1.2998 \cdot 10^{-2}$	2.0218	$1.2996 \cdot 10^{-2}$	2.0222	$9.4608 \cdot 10^{-3}$	2.0547
1600	$7.2659 \cdot 10^{-3}$	2.0281	$7.2636 \cdot 10^{-3}$	2.0269	$5.2386 \cdot 10^{-3}$	2.0459
2000	$4.6211 \cdot 10^{-3}$		$4.6209 \cdot 10^{-3}$		$3.3186 \cdot 10^{-3}$

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References

[1]	S. J. Altschuler, Singularities for the curve shortening flow for space curves, Journal of Differential Geometry, 34 (1991), 491-514. doi: 10.4310/jdg/1214447218.
[2]	S. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, Journal of Differential Geometry, 35 (1992), 283-298. doi: 10.4310/jdg/1214448076.
[3]	L. Ambrosio and H. M. Soner, A level set approach to the evolution of surfaces of any codimension, Journal of Differential Geometry, 43 (1996), 693-737. doi: 10.4310/jdg/1214458529.
[4]	M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 3 (2008), 509-523.
[5]	P. Burchard, L. T. Cheng, B. Merriman and S. Osher, Motion of curves in three spatial dimensions using a level set approach, Journal of Computational Physics, 170 (2001), 720-741. doi: 10.1006/jcph.2001.6758.
[6]	J. Christiansen, Numerical solution of ordinary simultaneous differential equations of the 1st order using a method for automatic step change, Numerische Mathematik, 14 (1970), 317-324. doi: 10.1007/BF02165587.
[7]	K. Corrales, Non existence of type Ⅱ singularities for embedded and unknotted space curves, preprint, arXiv: 1605.03100v1, 2016.
[8]	G. Dziuk, Convergence of a semi-discrete scheme for the curve shortening flow, Mathematical Models and Methods in Applied Sciences, 4 (1994), 589-606. doi: 10.1142/S0218202594000339.
[9]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
[10]	M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, Journal of Differential Geometry, 23 (1986), 69-96. doi: 10.4310/jdg/1214439902.
[11]	M. Grayson, The heat equation shrinks embedded plane curves to round points, Journal of Differential Geometry, 26 (1987), 285-314. doi: 10.4310/jdg/1214441371.
[12]	S. He, Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow, preprint, arXiv: 1209.5146v1, 2012.
[13]	M. Holodniok, A. Klíč, M. Kubíček and M. Marek, Methods of Analysis of Nonlinear Dynamical Models, Academia Praha, 1986.
[14]	G. Huisken, Flow by mean curvature of convex surfaces into spheres, Journal of Differential Geometry, 20 (1984), 237-266. doi: 10.4310/jdg/1214438998.
[15]	G. Khan, A condition ensuring spatial curves develop type-Ⅱ singularities under curve shortening flow, preprint, arXiv: 1209.4072v3, 2015.
[16]	M. Kimura, Geometry of hypersurfaces and moving hypersurfaces in $\mathbb{R}^m$ for the study of moving boundary problems, Topics on Partial Differential Equations, Jindřich Nečas Center for Mathematical Modeling, Lecture notes, 4 (2008), 39–93.
[17]	M. Kolář, M. Beneš and D. Ševčovič, Area preserving geodesic curvature driven flow of closed curves on a surface, Discrete and Continuous Dynamical Systems - Series B, 22 (2017), 3671-3689. doi: 10.3934/dcdsb.2017148.
[18]	K. Mikula, Image processing with partial differential equations, Nato Science Series, 75 (2002), 283-322.
[19]	T. Mura, Micromechanics of Defects in Solids, Springer Netherlands, 1987.
[20]	S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.
[21]	J. R. Rice and M. Mu, An experimental performance analysis for the rate of convergence of collocation on general domains, Numerical Methods for Partial Differential Equations, 5 (1989), 45-52. doi: 10.1002/num.1690050105.
[22]	J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1996.
[23]	A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.
[24]	Y. Y. Yang and X. X. Jiao, Curve shortening flow in arbitrary dimensional Euclidian space, Acta Mathematica Sinica, 21 (2005), 715-722. doi: 10.1007/s10114-004-0426-z.