# American Institute of Mathematical Sciences

December  2017, 22(10): 3671-3689. doi: 10.3934/dcdsb.2017148

## Area preserving geodesic curvature driven flow of closed curves on a surface

 1 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic 2 Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská Dolina, 842 48, Bratislava, Slovakia

* Corresponding author: Miroslav Kolář

Received  December 2016 Revised  March 2017 Published  April 2017

Fund Project: The first author is supported by the grant No. 14-36566G of the Czech Science Foundation and by the grant No. 15-27178A of Ministry of Health of the Czech Republic.

We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.

Citation: Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
##### References:
 [1] S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2. [2] S. Angenent, Nonlinear analytic semiflows, Proc. R. Soc. Edinb., Sect. A, 115 (1990), 91-107.  doi: 10.1017/S0308210500024598. [3] M. Beneš, Diffuse-interface treatment of the anisotropic mean-curvature flow, Appl. Math., 48 (2003), 437-453.  doi: 10.1023/B:APOM.0000024485.24886.b9. [4] M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bull. Inst. Math. Acad. Sinica (N. S.), 3 (2008), 509-523. [5] M. Beneš, J. Kratochvíl, J. Křišt'an, V. Minárik and P. Pauš, A parametric simulation method for discrete dislocation dynamics, Eur. Phys. J. ST, 177 (2009), 177-192. [6] M. Beneš, S. Yazaki and M. Kimura, Computational studies of non-local anisotropic Allen-Cahn equation, Math. Bohemica, 136 (2011), 429-437. [7] L. Bronsard and B. Stoth, Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807.  doi: 10.1137/S0036141094279279. [8] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅲ. Nucleation of a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699.  doi: 10.1002/9781118788295.ch5. [9] M. C. Dallaston and S. W. McCue, A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area Proc. R. Soc. A 472 (2016), 20150629, 15 pp. doi: 10.1098/rspa.2015.0629. [10] K. Deckelnick, Parametric mean curvature evolution with a Dirichlet boundary condition, J. Reine Angew. Math., 459 (1995), 37-60.  doi: 10.1515/crll.1995.459.37. [11] I. C. Dolcetta, S. F. Vita and R. March, Area preserving curve shortening flows: From phase separation to image processing, Interfaces Free Bound., 4 (2002), 325-343.  doi: 10.4171/IFB/64. [12] 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. [13] S. Esedoḡlu, S. Ruuth and R. Tsai, Threshold dynamics for high order geometric motions, Interfaces Free Bound., 10 (2008), 263-282.  doi: 10.4171/IFB/189. [14] M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math., 51 (1986), 51-62.  doi: 10.1090/conm/051/848933. [15] M. Henry, D. Hilhorst and M. Mimura, A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 125-154.  doi: 10.3934/dcdss.2011.4.125. [16] M. Kolář, M. Beneš and D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane, Math. Comput. Simulation, 126 (2016), 1-13. [17] C. Kublik, S. Esedoḡlu and J. A. Fessler, Algorithms for area preserving flows, SIAM J. Sci. Comput., 33 (2011), 2382-2401.  doi: 10.1137/100815542. [18] A. Lunardi, Abstract quasilinear parabolic equations, Math. Ann., 267 (1984), 395-416.  doi: 10.1007/BF01456097. [19] I. V. Markov, Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy, 2 edition, World Scientific Publishing Company, 2004. [20] J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.  doi: 10.4310/AJM.2003.v7.n1.a2. [21] K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), 1473-1501.  doi: 10.1137/S0036139999359288. [22] K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Vis. Sci., 6 (2004), 211-225.  doi: 10.1007/s00791-004-0131-6. [23] K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Math. Methods Appl. Sci., 27 (2004), 1545-1565.  doi: 10.1002/mma.514. [24] P. Pauš, M. Beneš, M. Kolář and J. Kratochvíl, Dynamics of dislocations described as evolving curves interacting with obstacles, Model. Simul. Mater. Sc., 24 (2016), 035003. [25] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264.  doi: 10.1093/imamat/48.3.249. [26] D. Ševčovič, Qualitative and quantitative aspects of curvature driven flows of planar curves, in Topics on Partial Differential Equations, Jindřich Nečas Cent. Math. Model. Lect. Notes, 2, Matfyzpress, Prague, 2007, 55-119. [27] D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan. J. Ind. Appl. Math., 28 (2011), 413-442.  doi: 10.1007/s13160-011-0046-9. [28] 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. [29] S. Yazaki, On the tangential velocity arising in a crystalline approximation of evolving plane curves, Kybernetika, 43 (2007), 913-918.

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##### References:
 [1] S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2. [2] S. Angenent, Nonlinear analytic semiflows, Proc. R. Soc. Edinb., Sect. A, 115 (1990), 91-107.  doi: 10.1017/S0308210500024598. [3] M. Beneš, Diffuse-interface treatment of the anisotropic mean-curvature flow, Appl. Math., 48 (2003), 437-453.  doi: 10.1023/B:APOM.0000024485.24886.b9. [4] M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bull. Inst. Math. Acad. Sinica (N. S.), 3 (2008), 509-523. [5] M. Beneš, J. Kratochvíl, J. Křišt'an, V. Minárik and P. Pauš, A parametric simulation method for discrete dislocation dynamics, Eur. Phys. J. ST, 177 (2009), 177-192. [6] M. Beneš, S. Yazaki and M. Kimura, Computational studies of non-local anisotropic Allen-Cahn equation, Math. Bohemica, 136 (2011), 429-437. [7] L. Bronsard and B. Stoth, Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807.  doi: 10.1137/S0036141094279279. [8] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅲ. Nucleation of a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699.  doi: 10.1002/9781118788295.ch5. [9] M. C. Dallaston and S. W. McCue, A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area Proc. R. Soc. A 472 (2016), 20150629, 15 pp. doi: 10.1098/rspa.2015.0629. [10] K. Deckelnick, Parametric mean curvature evolution with a Dirichlet boundary condition, J. Reine Angew. Math., 459 (1995), 37-60.  doi: 10.1515/crll.1995.459.37. [11] I. C. Dolcetta, S. F. Vita and R. March, Area preserving curve shortening flows: From phase separation to image processing, Interfaces Free Bound., 4 (2002), 325-343.  doi: 10.4171/IFB/64. [12] 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. [13] S. Esedoḡlu, S. Ruuth and R. Tsai, Threshold dynamics for high order geometric motions, Interfaces Free Bound., 10 (2008), 263-282.  doi: 10.4171/IFB/189. [14] M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math., 51 (1986), 51-62.  doi: 10.1090/conm/051/848933. [15] M. Henry, D. Hilhorst and M. Mimura, A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 125-154.  doi: 10.3934/dcdss.2011.4.125. [16] M. Kolář, M. Beneš and D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane, Math. Comput. Simulation, 126 (2016), 1-13. [17] C. Kublik, S. Esedoḡlu and J. A. Fessler, Algorithms for area preserving flows, SIAM J. Sci. Comput., 33 (2011), 2382-2401.  doi: 10.1137/100815542. [18] A. Lunardi, Abstract quasilinear parabolic equations, Math. Ann., 267 (1984), 395-416.  doi: 10.1007/BF01456097. [19] I. V. Markov, Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy, 2 edition, World Scientific Publishing Company, 2004. [20] J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.  doi: 10.4310/AJM.2003.v7.n1.a2. [21] K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), 1473-1501.  doi: 10.1137/S0036139999359288. [22] K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Vis. Sci., 6 (2004), 211-225.  doi: 10.1007/s00791-004-0131-6. [23] K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Math. Methods Appl. Sci., 27 (2004), 1545-1565.  doi: 10.1002/mma.514. [24] P. Pauš, M. Beneš, M. Kolář and J. Kratochvíl, Dynamics of dislocations described as evolving curves interacting with obstacles, Model. Simul. Mater. Sc., 24 (2016), 035003. [25] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264.  doi: 10.1093/imamat/48.3.249. [26] D. Ševčovič, Qualitative and quantitative aspects of curvature driven flows of planar curves, in Topics on Partial Differential Equations, Jindřich Nečas Cent. Math. Model. Lect. Notes, 2, Matfyzpress, Prague, 2007, 55-119. [27] D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan. J. Ind. Appl. Math., 28 (2011), 413-442.  doi: 10.1007/s13160-011-0046-9. [28] 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. [29] S. Yazaki, On the tangential velocity arising in a crystalline approximation of evolving plane curves, Kybernetika, 43 (2007), 913-918.
Illustration of a curve $\mathcal{G}_t$ on a given surface $\mathcal{M}$ and its projection $\Gamma_t$ to plane
Discretization of a segment of a curve by flowing finite volumes
Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 10$ (solid) and several intermediate curves $\mathcal{G}_t$ (dotted). The underlying surface $\mathcal{M}$ is plotted in gray color. Right: time evolution of the projected planar curves $\Gamma_t$ (see Example 1)
Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 30$ (solid). The underlying surface $\mathcal{M}$ is plotted in gray color. Right: time evolution of the projected planar curves $\Gamma_t$ (see Example 2)
Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 8$ (solid) are presented. The surface $\mathcal{M}$ is plotted in gray color. Right: time evolution of the projected planar curves $\Gamma_t$ (see Example 3)
Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 15$ (solid) are shown. The surface $\mathcal{M}$ is plotted in gray. Right: Time evolution of the projected planar curves $\Gamma_t$ (see Example 4)
Settings of computational examples
 Ex. $\mathbf{X}_{ini}, u \in [0,1]$ $\varphi$ 1 $\mathbf{X}_{ini} = (\frac14 + r(u) \cos(2 \pi u), -\frac14 + r(u) \sin(2 \pi u))^T$ $\varphi(x,y) = \sqrt{4 - x^2 - y^2}$ 2 $\mathbf{X}_{ini} = (\cos(2 \pi u), \frac1{10} + \sin(2 \pi u))^T$ $\varphi(x,y) = y^2$ 3 $\mathbf{X}_{ini} = (\cos(2 \pi u), \frac15 + \sin(2 \pi u))^T$ $\varphi(x,y) = \sin(\pi y)$ 4 $\mathbf{X}_{ini} = (\frac12 \cos(2 \pi u), \sin(2 \pi u))^T$ $\varphi(x,y) = x^2 - y^4$
 Ex. $\mathbf{X}_{ini}, u \in [0,1]$ $\varphi$ 1 $\mathbf{X}_{ini} = (\frac14 + r(u) \cos(2 \pi u), -\frac14 + r(u) \sin(2 \pi u))^T$ $\varphi(x,y) = \sqrt{4 - x^2 - y^2}$ 2 $\mathbf{X}_{ini} = (\cos(2 \pi u), \frac1{10} + \sin(2 \pi u))^T$ $\varphi(x,y) = y^2$ 3 $\mathbf{X}_{ini} = (\cos(2 \pi u), \frac15 + \sin(2 \pi u))^T$ $\varphi(x,y) = \sin(\pi y)$ 4 $\mathbf{X}_{ini} = (\frac12 \cos(2 \pi u), \sin(2 \pi u))^T$ $\varphi(x,y) = x^2 - y^4$
Table of EOCs for Example 1
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $3.2397 \cdot 10^{-2}$ - $3.2516 \cdot 10^{-2}$ - 200 $8.2467 \cdot 10^{-3}$ 1.9740 8.2767 $\cdot 10^{-3}$ 1.9740 300 $3.6408 \cdot 10^{-3}$ 2.0165 3.6542 $\cdot 10^{-3}$ 2.0164 400 $2.0411 \cdot 10^{-3}$ 2.0118 2.0485 $\cdot 10^{-3}$ 2.0117 500 $1.3033 \cdot 10^{-3}$ 2.0103 1.3081 $\cdot 10^{-3}$ 2.0102
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $3.2397 \cdot 10^{-2}$ - $3.2516 \cdot 10^{-2}$ - 200 $8.2467 \cdot 10^{-3}$ 1.9740 8.2767 $\cdot 10^{-3}$ 1.9740 300 $3.6408 \cdot 10^{-3}$ 2.0165 3.6542 $\cdot 10^{-3}$ 2.0164 400 $2.0411 \cdot 10^{-3}$ 2.0118 2.0485 $\cdot 10^{-3}$ 2.0117 500 $1.3033 \cdot 10^{-3}$ 2.0103 1.3081 $\cdot 10^{-3}$ 2.0102
Table of EOCs for Example 2
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $1.4812 \cdot 10^{-3}$ - $1.4839 \cdot 10^{-3}$ - 200 $3.7049 \cdot 10^{-4}$ 1.9993 $3.7092 \cdot 10^{-4}$ 2.0002 300 $1.6453 \cdot 10^{-4}$ 2.0019 $1.6471 \cdot 10^{-4}$ 2.0022 400 $8.2431 \cdot 10^{-5}$ 2.0045 $9.2525 \cdot 10^{-5}$ 2.0046 500 $5.9055 \cdot 10^{-5}$ 2.0077 $5.9114 \cdot 10^{-5}$ 2.0077
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $1.4812 \cdot 10^{-3}$ - $1.4839 \cdot 10^{-3}$ - 200 $3.7049 \cdot 10^{-4}$ 1.9993 $3.7092 \cdot 10^{-4}$ 2.0002 300 $1.6453 \cdot 10^{-4}$ 2.0019 $1.6471 \cdot 10^{-4}$ 2.0022 400 $8.2431 \cdot 10^{-5}$ 2.0045 $9.2525 \cdot 10^{-5}$ 2.0046 500 $5.9055 \cdot 10^{-5}$ 2.0077 $5.9114 \cdot 10^{-5}$ 2.0077
Table of EOCs for Example 3
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $4.3505 \cdot 10^{-3}$ - $4.7156 \cdot 10^{-3}$ - 200 $9.4649 \cdot 10^{-4}$ 2.2005 $9.5944 \cdot 10^{-4}$ 2.2972 300 $4.1813 \cdot 10^{-4}$ 2.0149 $4.2481 \cdot 10^{-4}$ 2.0082 400 $2.3506 \cdot 10^{-4}$ 2.0021 $2.3885 \cdot 10^{-4}$ 2.0015 500 $1.5050 \cdot 10^{-4}$ 1.9980 $1.5293 \cdot 10^{-4}$ 1.9980
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $4.3505 \cdot 10^{-3}$ - $4.7156 \cdot 10^{-3}$ - 200 $9.4649 \cdot 10^{-4}$ 2.2005 $9.5944 \cdot 10^{-4}$ 2.2972 300 $4.1813 \cdot 10^{-4}$ 2.0149 $4.2481 \cdot 10^{-4}$ 2.0082 400 $2.3506 \cdot 10^{-4}$ 2.0021 $2.3885 \cdot 10^{-4}$ 2.0015 500 $1.5050 \cdot 10^{-4}$ 1.9980 $1.5293 \cdot 10^{-4}$ 1.9980
Table of EOCs for Example 4
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $1.8882 \cdot 10^{-3}$ - $1.9422 \cdot 10^{-3}$ - 200 $4.7176 \cdot 10^{-4}$ 2.0009 $4.8494 \cdot 10^{-4}$ 2.0018 300 $2.0979 \cdot 10^{-4}$ 1.9986 $2.1563 \cdot 10^{-4}$ 1.9988 400 $1.1808 \cdot 10^{-4}$ 1.9978 $1.2136 \cdot 10^{-4}$ 1.9980 500 $7.5628 \cdot 10^{-5}$ 1.9966 $7.7728 \cdot 10^{-5}$ 1.9968
 $M$ $error_{max}$ EOC $error_{L1}$ EOC 100 $1.8882 \cdot 10^{-3}$ - $1.9422 \cdot 10^{-3}$ - 200 $4.7176 \cdot 10^{-4}$ 2.0009 $4.8494 \cdot 10^{-4}$ 2.0018 300 $2.0979 \cdot 10^{-4}$ 1.9986 $2.1563 \cdot 10^{-4}$ 1.9988 400 $1.1808 \cdot 10^{-4}$ 1.9978 $1.2136 \cdot 10^{-4}$ 1.9980 500 $7.5628 \cdot 10^{-5}$ 1.9966 $7.7728 \cdot 10^{-5}$ 1.9968
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