Evolution of multiple closed knotted curves in space
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Published
Sep 21, 2024
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Abstract
We investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semi-flows, we are able to prove local existence, uniqueness, and continuation of classical H\"older smooth solutions to the governing system of non-linear parabolic equations modelling $n$ evolving curves with mutual nonlocal interactions. We present several computational studies of the flow that combine the normal or binormal velocity and considering nonlocal interaction.
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Kolář, M., & Ševčovič, D.
(2024).
Evolution of multiple closed knotted curves in space.
Proceedings Of The Conference Algoritmy, , 129 - 138.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2162/1034
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References
[1] S. Angenent, Parabolic equations for curves on surfaces. I: Curves with p-integrable curvature, Ann. Math., 132(2) (1990), pp. 451–483.
[2] S. Angenent, Nonlinear analytic semi-flows. Proc. R. Soc. Edinb., Sect. A, 115 (1990), pp. 91–107.
[3] R.J. Arms and F. R. Hama, Localized Induction Concept on a Curved Vortex and Motion of an Elliptic Vortex Ring, Phys. Fluids, 8 (1965), pp. 553-559. [4] J. W. Barrett, H. Garcke, and R. Nürnberg, Numerical approximation of gradient flows for closed curves in Rd, IMA J. Numer. Anal., 30(1), (2010), pp. 4–60.
[5] J. W. Barrett, H. Garcke, and R. Nürnberg, Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves, Numer. Math., 120(3), (2012), pp. 489–542.
[6] R. Betchov, On the curvature and torsion of an isolated vortex filament, J. Fluid Mech., 22 (1965), pp. 471-479.
[7] G. P. Bewley, M. S. Paoletti, K. R. Sreenivasan, and D. P. Lathrop, Characterization of reconnecting vortices in super-fluid helium, P. Natl. Acad. Sci. USA., 105(37) (2008), pp. 13707-13710.
[8] M. Beneš, M. Kolář M, and D. Ševčovič, Curvature driven flow of a family of interacting curves with applications, Math. Method. Appl. Sci., 43 (2020), pp. 4177-4190.
[9] M. Beneš, M. Kolář M, and D. Ševčovič, Qualitative and numerical aspects of a motion of a family of interacting curves in space, SIAM Journal on Applied Mathematics, 82(2), (2022), 549-575.
[10] M. Minárik, M. Beneš, and J. Kratochvı́l, Simulation of dynamical interaction between dislocations and dipolar loops, Journal of Applied Physics, 107(2), (2010), 061802.
[11] J. Křišťan, J. Kratochvı́l, M. Minárik and M. Beneš, Simulation of dynamical interacting dislocations glide in a channel of persistent slip band, Modeling and Simulation in Material Science and Engineering, 17, (2009), 045009.
[12] Ch. M. Elliott and H. Fritz, On approximations of the curve shortening flow and of the mean curvature flow ased on the DeTurck trick, IMA J. Numer. Anal., 37(2), (2017), pp. 543–603.
[13] Y. Fukumoto, On Integral Invariants for Vortex Motion under the Localized Induction Approximation, J. Phys. Soc. Jpn., 56(12) (1987), pp. 4207–4209.
[14] Y. Fukumoto and T. Miyzaki Three-dimensional distortions of a vortex filament with axial velocity, J. Fluid Mech., 222 (1991), pp. 369–416.
[15] H. Garcke, Y. Kohsaka, and D. Ševčovič, Nonlinear stability of stationary solutions for curvature flow with triple junction, Hokkaido Math. J., 38(4) (2009), pp. 721–769.
[16] H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen ntsprechen, J. Reine Angew. Math., 55 (1858), pp. 25–55.
[17] T. Y. Hou, J. Lowengrub, and M. Shelley, Removing the stiffness from interfacial flows and surface tension, J. Comput. Phys., 114 (1994), pp. 312–338.
[18] F. de la Hoz and L. Vega, Vortex filament equation for a regular polygon, Nonlinearity 27(12) (2014), pp. 3031–3057.
[19] D. A. Kessler, J. Koplik, and H. Levine, Numerical simulation of two-dimensional snowflake growth, Phys. Rev. A, 30(5), (1984), pp. 2820–2823.
[20] M. Kimura, Numerical analysis for moving boundary problems using the boundary tracking method, Jpn. J. Indust. Appl. Math., 14 (1997), pp. 373–398.
[21] R. Mariani and K. Kontis, Experimental studies on coaxial vortex loops, Phys. Fluids, 22 (2010), p. 126102.
[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), pp. 211–225.
[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), pp. 1545–1565.
[24] L. S. Da Rios, Sul Moto di un filetto vorticoso di forma qualunque, Rend. Circ. Mat. Palermo, 22 (1906), pp. 117-135.
[25]P. Pauš, M. Beneš, M. Kolář, and J. Kratochvı́l, Dynamics of dislocations described as evolving curves interacting with obstacles, Model. Simul. Mater. Sci., 24 (2016), p. 035003.
[26] K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), pp.1473–1501.
[27] 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), pp. 1784–1798.
[28] J. Strain, A boundary integral approach to unstable solidification, J. Comput. Phys., 85(2), (1989), pp. 342–389.
[29] W. Thomson, On Vortex Atoms, Proc. R. Soc. Edinb., 6 (1867), 94–105.
[30] L. Vega, The dynamics of vortex filaments with corners, Commun. Pur. Appl. Anal., 14(4) (2015), 1581–1601.
[2] S. Angenent, Nonlinear analytic semi-flows. Proc. R. Soc. Edinb., Sect. A, 115 (1990), pp. 91–107.
[3] R.J. Arms and F. R. Hama, Localized Induction Concept on a Curved Vortex and Motion of an Elliptic Vortex Ring, Phys. Fluids, 8 (1965), pp. 553-559. [4] J. W. Barrett, H. Garcke, and R. Nürnberg, Numerical approximation of gradient flows for closed curves in Rd, IMA J. Numer. Anal., 30(1), (2010), pp. 4–60.
[5] J. W. Barrett, H. Garcke, and R. Nürnberg, Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves, Numer. Math., 120(3), (2012), pp. 489–542.
[6] R. Betchov, On the curvature and torsion of an isolated vortex filament, J. Fluid Mech., 22 (1965), pp. 471-479.
[7] G. P. Bewley, M. S. Paoletti, K. R. Sreenivasan, and D. P. Lathrop, Characterization of reconnecting vortices in super-fluid helium, P. Natl. Acad. Sci. USA., 105(37) (2008), pp. 13707-13710.
[8] M. Beneš, M. Kolář M, and D. Ševčovič, Curvature driven flow of a family of interacting curves with applications, Math. Method. Appl. Sci., 43 (2020), pp. 4177-4190.
[9] M. Beneš, M. Kolář M, and D. Ševčovič, Qualitative and numerical aspects of a motion of a family of interacting curves in space, SIAM Journal on Applied Mathematics, 82(2), (2022), 549-575.
[10] M. Minárik, M. Beneš, and J. Kratochvı́l, Simulation of dynamical interaction between dislocations and dipolar loops, Journal of Applied Physics, 107(2), (2010), 061802.
[11] J. Křišťan, J. Kratochvı́l, M. Minárik and M. Beneš, Simulation of dynamical interacting dislocations glide in a channel of persistent slip band, Modeling and Simulation in Material Science and Engineering, 17, (2009), 045009.
[12] Ch. M. Elliott and H. Fritz, On approximations of the curve shortening flow and of the mean curvature flow ased on the DeTurck trick, IMA J. Numer. Anal., 37(2), (2017), pp. 543–603.
[13] Y. Fukumoto, On Integral Invariants for Vortex Motion under the Localized Induction Approximation, J. Phys. Soc. Jpn., 56(12) (1987), pp. 4207–4209.
[14] Y. Fukumoto and T. Miyzaki Three-dimensional distortions of a vortex filament with axial velocity, J. Fluid Mech., 222 (1991), pp. 369–416.
[15] H. Garcke, Y. Kohsaka, and D. Ševčovič, Nonlinear stability of stationary solutions for curvature flow with triple junction, Hokkaido Math. J., 38(4) (2009), pp. 721–769.
[16] H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen ntsprechen, J. Reine Angew. Math., 55 (1858), pp. 25–55.
[17] T. Y. Hou, J. Lowengrub, and M. Shelley, Removing the stiffness from interfacial flows and surface tension, J. Comput. Phys., 114 (1994), pp. 312–338.
[18] F. de la Hoz and L. Vega, Vortex filament equation for a regular polygon, Nonlinearity 27(12) (2014), pp. 3031–3057.
[19] D. A. Kessler, J. Koplik, and H. Levine, Numerical simulation of two-dimensional snowflake growth, Phys. Rev. A, 30(5), (1984), pp. 2820–2823.
[20] M. Kimura, Numerical analysis for moving boundary problems using the boundary tracking method, Jpn. J. Indust. Appl. Math., 14 (1997), pp. 373–398.
[21] R. Mariani and K. Kontis, Experimental studies on coaxial vortex loops, Phys. Fluids, 22 (2010), p. 126102.
[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), pp. 211–225.
[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), pp. 1545–1565.
[24] L. S. Da Rios, Sul Moto di un filetto vorticoso di forma qualunque, Rend. Circ. Mat. Palermo, 22 (1906), pp. 117-135.
[25]P. Pauš, M. Beneš, M. Kolář, and J. Kratochvı́l, Dynamics of dislocations described as evolving curves interacting with obstacles, Model. Simul. Mater. Sci., 24 (2016), p. 035003.
[26] K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), pp.1473–1501.
[27] 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), pp. 1784–1798.
[28] J. Strain, A boundary integral approach to unstable solidification, J. Comput. Phys., 85(2), (1989), pp. 342–389.
[29] W. Thomson, On Vortex Atoms, Proc. R. Soc. Edinb., 6 (1867), 94–105.
[30] L. Vega, The dynamics of vortex filaments with corners, Commun. Pur. Appl. Anal., 14(4) (2015), 1581–1601.