American Institute of Mathematical Sciences

September  2016, 11(3): 509-526. doi: 10.3934/nhm.2016007

Evolution of spoon-shaped networks

 1 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy

Received  March 2015 Revised  September 2015 Published  August 2016

We consider a regular embedded network composed by two curves, one of them closed, in a convex and smooth domain $\Omega$. The two curves meet only at one point, forming angles of $120$ degrees. The non-closed curve has a fixed end--point on $\partial\Omega$. We study the evolution by curvature of this network. We show that the maximal time of existence is finite and depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.
Citation: Alessandra Pluda. Evolution of spoon-shaped networks. Networks and Heterogeneous Media, 2016, 11 (3) : 509-526. doi: 10.3934/nhm.2016007
References:
 [1] U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Diff. Geom., 23 (1986), 175-196. [2] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton, NJ, 1978. [3] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), 355-379. doi: 10.1007/BF00375607. [4] X. Chen and J. Guo, Motion by curvature of planar curves with end points moving freely on a line, Math. Ann., 350 (2011), 277-311. doi: 10.1007/s00208-010-0558-7. [5] K. S. Chou and X. P. Zhu, Shortening complete planar curves, J. Diff. Geom., 50 (1998), 471-504. [6] M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96. [7] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Diff. Geom., 26 (1987), 285-314. [8] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diff. Geom., 24 (1986), 153-179. [9] R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, in Modern Methods in Complex Analysis (Princeton, NJ, 1992), Princeton University Press, Princeton, NJ, 137 (1995), 201-222. doi: 10.1016/1053-8127(94)00130-3. [10] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 31 (1990), 285-299. [11] G. Huisken, A distance comparison principle for evolving curves, Asian J. Math., 2 (1998), 127-133. doi: 10.4310/AJM.1998.v2.n1.a2. [12] T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, preprint,, , (). [13] D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), 713-729. doi: 10.1142/S0218202501001069. [14] A. Magni and C. Mantegazza, A note on Grayson's theorem, Rend. Semin. Mat. Univ. Padova, 131 (2014), 263-279. doi: 10.4171/RSMUP/131-16. [15] A. Magni, C. Mantegazza and M. Novaga, Motion by curvature of planar networks II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 15 (2016), 117-144. [16] C. Mantegazza, M. Novaga, A. Pluda and F. Schulze, Evolution of network with multiple junctions, Preprint, 2015. [17] C. Mantegazza, M. Novaga and V. M. Tortorelli, Motion by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3 (2004), 235-324. [18] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 13 (1959), 115-162.

show all references

References:
 [1] U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Diff. Geom., 23 (1986), 175-196. [2] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton, NJ, 1978. [3] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), 355-379. doi: 10.1007/BF00375607. [4] X. Chen and J. Guo, Motion by curvature of planar curves with end points moving freely on a line, Math. Ann., 350 (2011), 277-311. doi: 10.1007/s00208-010-0558-7. [5] K. S. Chou and X. P. Zhu, Shortening complete planar curves, J. Diff. Geom., 50 (1998), 471-504. [6] M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96. [7] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Diff. Geom., 26 (1987), 285-314. [8] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diff. Geom., 24 (1986), 153-179. [9] R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, in Modern Methods in Complex Analysis (Princeton, NJ, 1992), Princeton University Press, Princeton, NJ, 137 (1995), 201-222. doi: 10.1016/1053-8127(94)00130-3. [10] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 31 (1990), 285-299. [11] G. Huisken, A distance comparison principle for evolving curves, Asian J. Math., 2 (1998), 127-133. doi: 10.4310/AJM.1998.v2.n1.a2. [12] T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, preprint,, , (). [13] D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), 713-729. doi: 10.1142/S0218202501001069. [14] A. Magni and C. Mantegazza, A note on Grayson's theorem, Rend. Semin. Mat. Univ. Padova, 131 (2014), 263-279. doi: 10.4171/RSMUP/131-16. [15] A. Magni, C. Mantegazza and M. Novaga, Motion by curvature of planar networks II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 15 (2016), 117-144. [16] C. Mantegazza, M. Novaga, A. Pluda and F. Schulze, Evolution of network with multiple junctions, Preprint, 2015. [17] C. Mantegazza, M. Novaga and V. M. Tortorelli, Motion by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3 (2004), 235-324. [18] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 13 (1959), 115-162.
 [1] Yang Xiang, Xiaodong Yan. Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2989-3021. doi: 10.3934/dcdsb.2017183 [2] Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124. [3] Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27 [4] Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure and Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549 [5] Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811 [6] Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062 [7] Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021 [8] Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks. Mathematical Biosciences & Engineering, 2017, 14 (3) : 607-624. doi: 10.3934/mbe.2017035 [9] Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks and Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41 [10] Meng Zhao. The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks. Electronic Research Archive, 2020, 28 (3) : 1143-1160. doi: 10.3934/era.2020063 [11] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Ghost effect by curvature in planar Couette flow. Kinetic and Related Models, 2011, 4 (1) : 109-138. doi: 10.3934/krm.2011.4.109 [12] Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441 [13] Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153 [14] Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016 [15] Lixia Yuan, Wei Zhao. On a curvature flow in a band domain with unbounded boundary slopes. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 261-283. doi: 10.3934/dcds.2021115 [16] Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137 [17] Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477 [18] Young-Pil Choi, Jinwook Jung. On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022016 [19] Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681 [20] Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1083-1096. doi: 10.3934/dcdsb.2019209

2020 Impact Factor: 1.213