# American Institute of Mathematical Sciences

January  2015, 14(1): 121-125. doi: 10.3934/cpaa.2015.14.121

## A counterexample to finite time stopping property for one-harmonic map flow

 1 Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914 2 Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

Received  February 2014 Revised  April 2014 Published  September 2014

For a very strong diffusion equation like total variation flow it is often observed that the solution stops at a steady state in a finite time. This phenomenon is called a finite time stopping or a finite time extinction if the steady state is zero. Such a phenomenon is also observed in one-harmonic map flow from an interval to a unit circle when initial data is piecewise constant. However, if the target manifold is a unit two-dimensional sphere, the finite time stopping may not occur. An explicit example is given in this paper.
Citation: Yoshikazu Giga, Hirotoshi Kuroda. A counterexample to finite time stopping property for one-harmonic map flow. Communications on Pure and Applied Analysis, 2015, 14 (1) : 121-125. doi: 10.3934/cpaa.2015.14.121
##### References:
 [1] F. Andreu, V. Caselles, J. I. Díaz and J. M. Mazón, Some Qualitative properties for the total variation flow, Journal of Functional Analysis, 188 (2) (2002), 516-547. doi: 10.1006/jfan.2001.3829. [2] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progress in Mathematics, 223, Birkhäuser Basel, 2004. doi: 10.1007/978-3-0348-7928-6. [3] J. W. Barrett, X. Feng and A. Prohl, On p-harmonic map heat flows for $1 \leq p<\infty$ and their finite element approximations, SIAM J. Math. Anal., 40 (2008), 1471-1498. doi: 10.1137/070680825. [4] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973. [5] R. Dal Passo, L. Giacomelli and S. Moll, Rotationally symmetric 1-harmonic maps from $D^2$ to $S^2$, Calc. Var. PDEs, 32 (2008), 533-554. doi: 10.1007/s00526-007-0153-2. [6] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [7] X. Feng, Divergence-$L^q$ and divergence-measure tensor fields and gradient flows for linear growth functionals of maps into the unit sphere, Calc. Var. PDEs, 37 (2010), 111-139. doi: 10.1007/s00526-009-0255-0. [8] L. Giacomelli, J. M. Mazón and S. Moll, The 1-harmonic flow with values into $\mathbbS^1$, SIAM J. Math. Anal., 45 (2013), 1723-1740. doi: 10.1137/12088402X. [9] L. Giacomelli, J. M. Mazón and S. Moll, The 1-harmonic flow with values in a hyperoctant of the $N$-sphere, Analysis and PDEs, 7 (2014), 627-671. doi: 10.1016/j.aml.2013.05.016. [10] L. Giacomelli and S. Moll, Rotationally symmetric 1-harmonic flows from $D^2$ to $S^2$: local well-posedness and finite time blowup, SIAM J. Math. Anal., 42 (2010), 2791-2817. doi: 10.1137/090764293. [11] Y. Giga, Y. Kashima and N. Yamazaki, Local solvability of a constrained gradient system of total variation, Abstr. Appl. Anal., 8 (2004), 651-682. doi: 10.1155/S1085337504311048. [12] Y. Giga and R. Kobayashi, On constrained equations with singular diffusivity, Methods Appl. Anal., 10 (2003), 253-277. [13] Y. Giga and R. Kohn, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst., 30 (2011), 509-535. doi: 10.3934/dcds.2011.30.509. [14] Y. Giga and H. Kuroda, On breakdown of solutions of a constrained gradient system of total variation, Bol. Soc. Parana. Mat., 22 (2004), 9-20. doi: 10.5269/bspm.v22i1.7491. [15] Y. Giga, H. Kuroda and N. Yamazaki, An existence result for a discretized constrained gradient system of total variation flow in color image processing, Interdiscip. Inform. Sci., 11 (2005), 199-204. doi: 10.4036/iis.2005.199. [16] Y. Giga, H. Kuroda and N. Yamazaki, Global solvability of constrained singular diffusion equation associated with essential variation, International Series of Numerical Mathematics, 154, Free Boundary Problems: Theory and Applications, Birkhäuser Verlag Basel (2007), 209-218. doi: 10.1007/978-3-7643-7719-9_21. [17] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Stat. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372. [18] Y. Kōmura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. [19] B. Tang, G. Sapiro and V. Caselles, Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case, Int. J. Computer Vision, 36 (2000), 149-161. [20] B. Tang, G. Sapiro and V. Caselles, Color image enhancement via chromaticity diffusion, IEEE Transactions on Image Processing, 10 (2001), 701-707.

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##### References:
 [1] F. Andreu, V. Caselles, J. I. Díaz and J. M. Mazón, Some Qualitative properties for the total variation flow, Journal of Functional Analysis, 188 (2) (2002), 516-547. doi: 10.1006/jfan.2001.3829. [2] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progress in Mathematics, 223, Birkhäuser Basel, 2004. doi: 10.1007/978-3-0348-7928-6. [3] J. W. Barrett, X. Feng and A. Prohl, On p-harmonic map heat flows for $1 \leq p<\infty$ and their finite element approximations, SIAM J. Math. Anal., 40 (2008), 1471-1498. doi: 10.1137/070680825. [4] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973. [5] R. Dal Passo, L. Giacomelli and S. Moll, Rotationally symmetric 1-harmonic maps from $D^2$ to $S^2$, Calc. Var. PDEs, 32 (2008), 533-554. doi: 10.1007/s00526-007-0153-2. [6] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [7] X. Feng, Divergence-$L^q$ and divergence-measure tensor fields and gradient flows for linear growth functionals of maps into the unit sphere, Calc. Var. PDEs, 37 (2010), 111-139. doi: 10.1007/s00526-009-0255-0. [8] L. Giacomelli, J. M. Mazón and S. Moll, The 1-harmonic flow with values into $\mathbbS^1$, SIAM J. Math. Anal., 45 (2013), 1723-1740. doi: 10.1137/12088402X. [9] L. Giacomelli, J. M. Mazón and S. Moll, The 1-harmonic flow with values in a hyperoctant of the $N$-sphere, Analysis and PDEs, 7 (2014), 627-671. doi: 10.1016/j.aml.2013.05.016. [10] L. Giacomelli and S. Moll, Rotationally symmetric 1-harmonic flows from $D^2$ to $S^2$: local well-posedness and finite time blowup, SIAM J. Math. Anal., 42 (2010), 2791-2817. doi: 10.1137/090764293. [11] Y. Giga, Y. Kashima and N. Yamazaki, Local solvability of a constrained gradient system of total variation, Abstr. Appl. Anal., 8 (2004), 651-682. doi: 10.1155/S1085337504311048. [12] Y. Giga and R. Kobayashi, On constrained equations with singular diffusivity, Methods Appl. Anal., 10 (2003), 253-277. [13] Y. Giga and R. Kohn, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst., 30 (2011), 509-535. doi: 10.3934/dcds.2011.30.509. [14] Y. Giga and H. Kuroda, On breakdown of solutions of a constrained gradient system of total variation, Bol. Soc. Parana. Mat., 22 (2004), 9-20. doi: 10.5269/bspm.v22i1.7491. [15] Y. Giga, H. Kuroda and N. Yamazaki, An existence result for a discretized constrained gradient system of total variation flow in color image processing, Interdiscip. Inform. Sci., 11 (2005), 199-204. doi: 10.4036/iis.2005.199. [16] Y. Giga, H. Kuroda and N. Yamazaki, Global solvability of constrained singular diffusion equation associated with essential variation, International Series of Numerical Mathematics, 154, Free Boundary Problems: Theory and Applications, Birkhäuser Verlag Basel (2007), 209-218. doi: 10.1007/978-3-7643-7719-9_21. [17] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Stat. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372. [18] Y. Kōmura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. [19] B. Tang, G. Sapiro and V. Caselles, Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case, Int. J. Computer Vision, 36 (2000), 149-161. [20] B. Tang, G. Sapiro and V. Caselles, Color image enhancement via chromaticity diffusion, IEEE Transactions on Image Processing, 10 (2001), 701-707.
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