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

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