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

show all references

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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