-
Previous Article
Phragmén--Lindelöf theorem for infinity harmonic functions
- CPAA Home
- This Issue
-
Next Article
Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation
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 |
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. |
show all references
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. |
[1] |
Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390 |
[2] |
Yoshikazu Giga, Robert V. Kohn. Scale-invariant extinction time estimates for some singular diffusion equations. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 509-535. doi: 10.3934/dcds.2011.30.509 |
[3] |
Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237 |
[4] |
Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940 |
[5] |
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 |
[6] |
Konstantinos Papafitsoros, Kristian Bredies. A study of the one dimensional total generalised variation regularisation problem. Inverse Problems and Imaging, 2015, 9 (2) : 511-550. doi: 10.3934/ipi.2015.9.511 |
[7] |
Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066 |
[8] |
Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems and Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565 |
[9] |
Jun Li, Fubao Xi. Exponential ergodicity for regime-switching diffusion processes in total variation norm. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021309 |
[10] |
Jesus Ildefonso Díaz, David Gómez-Castro, Jean Michel Rakotoson, Roger Temam. Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 509-546. doi: 10.3934/dcds.2018023 |
[11] |
Anna Maria Cherubini, Giorgio Metafune, Francesco Paparella. On the stopping time of a bouncing ball. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 43-72. doi: 10.3934/dcdsb.2008.10.43 |
[12] |
Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265 |
[13] |
Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations and Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042 |
[14] |
Jiaxi Huang, Youde Wang, Lifeng Zhao. Equivariant Schrödinger map flow on two dimensional hyperbolic space. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184 |
[15] |
Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867 |
[16] |
Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191 |
[17] |
Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431 |
[18] |
Jeremy LeCrone, Yuanzhen Shao, Gieri Simonett. The surface diffusion and the Willmore flow for uniformly regular hypersurfaces. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3503-3524. doi: 10.3934/dcdss.2020242 |
[19] |
Feng Li, Erik Lindgren. Large time behavior for a nonlocal nonlinear gradient flow. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022079 |
[20] |
Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]