November  2006, 15(4): 1215-1236. doi: 10.3934/dcds.2006.15.1215

Stability for steady-state patterns in phase field dynamics associated with total variation energies


Department of Applied Mathematics, Faculty of Engineering, Kobe University, 1-1 Rokkodai, Nada, Kobe, 657-8501, Japan

Received  April 2004 Revised  March 2006 Published  May 2006

In this paper, we shall deal with a mathematical model to represent the dynamics of solid-liquid phase transitions, which take place in a two-dimensional bounded domain. This mathematical model is formulated as a coupled system of two kinetic equations.
    The first equation is a kind of heat equation, however a time-relaxation term is additionally inserted in the heat flux. Since the additional term guarantees some smoothness of the velocity of the heat diffusion, it is expected that the behavior of temperature is estimated in stronger topology than that as in the usual heat equation.
    The second equation is a type of the so-called Allen-Cahn equation, namely it is a kinetic equation of phase field dynamics derived as a gradient flow of an appropriate functional. Such functional is often called as "free energy'', and in case of our model, the free energy is formulated with use of the total variation functional. Therefore, the second equation involves a singular diffusion, which formally corresponds to a function of (mean) curvature on the free boundary between solid-liquid states (interface). It implies that this equation can be a modified expression of Gibbs-Thomson law.
    In this paper, we will focus on the geometry of the pattern drawn by solid-liquid phases in steady-state (steady-state pattern), which will be expected to have some stability in dynamical system generated by our mathematical model. Consequently, various geometric patterns, parted by gradual curves, will be shown as representative examples of such steady-state patterns.
Citation: Ken Shirakawa. Stability for steady-state patterns in phase field dynamics associated with total variation energies. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1215-1236. doi: 10.3934/dcds.2006.15.1215

Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997


Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373


Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147


Shihe Xu, Fangwei Zhang, Meng Bai. Stability of positive steady-state solutions to a time-delayed system with some applications. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 5561-5572. doi: 10.3934/dcdsb.2021286


Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565


Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545


Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317


Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks and Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011


Tomoyuki Miyaji, Yoshio Tsutsumi. Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1633-1650. doi: 10.3934/cpaa.2018078


Wing-Cheong Lo. Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 775-787. doi: 10.3934/dcdsb.2014.19.775


Li Ma, Youquan Luo. Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2555-2582. doi: 10.3934/dcdsb.2020022


Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643


Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135


Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 25-44. doi: 10.3934/jimo.2020141


Jiayan Yang, Dongpei Zhang. Superfluidity phase transitions for liquid $ ^{4} $He system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5107-5120. doi: 10.3934/dcdsb.2019045


Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159


Shu-Yi Zhang. Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2221-2239. doi: 10.3934/dcds.2013.33.2221


Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691


Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045


Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587

2021 Impact Factor: 1.588


  • PDF downloads (67)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]