
Previous Article
Dynamics of shape memory alloys patches with mechanically induced transformations
 DCDS Home
 This Issue

Next Article
Global attractor for a parabolichyperbolic PenroseFife phase field system
Stability for steadystate patterns in phase field dynamics associated with total variation energies
1.  Department of Applied Mathematics, Faculty of Engineering, Kobe University, 11 Rokkodai, Nada, Kobe, 6578501, Japan 
The first equation is a kind of heat equation, however a timerelaxation 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 socalled AllenCahn 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 solidliquid states (interface). It implies that this equation can be a modified expression of GibbsThomson law.
In this paper, we will focus on the geometry of the pattern drawn by solidliquid phases in steadystate (steadystate 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 steadystate patterns.
[1] 
Sylvie BenzoniGavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solidliquid phase transitions. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 19972025. doi: 10.3934/dcds.2012.32.1997 
[2] 
Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steadystate hotspot patterns for a reactiondiffusion model of urban crime. Discrete and Continuous Dynamical Systems  B, 2014, 19 (5) : 13731410. doi: 10.3934/dcdsb.2014.19.1373 
[3] 
Meihua Wei, Jianhua Wu, Yinnian He. Steadystate solutions and stability for a cubic autocatalysis model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 11471167. doi: 10.3934/cpaa.2015.14.1147 
[4] 
Shihe Xu, Fangwei Zhang, Meng Bai. Stability of positive steadystate solutions to a timedelayed system with some applications. Discrete and Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021286 
[5] 
Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropicnematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems  S, 2011, 4 (3) : 565579. doi: 10.3934/dcdss.2011.4.565 
[6] 
Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 545556. doi: 10.3934/cpaa.2004.3.545 
[7] 
Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steadystate mathematical model for an EOS capacitor: The effect of the size exclusion. Networks and Heterogeneous Media, 2016, 11 (4) : 603625. doi: 10.3934/nhm.2016011 
[8] 
Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquidvapour phase transitions. Discrete and Continuous Dynamical Systems  S, 2013, 6 (2) : 317330. doi: 10.3934/dcdss.2013.6.317 
[9] 
Tomoyuki Miyaji, Yoshio Tsutsumi. Steadystate mode interactions of radially symmetric modes for the LugiatoLefever equation on a disk. Communications on Pure and Applied Analysis, 2018, 17 (4) : 16331650. doi: 10.3934/cpaa.2018078 
[10] 
WingCheong Lo. Morphogen gradient with expansionrepression mechanism: Steadystate and robustness studies. Discrete and Continuous Dynamical Systems  B, 2014, 19 (3) : 775787. doi: 10.3934/dcdsb.2014.19.775 
[11] 
Li Ma, Youquan Luo. Dynamics of positive steadystate solutions of a nonlocal dispersal logistic model with nonlocal terms. Discrete and Continuous Dynamical Systems  B, 2020, 25 (7) : 25552582. doi: 10.3934/dcdsb.2020022 
[12] 
Yacine Chitour, JeanMichel Coron, Mauro Garavello. On conditions that prevent steadystate controllability of certain linear partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 643672. doi: 10.3934/dcds.2006.14.643 
[13] 
Zhenzhen Zheng, ChingShan Chou, TauMu Yi, Qing Nie. Mathematical analysis of steadystate solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 11351168. doi: 10.3934/mbe.2011.8.1135 
[14] 
Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steadystate analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 2544. doi: 10.3934/jimo.2020141 
[15] 
Jiayan Yang, Dongpei Zhang. Superfluidity phase transitions for liquid $ ^{4} $He system. Discrete and Continuous Dynamical Systems  B, 2019, 24 (9) : 51075120. doi: 10.3934/dcdsb.2019045 
[16] 
Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for AllenCahn type equation associated with total variation energy. Discrete and Continuous Dynamical Systems  S, 2012, 5 (1) : 159181. doi: 10.3934/dcdss.2012.5.159 
[17] 
HsinYi Liu, Hsing Paul Luh. Kronecker productforms of steadystate probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 691711. doi: 10.3934/naco.2011.1.691 
[18] 
Na Min, Mingxin Wang. Hopf bifurcation and steadystate bifurcation for a LeslieGower preypredator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 10711099. doi: 10.3934/dcds.2019045 
[19] 
ShinYi Lee, ShinHwa Wang, ChiouPing Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a plaplacian steadystate reactiondiffusion problem. Conference Publications, 2005, 2005 (Special) : 587596. doi: 10.3934/proc.2005.2005.587 
[20] 
ShuYi Zhang. Existence of multidimensional nonisothermal phase transitions in a steady van der Waals flow. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 22212239. doi: 10.3934/dcds.2013.33.2221 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]