American Institute of Mathematical Sciences

Journals

DCDS-S
Discrete & Continuous Dynamical Systems - S 2011, 4(2): 483-504 doi: 10.3934/dcdss.2011.4.483
In this paper, a mathematical model, to represent the dynamics of two-dimensional solid-liquid phase transition, is considered. This mathematical model is formulated as a coupled system of a heat equation with a time-relaxation diffusion, and an Allen-Cahn equation such that the two-dimensional norm, of crystalline-type, is adopted as the mathematical expression of the anisotropy. Through the structural observations for steady-state solutions, some geometric conditions to guarantee their stability will be presented in the main theorem of this paper.
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DCDS-S
Discrete & Continuous Dynamical Systems - S 2014, 7(1): 139-159 doi: 10.3934/dcdss.2014.7.139
In this paper, a coupled system of two parabolic initial-boundary value problems is considered. The system presented is a one-dimensional version of the Kobayashi-Warren-Carter model of grain boundary motion [15,16], that is derived as a gradient system of a governing free energy including a weighted total variation. Due to the weighted total variation, some nonstandard terms appear in the mathematical expressions of this system, and such nonstandard terms have made the mathematical treatments to be quite delicate. Recently, a certain definition of the solution have been provided in [21], together with the solvability result. The main objective in this paper is to verify that the system reproduces the foundational rules as a gradient system of parabolic PDEs, such as smoothing effect'' and energy-dissipation''. Consequently, the existence of a special solution, called energy-dissipative solution'', will be demonstrated in the Main Theorem of this paper.
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CPAA
Communications on Pure & Applied Analysis 2010, 9(5): 1345-1361 doi: 10.3934/cpaa.2010.9.1345
In this paper, a class of minimization problems, associated with the micromagnetics of thin films, is dealt with. Each minimization problem is distinguished by the thickness of the thin film, denoted by $0 < h < 1$, and it is considered under spatial indefinite and degenerative setting of the material coefficients. On the basis of the fundamental studies of the governing energy functionals, the existence of minimizers, for every $0 < h < 1$, and the 3D-2D asymptotic analysis for the observing minimization problems, as $h \to 0$, will be demonstrated in the main theorem of this paper.
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PROC
Conference Publications 2011, 2011(Special): 624-633 doi: 10.3934/proc.2011.2011.624
In this paper, a class of minimization problems, labeled by an index 0 < $h$ < 1, is considered. Each minimization problem is for a free-energy, motivated by the magnetics in 3D-ferromagnetic thin film, and in the context, the index $h$ denotes the thickness of the observing film. The Main Theorem consists of two themes, which are concerned with the study of the solvability (existence of minimizers) and the 3D-2D asymptotic analysis for our minimization problems. These themes will be discussed under degenerate setting of the material coecients, and such degenerate situation makes the energy-domain be variable with respect to $h$. In conclusion, assuming some restrictive conditions for the domain-variation, a de nite association between our 3D-minimization problems, for very thin $h$, and a 2D-limiting problem, as $h \searrow$ 0, will be demonstrated with help from the theory of $\Gamma$-convergence.
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PROC
Conference Publications 2007, 2007(Special): 927-937 doi: 10.3934/proc.2007.2007.927
In this paper, we deal with systems of nonlinear evolution equations, which are mathematical models of phase transitions, classified as "Penrose-Fife type". In the presented models, $p$-Laplacians, with 1 $<= p$ < 2, are adopted to describe the diffusions in exchanges. As the main conclusions, some theorems, concerned with the existence and the uniqueness of solutions of our systems, will be proved, under appropriate assumptions.
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DCDS
Discrete & Continuous Dynamical Systems - A 2006, 15(4): 1215-1236 doi: 10.3934/dcds.2006.15.1215
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.
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PROC
Conference Publications 2009, 2009(Special): 697-707 doi: 10.3934/proc.2009.2009.697
This paper is devoted to the stability analysis for two dimensional interfaces in solid-liquid phase transitions, represented by some types of Allen-Cahn equations. Each Allen-Cahn equation is derived from a free energy, associated with a two dimensional Finsler norm, under the so-called crystalline type setting, and then the Wulff shape of the Finsler norm is supposed to correspond to the basic structural unit of masses of pure phases (crystals). Consequently, special piecewise smooth Jordan curves, based on Wulff shapes, will be exemplified in the main theorems, as the geometric representations of the stability condition.
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PROC
Conference Publications 2015, 2015(special): 1009-1018 doi: 10.3934/proc.2015.1009
In this paper, a system of parabolic initial-boundary value problems is considered as a possible PDE model of isothermal grain boundary motion. The solvability of this system was proved in [preprint, arXiv:1408.4204., by means of the notion of weighted total variation. In this light, we set our goal to prove two main theorems, which are concerned with the $\Gamma$-convergence for time-dependent versions of the weighted total variations, and the large-time behavior of solution.
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NHM
Networks & Heterogeneous Media 2014, 9(4): 655-668 doi: 10.3934/nhm.2014.9.655
In this paper we deal with a one-dimensional free boundary problem, which is a mathematical model for an adsorption phenomena appearing in concrete carbonation process. This model was proposed in line of previous studies of three dimensional concrete carbonation process. The main result in this paper is concerned with the existence and uniqueness of a time-local solution to the free boundary problem. This result will be obtained by means of the abstract theory of nonlinear evolution equations and Banach's fixed point theorem, and especially, the maximum principle applied to our problem will play a very important role to obtain the uniform estimate to approximate solutions.
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PROC
Conference Publications 2003, 2003(Special): 798-808 doi: 10.3934/proc.2003.2003.798
In this paper, an evolution dynamical system, which is generated by one-dimensional Frémond models of shape memory alloys, is considered. Assuming that forcing terms converge to some time-independent terms in appropriate senses as time goes to infinity, we shall characterize the asymptotic stability for our dynamical system by the global attractor for the limiting autonomous dynamical system.
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