# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2006 , Volume 15 , Issue 4

Special Issue
Mathematical Problems in Phase Transitions

Guest Editors:
A. Miranville, H. M. Yin and R. Showalter

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2006, 15(4): i-ii doi: 10.3934/dcds.2006.15.4i +[Abstract](121) +[PDF](28.8KB)
Abstract:
Phase transition phenomena are often encountered in real world situations and technological applications. Examples include solidification in complex alloys, melting, freezing or evaporation in food processing, glass formation and polymer crystallization in industrial applications. The modeling and analysis of problems involving such phenomena have attracted considerable attention in the scientific community over the past decades.
This special issue is an expansion from the papers presented at the special session "Mathematical Methods and Models in Phase Transitions" at the Fifth AIMS International Conference on Dynamical Systems and Differential Equations held at California State University at Pomona from June 17-21, 2004. This special session was organized by A. Miranville, R. Showalter and H.M. Yin. The papers presented at that conference have been supplemented with invited contributions from specialists. These papers include problems arising from industry and numerical analysis and computational issues arising in the simulation of solutions.

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2006, 15(4): 1017-1034 doi: 10.3934/dcds.2006.15.1017 +[Abstract](174) +[PDF](245.4KB)
Abstract:
We propose a phase field model that approximates its limiting sharp interface model (free boundary problem) up to second order in interface thickness. A broad range of double-well potentials can be utilized so long as the dynamical coefficient in the phase equation is adjusted appropriately. This model thereby assures that computation with particular value of interface thickness $\varepsilon$, will differ at most by $O(\varepsilon^2$) from the limiting sharp interface problem. As an illustration, the speed of a traveling wave of the phase field model is asymptotically expanded to demonstrate that it differs from the speed of the traveling wave of the limit problem by $O(\varepsilon^2)$.
2006, 15(4): 1035-1047 doi: 10.3934/dcds.2006.15.1035 +[Abstract](186) +[PDF](158.1KB)
Abstract:
Rapid solidification of a non-dilute binary alloy is studied using a phase-field model with a general formulation for different diffusion coefficients of the two alloy components. For high solidification velocities, we observe the effect of solute trapping in our simulations leading to the incorporation of solute into the growing solid at a composition significantly different from the predicted equilibrium value according to the phase diagram. The partition coefficient tends to unity and the concentration change across the interface progressively reduces as the solidification rate increases. For non-dilute binary alloys with a value of the partition coefficient close to unity, analytical solutions of the phase-field and of the concentration profiles are found in terms of power series expansions taking into account different diffusion coefficients of the alloy components. A new relation for the velocity dependence of the nonequilibrium partition coefficient $k(V)$ is derived and compared with predictions of continuous growth model by Aziz and Kaplan [1]. As a major result for applications, we obtain a steeper profile of the nonequilibrium partition coefficient in the rapid solidification regime for $V/V_D>1$ than previous sharp and diffuse interface models which is in better accordance with experimental measurements (e.g. [2]).
2006, 15(4): 1049-1078 doi: 10.3934/dcds.2006.15.1049 +[Abstract](177) +[PDF](704.3KB)
Abstract:
This paper studies spinodal decomposition in the Cahn-Hilliard model on the unit disk. It has previously been shown that starting at initial conditions near a homogeneous equilibrium on a rectangular domain, solutions to the linearized and the nonlinear Cahn-Hilliard equation behave indistinguishably up to large distances from the homogeneous state. In this paper we demonstrate how these results can be extended to nonrectangular domains. Particular emphasis is put on the case of the unit disk, for which interesting new phenomena can be observed. Our proof is based on vector-valued extensions of probabilistic methods used in Wanner [37]. These are the first results of this kind for domains more general than rectangular.
2006, 15(4): 1079-1088 doi: 10.3934/dcds.2006.15.1079 +[Abstract](201) +[PDF](218.0KB)
Abstract:
We consider a one-dimensional reaction-diffusion type equation with memory, originally proposed by W.E. Olmstead et al. to model the velocity $u$ of certain viscoelastic fluids. More precisely, the usual diffusion term $u_{x x}$ is replaced by a convolution integral of the form $\int_0^\infty k(s) u_{x x}(t-s)ds$, whereas the reaction term is the derivative of a double-well potential. We first reformulate the equation, endowed with homogeneous Dirichlet boundary conditions, by introducing the integrated past history of $u$. Then we replace $k$ with a time-rescaled kernel $k_\varepsilon$, where $\varepsilon>0$ is the relaxation time. The obtained initial and boundary value problem generates a strongly continuous semigroup $S_\varepsilon(t)$ on a suitable phase-space. The main result of this work is the existence of the global attractor for $S_\varepsilon(t)$, provided that $\varepsilon$ is small enough.
2006, 15(4): 1089-1117 doi: 10.3934/dcds.2006.15.1089 +[Abstract](182) +[PDF](321.5KB)
Abstract:
In the present paper we treat the system

(PFM) $u_t + \frac{l}{2} \phi_t =\int_{0}^t a_1(t-s)$Δ$u(s) ds$,
$\tau \phi_t = \int_{0}^t a_2(t-s)[\xi^2$Δ$\phi + \frac{1}{\eta}(\phi - \phi^3) + u](s) ds$,

for $(x, t) \in \Omega \times (0, T)$, $0 < T < \infty$, with the boundary conditions

n $\cdot \nabla u$= n $\cdot \nabla \phi=0, (x, t) \in \partial\Omega \times (0, T)$,

and initial conditions $u(x, 0)=u_0(x)$, $\phi(x, 0)=\phi_0(x)$, $x \in \Omega$, which was proposed in [36] to model phase transitions taking place in the presence of memory effects which arise as a result of slowly relaxing internal degrees of freedom, although in [36] the effects of past history were also included. This system has been shown to exhibit some intriguing effects such as grains which appear to rotate as they shrink [36]. Here the set of steady states of (PFM) and of an associated classical phase field model are shown to be the same. Moreover, under the assumption that $a_1$ and $a_2$ are both proportional to a kernel of positive type, the index of instability and the number of unstable modes for any given stationary state of the two systems can be compared and spectral instability is seen to imply instability. By suitably restricting further the memory kernels, the (weak) $\omega-$limit set of any initial condition can be shown to contain only steady states and linear stability can be shown to imply nonlinear stability.

2006, 15(4): 1119-1135 doi: 10.3934/dcds.2006.15.1119 +[Abstract](216) +[PDF](201.9KB)
Abstract:
A phase-field system, non-local in space and non-smooth in time, with heat flux proportional to the gradient of the inverse temperature, is shown to admit a unique strong thermodynamically consistent solution on the whole time axis. The temperature remains globally bounded both from above and from below, and its space gradient as well as the time derivative of the order parameter asymptotically vanish in $L^2$-norm as time tends to infinity.
2006, 15(4): 1137-1153 doi: 10.3934/dcds.2006.15.1137 +[Abstract](178) +[PDF](213.6KB)
Abstract:
We state an alternative for paths of equilibria of the Cahn-Hilliard equation on the square, bifurcating from the trivial solution at eigenfunctions of the form $w_{ij}=\cos(\pi ix)\cos(\pi j y)$, for $i,j \in \N$. We show that the paths either only connect the bifurcation point $m_{ij}$ with $-m_{ij}$ and are separated from all other paths with even more symmetry, or they contain a loop of nontrivial solutions connecting the bifurcation point $m_{ij}$ with itself. In any case the continua emerging at $m_{ij}$ and $-m_{ij}$ are equal. For fixed mass $m_0=0$ we furthermore prove that the continua bifurcating from the trivial solution at eigenfunctions of the form $w_{i0}+w_{0i}$ or $w_{ij}$, for $i,j \in \N$ are smooth curves parameterized over the interaction length related parameter $\lambda$.
2006, 15(4): 1155-1168 doi: 10.3934/dcds.2006.15.1155 +[Abstract](143) +[PDF](196.9KB)
Abstract:
In this paper we study a free boundary problem modeling a phase-change process by using microwave heating. The mathematical model consists of Maxwell's equations coupled with nonlinear heat conduction with a phase-change. The enthalpy form is used to characterize the phase-change process in the model. It is shown that the problem has a global solution.
2006, 15(4): 1169-1191 doi: 10.3934/dcds.2006.15.1169 +[Abstract](214) +[PDF](226.9KB)
Abstract:
The goal of this paper is to derive again the generalized Cahn-Hilliard and Allen-Cahn models in deformable continua introduced previously by E. Fried and M. E. Gurtin on the basis of a microforce balance. We use a~different approach based on the second law in the form of the entropy principle according to I. Müller and I. S. Liu which leads to the evaluation of the entropy inequality with multipliers.
Both approaches provide the same systems of field equations. In particular, our differential equation for the multiplier associated with the balance law for the order parameter turns out to be identical with the Fried-Gurtin microforce balance.
2006, 15(4): 1193-1214 doi: 10.3934/dcds.2006.15.1193 +[Abstract](208) +[PDF](279.4KB)
Abstract:
A singular nonlinear parabolic-hyperbolic PDE's system describing the evolution of a material subject to a phase transition is considered. The goal of the present paper is to analyze the asymptotic behaviour of the associated dynamical system from the point of view of global attractors. The physical variables involved in the process are the absolute temperature $\vartheta$ (whose evolution is governed by a parabolic singular equation coming from the Penrose-Fife theory) and the order parameter $\chi$ (whose evolution is ruled by a nonlinear damped hyperbolic relation coming from a hyperbolic relaxation of the Allen-Cahn equation). Dissipativity of the system and the existence of a global attractor are proved. Due to questions of regularity, the one space dimensional case (1D) and the 2D - 3D cases require different sets of hypotheses and have to be settled in slightly different functional spaces.
2006, 15(4): 1215-1236 doi: 10.3934/dcds.2006.15.1215 +[Abstract](184) +[PDF](274.6KB)
Abstract:
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.
2006, 15(4): 1237-1252 doi: 10.3934/dcds.2006.15.1237 +[Abstract](207) +[PDF](2515.1KB)
Abstract:
A mathematical model is constructed for the modelling of two dimensional thermo-mechanical behavior of shape memory alloy patches. The model is constructed on the basis of a modified Landau-Ginzburg theory and includes the coupling effect between thermal and mechanical fields. The free energy functional for the model is exemplified for the square to rectangular transformations. The model, based on nonlinear coupled partial differential equations, is reduced to a system of differential-algebraic equations and the backward differentiation methodology is used for its numerical analysis. Computational experiments with representative distributed mechanical loadings are carried out for patches of different sizes to analyze thermo-mechanical waves, coupling effects, and 2D phase transformations.

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