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Discrete & Continuous Dynamical Systems - S

2011 , Volume 4 , Issue 2

Issue on thermomechanics and phase change
Guest Editors: Alain Miranville and Ulisse Stefanelli

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Preface
Alain Miranville and Ulisse Stefanelli
2011, 4(2): i-ii doi: 10.3934/dcdss.2011.4.2i +[Abstract](249) +[PDF](27.5KB)
Abstract:
Phase transition phenomena arise in a variety of relevant real-world situations. For instance, one may think of melting and freezing in a solid-liquid system, evaporation, metal casting, solid-solid phase transitions, combustion, crystal growth, glass formation, phase transitions in polymers, plasticity, just to mention a few. Phase changes are very often triggered by thermomechanical actions. On the contrary, the overall thermomechanical behavior of a body is clearly influenced by its experienced phase change. Hence, Thermomechanics and phase change are to be regarded as twinned aspects of the same reality description.

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The position of the joint of shape memory alloy and bias springs
Toyohiko Aiki
2011, 4(2): 239-246 doi: 10.3934/dcdss.2011.4.239 +[Abstract](250) +[PDF](134.7KB)
Abstract:
In our previous work we proposed the mathematical model for a device made of the standard spring and the shape memory alloy spring. The model was given by the system of partial differential equations with the dynamic boundary condition. Also, we have proved the existence and the uniquess theorems for the model. The purpose of this paper is to improve the existence theorem.
Global and exponential attractors for a Ginzburg-Landau model of superfluidity
Alessia Berti, Valeria Berti and Ivana Bochicchio
2011, 4(2): 247-271 doi: 10.3934/dcdss.2011.4.247 +[Abstract](278) +[PDF](440.2KB)
Abstract:
The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techinques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.
Long-time behaviour of a thermomechanical model for adhesive contact
Elena Bonetti, Giovanna Bonfanti and Riccarda Rossi
2011, 4(2): 273-309 doi: 10.3934/dcdss.2011.4.273 +[Abstract](277) +[PDF](674.9KB)
Abstract:
This paper deals with the large-time analysis of a PDE system modelling contact with adhesion, in the case when thermal effects are taken into account. The phenomenon of adhesive contact is described in terms of phase transitions for a surface damage model proposed by M. Frémond. Thermal effects are governed by entropy balance laws. The resulting system is highly nonlinear, mainly due to the presence of internal constraints on the physical variables and the coupling of equations written in a domain and on a contact surface. We prove existence of solutions on the whole time interval $(0,+\infty)$ by a double approximation procedure. Hence, we are able to show that solution trajectories admit cluster points which fulfil the stationary problem associated with the evolutionary system, and that in the large-time limit dissipation vanishes.
Anisotropic phase field equations of arbitrary order
G. Caginalp and Emre Esenturk
2011, 4(2): 311-350 doi: 10.3934/dcdss.2011.4.311 +[Abstract](316) +[PDF](673.1KB)
Abstract:
We derive a set of higher order phase field equations using a microscopic interaction Hamiltonian with detailed anisotropy in the interactions of the form $a_{0}+\delta\sum_{n=1}^{N}{a_{n}\cos( 2n\theta) + b_{n}\sin( 2n\theta) }$ where $\theta$ is the angle with respect to a fixed axis, and $\delta$ is a parameter. The Hamiltonian is expanded using complex Fourier series, and leads to a free energy and phase field equation with arbitrarily high order derivatives in the spatial variable. Formal asymptotic analysis is performed on these phase field equation in terms of the interface thickness in order to obtain the interfacial conditions. One can capture $2N$-fold anisotropy by retaining at least $2N^{th}$ degree phase field equation. We derive, in the limit of small $\delta,$ the classical result $( T-T_{E} ) [s]_{E}=-\kappa {\sigma( \theta ) + \sigma^{''}( \theta) }$ where $T-T_{E}$ is the difference between the temperature at the interface and the equilibrium temperature between phases, $[s]_{E}$ is the entropy difference between phases, $\sigma$ is the surface tension and $\kappa$ is the curvature. If there is only one mode in the anisotropy [i.e., the sum contains only one term: $A_{n}\cos( 2n\theta) $] then this identity is exact (valid for any magnitude of $\delta$) if the surface tension is interpreted as the sharp interface limit of excess free energy obtained by the solution of the $2N^{th}$ degree differential equation. The techniques rely on rewriting the sums of derivatives using complex variables and combinatorial identities, and performing formal asymptotic analyses for differential equations of arbitrary order.
Asymptotics of the Coleman-Gurtin model
Mickaël D. Chekroun, Francesco di Plinio, Nathan Glatt-Holtz and Vittorino Pata
2011, 4(2): 351-369 doi: 10.3934/dcdss.2011.4.351 +[Abstract](259) +[PDF](488.2KB)
Abstract:
This paper is concerned with the integrodifferential equation

$\partial_{t} u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\d s + \varphi(u)=f$

arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $\varphi$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.

Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes
M. Hassan Farshbaf-Shaker and Harald Garcke
2011, 4(2): 371-389 doi: 10.3934/dcdss.2011.4.371 +[Abstract](313) +[PDF](422.8KB)
Abstract:
In this paper we derive thermodynamically consistent higher order phase field models for the dynamics of biomembranes in incompressible viscous fluids. We start with basic conservation laws and an appropriate version of the second law of thermodynamics and obtain generalizations of models introduced by Du, Li and Liu [3] and Jamet and Misbah [11]. In particular we derive a stress tensor involving higher order derivatives of the phase field and generalize the classical Korteweg capillarity tensor.
Phase separation in a gravity field
Pavel Krejčí, Elisabetta Rocca and Jürgen Sprekels
2011, 4(2): 391-407 doi: 10.3934/dcdss.2011.4.391 +[Abstract](320) +[PDF](411.5KB)
Abstract:
We prove here well-posedness and convergence to equilibria for the solution trajectories associated with a model for solidification of the liquid content of a rigid container in a gravity field. We observe that the gravity effects, which can be neglected without considerable changes of the process on finite time intervals, have a substantial influence on the long time behavior of the evolution system. Without gravity, we find a temperature interval, in which all phase distributions with a prescribed total liquid content are admissible equilibria, while, under the influence of gravity, the only equilibrium distribution in a connected container consists in two pure phases separated by one plane interface perpendicular to the gravity force.
Variational inequalities for a non-isothermal phase field model
Kota Kumazaki and Masahiro Kubo
2011, 4(2): 409-421 doi: 10.3934/dcdss.2011.4.409 +[Abstract](233) +[PDF](352.4KB)
Abstract:
We study variational inequalities for a non-isothermal phase field model of the Penrose-Fife type. We consider time-dependent constraints for the order parameter and the Dirichlet boundary condition for the temperature.
Complete-damage evolution based on energies and stresses
Alexander Mielke
2011, 4(2): 423-439 doi: 10.3934/dcdss.2011.4.423 +[Abstract](356) +[PDF](470.1KB)
Abstract:
The rate-independent damage model recently developed in [1] allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized $\Gamma$-convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing $\Gamma$-convergence of energetic solutions from partial to complete damage under rather general conditions.
Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm
Irena Pawłow and Wojciech M. Zajączkowski
2011, 4(2): 441-466 doi: 10.3934/dcdss.2011.4.441 +[Abstract](279) +[PDF](447.2KB)
Abstract:
In this paper we study a nonlinear thermoviscoelasticity system within the framework of parabolic theory in anisotropic Sobolev spaces with a mixed norm. The application of such a framework allows to generalize the previous results by admitting stronger thermomechanical nonlinearity and a broader class of solution spaces.
On certain convex compactifications for relaxation in evolution problems
Tomáš Roubíček
2011, 4(2): 467-482 doi: 10.3934/dcdss.2011.4.467 +[Abstract](247) +[PDF](478.2KB)
Abstract:
A general-topological construction of limits of inverse systems is applied to convex compactifications and furthermore to special convex compactifications of Lebesgue-space-valued functions parameterized by time. Application to relaxation of quasistatic evolution in phase-change-type problems is outlined.
Stability analysis for phase field systems associated with crystalline-type energies
Ken Shirakawa
2011, 4(2): 483-504 doi: 10.3934/dcdss.2011.4.483 +[Abstract](258) +[PDF](612.4KB)
Abstract:
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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