Dissipative Phase Transitions

The following sections are included:

• PREFACE

• CONTENTS

The purpose of the present paper is to summarize our recent works dealing with hysteresis. Although there are several approaches to study hysteresis operators, here we focus on the characterization of hysteresis operators by using ordinary differential equations. Applying this method, we have made some progress in the fields related to shape memory alloys, solid-liquid phase transitions, magnetic materials and biological models. In this paper, we introduce our mathematical models consisting of partial differential equations and ordinary differential equations, and show theorems concerned with the well-posedness of the models.

The paper deals with a model of phase transitions with the possibility of thermal memory described via an entropy equation, and a generalized version of the principle of virtual power including microscopic movements, which are responsible for the thermomechanical process. We first recall an existence result for the general situation. Hence, we focus our attention on a simplified version of the problem, in which neither the thermal memory nor the diffusive effects in the phase transition are accounted for. After recalling an existence and uniqueness result for the corresponding initial and boundary value problem, we investigate the long-time behaviour of the solution. In particular, we characterize the ω-limit set.

Michel Frémond has recently proposed a new model for phase transition phenomena. When the process is very fast, the power of the acceleration forces cannot be neglected. This fact leads to a nonlinear parabolic-hyperbolic system. The global existence of solutions of the related Cauchy-Neumann problem is still open in the three-dimensional framework. Here we establish the well-posedness in the one-dimensional case with strong dissipation.

The problem we deal with arises in the theory of heat propagation in materials with memory. We consider the identification of both the relaxation kernel and the time dependence of the heat source for an integro–differential equation of parabolic type. We prove an existence and uniqueness theorem, global in time, for the abstract version of the problem and we give an application to the concrete case. The novelties of this work are: the choice of the functional setting (spaces of bounded functions with values in an interpolation space) and the existence and uniqueness results, global in time, which are not easy to obtain for inverse problems.

In this paper we introduce some nonlinear partial differential equations and discuss the solvability of their initial and boundary value problems. These are called the enthalpy formulations of the Stefan problem, which describes the solid-liquid phase transitions. Firstly, we tackle a problem with convection governed by Navier-Stokes equations in the unknown liquid region. The weak solution, which is a limit of an approximate procedure of the solid-liquid interface in some sense, is constructed. Secondly, we introduce a coupled system of Stefan and Navier-Stokes equations. This is a transmission problem between the solid-liquid and gas regions.

We consider the memory relaxation of the one-dimensional Cahn-Hilliard equation, endowed with the no-flux boundary conditions. The resulting integrodifferential equation is characterized by a memory kernel, which is the rescaling of a given positive decreasing function. The Cahn-Hilliard equation is then viewed as the formal limit of the relaxed equation, when the scaling parameter (or relaxation time) ε tends to zero. In particular, if the memory kernel is the decreasing exponential, then the relaxed equation is equivalent to the standard hyperbolic relaxation. The main result of this paper is the existence of a family of robust exponential attractors for the one-parameter dissipative dynamical system generated by the relaxed equation. Such a family is stable with respect to the singular limit ε → 0.

Thermally induced phase transitions in materials with thermal memory are modelled by means of an order parameter, or phase–field, that changes smoothly in a fixed region. In this paper, two thermodynamic approaches are developed by regarding the phase–field evolution equation as a supplementary balance law. In the first one, the kinetic equation is regarded as the balance of some microscopic or accretive forces, and the first law of thermodynamics is modified taking into account their mechanical power. The second approach takes any mechanical meaning off the phase–field evolution, and modifies the second law of thermodynamics through some entropy extra–flux. Restrictions due to thermodynamics are derived and a general form of the kinetic equation is obtained in both cases. A simplified model ruling the system evolution around the transition temperature is derived according to the first approach.

We study a hyperbolic equation of first order with a hysteresis nonlinearity. For the integral solution of this equation we derive an entropy condition of the type introduced by Kružkov.

In this paper, we approximate an identification problem for a partial integro-differential system, related to Caginalp's model for phase transitions, when the data are affected by a known error δ. Since such a problem is weakly ill–posed, we regularize it by a family of well-posed identification problems, depending on a small parameter ε. We prove that, as δ and ε accordingly tend to 0, the regularized solution converges to the exact one. We also compute the conditioned convergence rate.

In this article we consider a nonlinear large reaction small diffusion problem which has two (or more) stable states, and we analyse it using two different methods.

-First method: an approach based on (asymptotic) expansions;

-and second method: an approach based on the notion of Γ-convergence.

The analysis of such a problem shows that the two methods are complementary. It is well-known that, for such a problem, time-dependent solutions are characterised by (moving) layers or vortices. Here, we are specially interested in the existence, the shape and the motion of such layers or vortices, with respect to the inhomogeneous coefficients appearing in the problem as well as the domain Ω. We generalise, in the limit of small diffusion, the usual motion by mean curvature laws found for homogeneous problems.

This paper presents a new existence result for a three-dimensional (3-D) shape memory model which has the form of a nonlinear thermoelasticity system, with viscosity ν > 0 and capillarity ϰ > 0. In contrast to the authors' previous results, proved under the assumption , here we admit ϰ > 0 and ν > 0 possibly arbitrarily small. Thus, the existence result obtained becomes more adequate for shape memory problems where viscosity effects are negligible small. Moreover, we consider a broader class of boundary conditions. The main new part of the present paper is given by the solvability analysis of the initial-boundary-value problem for the viscoelasticity system with capillarity.

This note addresses the analysis of a thermoviscoelastic model with a temperature-dependent viscous modulus. We provide the well-posedness of a related initial and boundary value problem and detail a suitable fully implicit variable time-step discretization. The latter is proved to be conditionally stable and convergent. Moreover, some a priori error estimates of optimal order are established.

We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase separation in mixtures and alloys. Specifically, we prove the existence of (a suitable notion of) the global attractor for the weak solutions of the so-called generalized viscous Cahn-Hilliard equation.

In this paper, we shall consider a mathematical model of solid-liquid phase transition, which is described as a coupled system of two evolution equations. Here, the first equation is a kinetic equation of heat exchanges, and the second one is a type of Allen-Cahn equation, involving a spatially indefinite surface tension coefficient. Recently, in a known setting of the temperature, the author of [14] has studied the geometry of pattern formations which give some stability in the dynamical system generated by the single Allen-Cahn equation. In the main theorem of this paper, the result of [14] will be somehow extended to the framework of our model, in which the temperature is unknown.

This note gives an informal presentation of a few recent results obtained by some (relatively) young researchers. We will not go into the full details of the (quite technical) proofs here; rather, we will try to highlight some ideas in a concrete case. The reader willing to go into the matter more thoroughly may look at [43, 39, 35, 36, 40], and [46].