Phase-field System Research Articles

On a phase-field system based on the Cattaneo law

Our aim in this paper is to study a generalization of the Caginalp phase-field system based on the Maxwell–Cattaneo law for heat conduction and endowed with Neumann boundary conditions. In particular, we obtain well-posedness results and study the dissipativity of the associated solution operators. We also prove, when the enthalpy is conserved, the existence of the global attractor. We finally study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist and have a proper (spatial) decay at infinity.

Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions

This paper deals with the longtime behavior of the Caginalp phase-field systemwith coupled dynamic boundary conditions on both state variables.We prove that the system generates a dissipative semigroup in a suitable phase-spaceand possesses the finite-dimensional smooth global attractor and an exponential attractor.

Regularity and separation from potential barriers for a non-local phase-field system

We show that solutions of a two-phase model involving a non-local interactive term become more regular immediately after the moment they separate from the pure phases. This result allows us to prove stronger convergence to equilibria. A new proof of the separation property is also given.

A type III phase-field system with a logarithmic potential

Our aim in this note is to study the existence and uniqueness of solutions to a phase-field system, based on type III heat conduction, with a logarithmic potential. The main difficulty is to prove that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.

Stability analysis for phase field systems associated with crystalline-type energies

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.

Convergence of solutions of a non-local phase-field system

We show that solutions of a two-phase model involving a non-local interactive term separate from the pure phases from a certain time on, even if this is not the case initially. This result allows us to apply a generalized Lojasiewicz-Simon theorem and to establish the convergence of solutions to asingle stationary state as time goes to infinity.

A Product Formula Approach to a Nonhomogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System

In this paper we prove the convergence of an iterative scheme of fractional steps type for a non-homogeneous Cauchy-Neumann boundary optimal control problem governed by non-linear phase-field system, when the boundary control is dependent both on time and spatial variables. Moreover, necessary optimality conditions are established for the approximating process. The advantage of such approach leads to a numerical algorithm in order to approximate the original optimal control problem.

Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows

Numerical approximations of Cahn-Hilliard phase-field model for the two-phase incompressible flows are considered in this paper. Several efficient and energy stable time discretization schemes for the coupled nonlinear Cahn-Hilliard phase-field system for both the matched density case and the variable density case are constructed, and are shown to satisfy discrete energy laws which are analogous to the continuous energy laws.

A Phase-Field Model Based on a Three-Phase-Lag Heat Conduction

Our aim in this article is to study a phase-field system based on a three-phase-lag for the thermal flux vector. In particular, we prove the existence and uniqueness of solutions and then study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist.

On structures in instability zones

Two mathematical crystallization models describing structure formations in instability zones are proposed and justified. The first model, based on a phase field system, describes crystallization processes in binary alloys. The second model, based on a modified Biot model of a porous medium and the convective Cahn-Hilliard model, governs oriented crystallization. Physical interpretation and numerical analysis are discussed. Bibliography: 23 titles. Illustration: 5 figures.

Solidification and Structuresation of Instability Zones

Two mathematical crystallization models describing structure formations in instability zones are proposed and justified. The first model, based on a phase field system, describes crystallization processes in binary alloys. The second model, based on a modified Biot model of a porous medium and the convective Cahn–Hilliard model, governs oriented crystallization. Physical interpretation and numerical analysis are discussed.

The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials

We study in this paper the well-posedness and the asymptotic behavior, in terms of global attractors, of the Caginalp system with coupled dynamic boundary conditions and possibly singular potentials (e.g., of logarithmic type).

Reconstruction of two time independent coefficients in an inverse problem for a phase field system

In this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain Ω ⊂ R n for the dimension n ≤ 3 with a single observation on a subdomain ω ⋐ Ω and the Sobolev norm of certain partial derivatives of the solutions at a fixed positive time θ ∈ ( 0 , T ) over the whole spatial domain. The proof of these results relies on an appropriate Carleman estimate for the phase field system.

Phase-field systems with nonlinear coupling and dynamic boundary conditions

We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter fulfills a dynamic boundary condition, while the (relative) temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend several results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to a single equilibrium.

A Caginalp phase-field system with a nonlinear coupling

Our aim in this article is to study a Caginalp phase-field system based on a thermomechanical theory of deformable continua proposed by Green and Naghdi (called type III thermoelasticity) and with a nonlinear coupling. In particular, we prove the existence and uniqueness of solutions and the dissipativity of the associated semigroup. We then study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist.

Some generalizations of the Caginalp phase-field system

Our aim in this article is to study generalizations of the Caginalp phase-field system based on a thermomechanical theory of deformable continua proposed by Green and Naghdi. In particular, we obtain well-posedness results and study the asymptotic and spatial behaviours of the solutions.

Fractional Steps Scheme to Approximate the Phase-Field Transition System with Non-Homogeneous Cauchy–Neumann Boundary Conditions

The paper presents a proof of the convergence for an iterative scheme of fractional steps type associated to the phase-field transition system (a nonlinear parabolic system) with non-homogeneous Cauchy–Neumann boundary conditions. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of a nonlinear parabolic system. On the basis of this approach, a numerical algorithm in the two dimensional case is introduced and an industrial implementation is made.

On the well-posedness of the prediction-control problem for certain systems of equations

We consider the inverse problem for equations of Sobolev type and their applications to linearized Navier-Stokes systems and phase-field systems. We obtain conditions for the well-defined solvability of these systems.

Inverse problems for the phase field system with one observation

First we establish a Carleman estimate with a single observation acting on a subdomain for the phase field system in a bounded domain Then this estimate is successfully used along with certain energy estimates to obtain the stability result for the inverse problem consisting of retrieving a smooth diffusion coefficient in the given system for the dimension

On the Caginalp system with dynamic boundary conditions and singular potentials

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H 2, the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.