Phase-field System Research Articles

Time discretization of an abstract problem from linearized equations of a coupled sound and heat flow

Recently, a time discretization of simultaneous abstract evolution equations applied to parabolic-hyperbolic phase-field systems has been studied. This article focuses on a time discretization of an abstract problem that has application to linearized equations of coupled sound and heat flow. As examples, we also study some parabolic-hyperbolic phase-field systems.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/96/abstr.html

A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions

<p style="text-indent:20px;">We propose a time semi-discrete scheme for the Caginalp phase-field system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution is shown to converge to the energy solution of the problem as the time step tends to <inline-formula><tex-math id="M1">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. The proof involves a multivalued operator and a monotonicity argument. This approach allows us to compute numerically singular solutions to the problem.

Sliding Mode Control for a Generalization of the Caginalp Phase-Field System

In the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution $\varphi^*$. We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time.

Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems

This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped wave equation for the order parameter (see e.g., Grasselli and Pata [Adv. Math. Sci. Appl. 13 (2003) 443–459, Comm. Pure Appl. Anal. 3 (2004) 849–881], Grasselli et al. [Comm. Pure Appl. Anal. 5 (2006) 827–838], Wu et al. [Math. Models Methods Appl. Sci. 17 (2007) 125–153, J. Math. Anal. Appl. 329 (2007) 948–976]). On the other hand, a time discretization of an initial value problem for an abstract evolution equation has been studied (see e.g., Colli and Favini [Int. J. Math. Math. Sci. 19 (1996) 481–494]) and Schimperna [J. Differ. Equ. 164 (2000) 395–430] has established existence of solutions to an abstract problem applying to a nonlinear phase-field system of Caginalp type on a bounded domain by employing a time discretization scheme. In this paper we focus on a time discretization of a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems. Moreover, we can establish an error estimate for the difference between continuous and discrete solutions.

On the Conserved Caginalp Phase-Field System with Logarithmic Potentials Based on the Maxwell–Cattaneo Law with Two Temperatures

Our aim in this article is to study generalizations of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the associated system. In particular, we prove the existence of the global attractor and prove the strict separation to the pure phases in two space dimensions. Furthermore, we give some numerical simulations, obtained with the FreeFem++ software [23], comparing the conserved Caginalp phase-field type model with regular and with logarithmic nonlinear terms.

Boundary controllability of a one-dimensional phase-field system with one control force

In this paper, we present some controllability results for linear and nonlinear phase-field systems of Caginalp type considered in a bounded interval of R when the scalar control force acts on the temperature equation of the system by means of the Dirichlet condition on one of the endpoints of the interval. In order to prove the linear result we use the moment method providing an estimate of the cost of fast controls. Using this estimate and following the methodology developed in [19], we prove a local exact boundary controllability result to constant trajectories of the nonlinear phase-field system. To the authors' knowledge, this is the first nonlinear boundary controllability result in the framework of non-scalar parabolic systems, framework in which some “hyperbolic” behaviors could arise.

Vanishing parameter for an optimal control problem modeling tumor growth

A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn-Hilliard equation for the phase variable (relative concentration of the tumor), coupled with a reaction-diffusion equation for the nutrient. The cost functional is of standard tracking-type and the control variable models the intensity with which it is possible to dispense a medication. The model we deal with presents two small and positive parameters which are introduced in previous contributions as relaxation terms. Here, starting from the already investigated optimal control problem for the relaxed model, we aim at confirming the existence of optimal control and characterizing the first-order optimality condition, via asymptotic schemes, when one of the two occurring parameters goes to zero.

Long time behavior of higher-order anisotropic perturbed phase field system with regular potentials

Long time behavior of higher-order anisotropic perturbed phase field system with regular potentials

Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard type equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, further assumptions have to be required. Mainly, we ought to impose some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide existence of optimal controls and a necessary condition that an optimal control has to satisfy has been characterized as well.

Study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

In this paper, we study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms, with initial and homogenous Dirichlet boundary condition.

Existence and uniqueness solution for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

Existence and uniqueness solution for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fr\'echet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

A Distributed Control Problem for a Fractional Tumor Growth Model

In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach

In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn–Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into account a non-smooth singular nonlinearity, namely the double obstacle potential. Due to its non-smoothness behavior, the standard procedure to characterize the necessary conditions for the optimality cannot be performed. Therefore, we follow a different strategy which in the literature is known as the 'deep quench' approach in order to obtain some optimality conditions that have to be interpreted in a more general framework. We establish the existence of optimal controls and some first-order optimality conditions for the system are derived by employing suitable approximation schemes.

Well-posedness, regularity and asymptotic analyses for a fractional phase field system

This paper is concerned with a non-conserved phase field system of Caginalp type in which the main operators are fractional versions of two fixed linear operators $A$ and $B$. The operators $A$ and $B$ are supposed to be densely defined, unbounded, self-adjoint, monotone in the Hilbert space $L^2(\Omega)$, for some bounded and smooth domain $\Omega$, and have compact resolvents. Our definition of the fractional powers of operators uses the approach via spectral theory. A nonlinearity of double-well type occurs in the phase equation and either a regular or logarithmic potential, as well as a non-differentiable potential involving an indicator function, is admitted in our approach. We show general well-posedness and regularity results, extending the corresponding results that are known for the non-fractional elliptic operators with zero Neumann conditions or other boundary conditions like Dirichlet or Robin ones. Then, we investigate the longtime behavior of the system, by fully characterizing every element of the $\omega$-limit as a stationary solution. In the final part of the paper we study the asymptotic behavior of the system as the parameter $\sigma$ appearing in the operator $B^{2\sigma}$ that plays in the phase equation decreasingly tends to zero. We can prove convergence to a phase relaxation problem at the limit, in which an additional term containing the projection of the phase variable on the kernel of $B$ appears.

Stabilisation of a linearised Cahn–Hilliard system for phase separation by proportional boundary feedbacks

ABSTRACT This work represents a first contribution on the problem of boundary stabilisation for the phase field system of Cahn–Hilliard type, which models the phase separation in a binary mixture. The feedback controller we design here is with actuation only on the temperature flow of the system, on one part of the boundary only. Moreover, it is of proportional type, given in an explicit form, expressed only in terms of the eigenfunctions of the Laplace operator, being easy to manipulate from the computational point of view. Furthermore, it ensures that the closed loop nonlinear system exponentially reaches the prescribed stationary solution provided that the initial datum is close enough to it.

The yield effect in viscoplastic materials. A mathematical model

We propose a phase field system that describes an elastic–plastic transition in which we observe a performance force linked to a stress that has a permanent deformation. This yield behavior is described by a nonlinear fractional model, and its thermodynamic compatibility is demonstrated. Finally, some simulations are presented describing the phase diagram of deformation and deformation cycles.

Time discretization of a nonlinear phase field system in general domains

This paper deals with the nonlinear phase field system \begin{document}$ \begin{equation*} \begin{cases} \partial_t (\theta +\ell \varphi) - \Delta\theta = f &amp; \mbox{in}\ \Omega\times(0, T), \\ \partial_t \varphi - \Delta\varphi + \xi + \pi(\varphi) = \ell \theta,\ \xi\in\beta(\varphi) &amp; \mbox{in}\ \Omega\times(0, T) \end{cases} \end{equation*} $\end{document} in a general domain $ \Omega\subseteq\mathbb{R}^{d} $. Here $ d \in \mathbb{N} $, $ T&gt;0 $, $ \ell&gt;0 $, $ f $ is a source term, $ \beta $ is a maximal monotone graph and $ \pi $ is a Lipschitz continuous function. We note that in the above system the nonlinearity $ \beta+\pi $ replaces the derivative of a potential of double well type. Thus it turns out that the system is a generalization of the Caginalp phase field model and it has been studied by many authors in the case that $ \Omega $ is a bounded domain. However, for unbounded domains the analysis of the system seems to be at an early stage. In this paper we study the existence of solutions by employing a time discretization scheme and passing to the limit as the time step $ h $ goes to $ 0 $. In the limit procedure we face with the difficulty that the embedding $ H^1(\Omega) \hookrightarrow L^2(\Omega) $ is not compact in the case of unbounded domains. Moreover, we can prove an interesting error estimate of order $ h^{1/2} $ for the difference between continuous and discrete solutions.

On higher-order anisotropic perturbed Caginalp phase field systems

Our aim in this paper is to study the existence and uniqueness of solution for hyperbolic relaxations of higher-order anisotropic Caginalp phase field systems with homogeous Dirichlet boundary conditions with regular potentials.

ON THE CAGINALP PHASE-FIELD SYSTEM BASED ON TYPE Ⅲ WITH TWO TEMPERATURES AND NONLINEAR COUPLING

This paper is devoted to the study of a generalization of the Caginalp phase-field system based on the theory of type Ⅲ thermomechanics with two temperatures for the heat conduction with a nonlinear coupling term. We start our analysis by establishing existence of the solutions. Then, we discuss dissipativity and uniqueness of the solutions. We finish our analysis by studying the spatial behavior of the solutions in a semi-infinite cylinder, assuming the existence of such solutions.