Global and exponential attractors for a Caginalp type phase-field problem

Bangola, Brice Doumbé

doi:10.2478/s11533-013-0258-0

Cited by 7 publications

(6 citation statements)

References 25 publications

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“…Let us point out that questions related to the well-posedness, long-time behaviour of solutions and optimal control problems have been investigated for the Caginalp system (1.1)-(1.2) and for some variation or extension of this phase field system. Without any sake of completeness, we mention the contributions [2,10,11,14,18,20,22,25,29,36,37,40] for various qualitative analyses and [5,16,17,27,28] for some related control problems.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Time discretization of a nonlinear phase field system in general domains

Colli¹,

Kurima²

2019

Communications on Pure &Amp; Applied Analysis

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This paper deals with the nonlinear phase field system ∂ t (θ + ℓϕ) − ∆θ = fin Ω × (0, T ),Here N ∈ N, T > 0, ℓ > 0, f is a source term, β is a maximal monotone graph and π is a Lipschitz continuous function. We note that in the above system the nonlinearity β + π 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 Ω 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 (Ω) ֒→ L 2 (Ω) 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.2010 Mathematics Subject Classification: 93C20, 34B15, 35A35, 82B26.

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Section: Introduction and Resultsmentioning

confidence: 99%

Time discretization of a nonlinear phase field system in general domains

Colli¹,

Kurima²

2019

Communications on Pure &Amp; Applied Analysis

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“…In the paper [6], several results have been proven. These include the existence and uniqueness of the solution obtained, respectively, using the Galerkin approximation scheme and Gronwall's lemma (see [11,12]), and the regularity of H 2 (Ω) acquired by Agmon, Holder, and Sobolev injections (see [18,34]).…”

Section: H U Umentioning

confidence: 91%

“…Taking into account (7), ( 8), (9), and (10), the derivation of the model ( 1)-( 4) is easily shown (see [6]).…”

Section: H U Umentioning

confidence: 99%

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Stability of Solutions to a Caginalp Phase-Field Type Equations

Ipopa,

Doumbé Bangola,

Andami Ovono

2024

Contemp. Math.

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This paper is concerned with the study of the asymptotic behavior of a generalization of the Caginalp phase-field model subject to homogeneous Neumann boundary conditions and regular potentials involving two temperatures. This work follows on from a paper in which the well-posedness of the problem, the dissipativity of the system, and the existence of global and exponential attractors were demonstrated. In addition, a study on the semi-infinite cylinder was also carried out. Indeed, if it is true that the existence of a global attractor makes it possible to predict the asymptotic behavior of solutions on a bounded domain, it does not say that these solutions converge. After having shown the existence of the global attractor, it is therefore important to look at the convergence of the solutions over time. There are several methods for determining the asymptotic behavior of the solutions of a differential system. We can mention the one that consists of transforming the given differential equations into integral equations and then applying the classical Picard successive approximation procedure to them. This work is devoted to the study of the convergence of solutions to steady states, adapting a well-known result concerning Lojasiewicz-Simon’s inequality.

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“…In this work, we make the hypothesis of being in a non-simple material. The two temperatures are then linked as follows (see [34], [35]):…”

Section: Introductionmentioning

confidence: 99%

Attractors and Numerical Simulations for a Two-Temperature Phase Transition System

Ipopa

Bangola²,

Ovono³

2022

JAMCS

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In this paper, we study a generalisation of the Caginalp phase field system obtained by considering the Fourier law as the heat conduction law and involving two temperatures. In the first section, we are interested in the existence and uniqueness of solutions. In section 2, we address the question of the asymptotic behaviour of the solutions, in particular with the demonstration of the existence of a global attractor. For this, we needed to show that the system is dissipative and to know the regularity of the solutions. Finally, in the last section, we turn to the numerical solution of our problem.

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Global and exponential attractors for a Caginalp type phase-field problem

Cited by 7 publications

References 25 publications

Time discretization of a nonlinear phase field system in general domains

Time discretization of a nonlinear phase field system in general domains

Stability of Solutions to a Caginalp Phase-Field Type Equations

Attractors and Numerical Simulations for a Two-Temperature Phase Transition System

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