On the Caginalp Phase-Field System Based on Type Ⅲ With Two Temperatures and Nonlinear Coupling

Ovono, Armel Andami; Bangola, Brice Doumbé; Ipopa, Mohamed Ali

doi:10.11948/2019.345

Cited by 4 publications

(2 citation statements)

References 38 publications

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“…For this reason, several authors have proposed alternative laws for heat flow that overcome this drawback. We can mention, among others, the Maxwell-Cattaneo law, the Green and Naghdi law, and other constitutive laws for heat flux coming from thermomechanics; see, e.g., [20,[23][24][25][26][27][28][29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

Stability of Solutions to a Caginalp Phase-Field Type Equations

Ipopa,

Doumbé Bangola,

Andami Ovono

2024

Contemp. Math.

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“…Several alternatives to Fourier's law have been explored. We can mention for example the Maxwell-Cattaneo law or laws from thermomechanics (see, for example, [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] and [29]).…”

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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Asymptotic behaviour of a phase field system derived from a generalization of Maxwell–Cattaneo’s law with a singular potential

Landry Doumbé Bangola,

Ali Ipopa,

Andami Ovono

2024

J. Phys. A: Math. Theor.

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In this paper, we study the asymptotic behaviour of a Caginalp-type phase ﬁeld system derived from a heat conduction law which is a generalization of the Maxwell-Cattano law and whose potential is singular. This type of law has the advantage of correcting the paradox of heat conduction that appears when the Fourier law is considered. The potential considered is typically logarithmic. Using such a potential makes the model much more relevant from a physical point of view. However, from a theoretical point of view, it is essential to obtain the property of strict separation of the order parameter in order to make sense of the equations. We ﬁrst prove the existence and uniqueness of the solution thanks to the separation property. We also address the question of the dissipativity of the system. Finally, we obtain the existence of the global attractor.

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On the Caginalp Phase-Field System Based on Type Ⅲ With Two Temperatures and Nonlinear Coupling

Cited by 4 publications

References 38 publications

Stability of Solutions to a Caginalp Phase-Field Type Equations

Stability of Solutions to a Caginalp Phase-Field Type Equations

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

Asymptotic behaviour of a phase field system derived from a generalization of Maxwell–Cattaneo’s law with a singular potential

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