Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

Jiang, Jie

doi:10.1002/mma.1092

Cited by 33 publications

(19 citation statements)

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“…Thus, several alternatives to (1.6) have been proposed in order to derive more realistic models: in particular, the Caginalp phase-field system, supplemented with either the Maxwell-Cattaneo law or other constitutive laws for the heat flux coming from thermomechanics, has been studied by several authors, see e.g. [13,38,39,41,42,43,44,46]. Still also the type III heat conduction theory, as well as the classical Fourier one, suffers from some theoretical drawbacks (see [28]) which are overcome by (1.1).…”

Section: Introductionmentioning

confidence: 99%

On a Caginalp Phase-Field System with Two Temperatures and Memory

et al. 2017

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Abstract. The Caginalp phase-field system has been proposed in [4] as a simple mathematical model for phase transition phenomena. In this paper, we are concerned with a generalization of this system based on the Gurtin-Pipkin law with two temperatures for heat conduction with memory, apt to describe transition phenomena in nonsimple materials. The model consists of a parabolic equation governing the order parameter which is linearly coupled with a nonclassical integrodifferential equation ruling the evolution of the thermodynamic temperature of the material. Our aim is to construct a robust family of exponential attractors for the associated semigroup, showing the stability of the system with respect to the collapse of the memory kernel. We also study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist.Mathematics Subject Classification (2010). 35B41, 35K55, 35L05, 80A22.

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Section: Introductionmentioning

confidence: 99%

On a Caginalp Phase-Field System with Two Temperatures and Memory

et al. 2017

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“…Now, one essential drawback of the Fourier law is that it predicts that thermal signals propagate at an infinite speed, which violates causality (the so-called paradox of heat conduction, see [13]). To overcome this drawback, or at least to account for more realistic features, several alternatives to the Fourier law, based, e.g., on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed and studied, in the context of the Caginalp phase-field system, in [29], [30], [33], [34], [35], [36] and [38].…”

Section: Introductionmentioning

confidence: 99%

A Caginalp phase-field system based on type III heat conduction with two temperatures

Miranville¹,

Quintanilla²

2016

Quart. Appl. Math.

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Abstract. Our aim in this paper is to study a generalization of the Caginalp phasefield system based on the theory of type III thermomechanics with two temperatures for the heat conduction. In particular, we obtain well-posedness results and study the dissipativity of the associated solution operators. We consider here both regular and singular nonlinear terms. Furthermore, we endow the equations with two types of boundary conditions, namely, Dirichlet and Neumann. Finally, we study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist.

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“…Now, one essential drawback of the Fourier law is that it predicts that thermal signals propagate at an infinite speed, which violates causality (the so-called paradox of heat conduction, see [17]). To overcome this drawback, or at least to account for more realistic features, several alternatives to the Fourier law, based, e.g., on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed and studied, in the context of the Caginalp phase-field systems, in [28], [29], [32], [34], [35], [36], [37] and [38].…”

Section: Introductionmentioning

confidence: 99%