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

Mavoungou, Urbain Cyriaque; Moukoko, Daniel; Langa, Franck Davhys Reval; Ampini, Dieudonne

doi:10.3233/asy-191539

Cited by 3 publications

(1 citation statement)

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“…Although the system was not originally related to the law of thermodynamics, the equations of Caginalp have been extensively investigated in literature, both from an analytical point of view [15,24,27,28], as well as from numerical simulations [2,19]. A variation of this model has been used to study dynamical undercooling at the liquid-solid interface, as well as asymptotically relating to the free boundary model (sharp interface) [11].…”

Section: A Brief Overview Of Related Literaturementioning

confidence: 99%

Temperature dependent extensions of the Cahn-Hilliard equation

Anna¹,

Li²,

Schlömerkemper³

et al. 2021

Preprint

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The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In this paper we focus on the dynamics of these binary media, when the underlying temperature is not constant. The aim of this paper is twofold. We first derive two distinct models that extend the classical Cahn-Hilliard equation with an evolutionary equation for the absolute temperature. Secondly, we analyse the local well-posedness of classical solution for one of these systems. Our modelling introduces the systems of PDEs by means of a general and unified formalism. This formalism couples standard principles of mechanics together with the main laws of thermodynamics. Our work highlights how certain assumptions on the transport of the temperature effect the overall physics of the systems. The variety of these thermodynamically consistent models opens the question of which one should be more appropriate. Our analysis shows that one of the derived models might be more desirable to the well-posedness theory of classical solutions, a property that might be natural as a selection criteria. We conclude our paper with an overview and comparison of our modelling formalism with some equations, which were previously derived in literature.

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Section: A Brief Overview Of Related Literaturementioning

confidence: 99%

Temperature dependent extensions of the Cahn-Hilliard equation

Anna¹,

Li²,

Schlömerkemper³

et al. 2021

Preprint

View full text Add to dashboard Cite

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Dual System Least-Squares Finite Element Method for a Hyperbolic Problem

Lee

2021

Computational Methods in Applied Mathematics

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This study investigates the dual system least-squares finite element method, namely the LL∗ method, for a hyperbolic problem. It mainly considers nonlinear hyperbolic conservation laws and proposes a combination of the LL∗ method and Newton’s iterative method. In addition, the inclusion of a stabilizing term in the discrete LL∗ minimization problem is proposed, which has not been investigated previously. The proposed approach is validated using the one-dimensional Burgers equation, and the numerical results show that this approach is effective in capturing shocks and provides approximations with reduced oscillations in the presence of shocks.

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