Differential and Integral Equations

On a doubly nonlinear phase-field model for first-order transitions with memory

V. Berti, M. Fabrizio, and C. Giorgi

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Abstract

Solid-liquid transitions in thermal insulators and weakly conducting media are modeled through a phase-field system with memory. The evolution of the phase variable ${\varphi}$ is ruled by a balance law which takes the form of a Ginzburg-Landau equation. A thermodynamic approach is developed starting from a special form of the internal energy and a nonlinear hereditary heat conduction flow of Coleman-Gurtin type. After some approximation of the energy balance, the absolute temperature ${\theta}$ obeys a doubly nonlinear ``heat equation" where a third-order nonlinearity in ${\varphi}$ appears in place of the (customarily constant) latent-heat. The related initial and boundary value problem is then formulated in a suitable setting and its well--posedness and stability is proved.

Article information

Source
Differential Integral Equations, Volume 21, Number 3-4 (2008), 325-350.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038783

Mathematical Reviews number (MathSciNet)
MR2484012

Zentralblatt MATH identifier
1224.35029

Subjects
Primary: 35G30: Boundary value problems for nonlinear higher-order equations
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 80A22: Stefan problems, phase changes, etc. [See also 74Nxx]

Citation

Berti, V.; Fabrizio, M.; Giorgi, C. On a doubly nonlinear phase-field model for first-order transitions with memory. Differential Integral Equations 21 (2008), no. 3-4, 325--350. https://projecteuclid.org/euclid.die/1356038783


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