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Weak solutions of a phase‐field model for phase change of an alloy with thermal properties
Author(s) -
Boldrini José Luiz,
Planas Gabriela
Publication year - 2002
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.334
Subject(s) - mathematics , degenerate energy levels , phase field models , partial differential equation , phase transition , field (mathematics) , phase (matter) , boundary value problem , parabolic partial differential equation , work (physics) , mathematical analysis , function (biology) , stefan problem , phase boundary , alloy , thermodynamics , boundary (topology) , pure mathematics , materials science , physics , quantum mechanics , evolutionary biology , biology , composite material
Abstract The phase‐field method provides a mathematical description for free‐boundary problems associated to physical processes with phase transitions. It postulates the existence of a function, called the phase‐field, whose value identifies the phase at a particular point in space and time. The method is particularly suitable for cases with complex growth structures occurring during phase transitions. The mathematical model studied in this work describes the solidification process occurring in a binary alloy with temperature‐dependent properties. It is based on a highly non‐linear degenerate parabolic system of partial differential equations with three independent variables: phase‐field, solute concentration and temperature. Existence of weak solutions for this system is obtained via the introduction of a regularized problem, followed by the derivation of suitable estimates and the application of compactness arguments. Copyright © 2002 John Wiley & Sons, Ltd.