Volume 35, Number 4, July-August 2001
|Page(s)||607 - 630|
|Published online||15 April 2002|
Estimates and Computations for Melting and Solidification Problems
Carnegie Mellon University,
Department of Mathematical Sciences,
Pittsburgh, PA 15213, USA. (email@example.com)
Revised: 25 January 2001
In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, δ, to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky . A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html
Mathematics Subject Classification: 35B25 / 35B40 / 35B45
Key words: Phase-field models / melting and solidification.
© EDP Sciences, SMAI, 2001
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