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Coarsening in an integro-differential model of phase transitions

Published online by Cambridge University Press:  26 February 2001

DUGALD B. DUNCAN ,
MICHAEL GRINFELD  and
IULIAN STOLERIU
DUGALD B. DUNCAN
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland
MICHAEL GRINFELD
Affiliation:
Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland
IULIAN STOLERIU
Affiliation:
Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland
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Abstract

Coarsening of solutions of the integro-differential equation

formula here

where Ω ⊂ ℝn, J(·) [ges ] 0, ε > 0 and f(u) = u3 − u (or similar bistable nonlinear term), is examined, and compared with results for the Allen–Cahn partial differential equation. Both equations are used as models of solid phase transitions. In particular, it is shown that when ε is small enough, solutions of this integro-differential equation do not coarsen, in contrast to those of the Allen–Cahn equation. The special case J(·) ≡ 1 is explored in detail, giving insight into the behaviour in the more general case J(·) [ges ] 0. Also, a numerical approximation method is outlined and used on tests in both one- and two-space dimensions to verify and illustrate the main result.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 11 , Issue 6 , December 2000 , pp. 561 - 572
Copyright
2000 Cambridge University Press

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