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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Maximum norms of random sums and transient pattern formation


Author: Thomas Wanner
Journal: Trans. Amer. Math. Soc. 356 (2004), 2251-2279
MSC (2000): Primary 35K35, 35B05, 42A61
Published electronically: October 8, 2003
MathSciNet review: 2048517
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Abstract | References | Similar Articles | Additional Information

Abstract: Many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes. In this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the Cahn-Hilliard equation. This model depends on a small parameter, and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero. Recent results on spinodal decomposition have produced such lower bounds. Unfortunately, for higher-dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments. The bounds exhibit a dependence on the dimension of the domain, which from a theoretical point of view seemed unavoidable, but which could not be observed in practice.

In this paper we resolve this apparent paradox. By employing probabilistic methods, we can improve the lower bounds for certain domains and remove the dimension dependence. We thereby obtain optimal results which close the gap between analytical methods and numerical observations, and provide more insight into the nature of the decomposition process. We also indicate how our results can be adapted to other situations.


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Additional Information

Thomas Wanner
Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
Email: wanner@math.gmu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03480-9
PII: S 0002-9947(03)03480-9
Keywords: Spinodal decomposition, Cahn-Hilliard equation, pattern formation, probabilistic aspects, random sums of functions
Received by editor(s): September 3, 2002
Published electronically: October 8, 2003
Article copyright: © Copyright 2003 American Mathematical Society