Universal bounds on coarsening rates for some models of phase transitions
Universal bounds on coarsening rates for some models of phase transitions
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Date
2005-04-04
Authors
Dai, Shibin
Advisor
Pego, Robert L
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Abstract
In this thesis, we prove one-sided,
universal bounds on coarsening rates for three models of phase transitions
by following
a strategy developed by Kohn and Otto (Comm. Math. Phys. 229(2002),375-395).
Our analysis for the phase-field model is performed in a regime in which
the ratio between the transition layer thickness and the length scale of the
pattern is small, and is also small compared to the square of the ratio between the
pattern scale and the system size. The analysis
extends the Kohn-Otto method to deal with
both temperature and phase fields.
For the mean-field models, we consider two kinds of them: one with a
coarsening rate
$l\sim t^{1/3}$ and the other with $l\sim t^{1/2}$. The $l\sim t^{1/2}$ rate is
proved using a new dissipation relation which extends the Kohn-Otto method. In both
cases, the dissipation relations are subtle and their proofs are based on a
residual lemma (Lagrange identity) for the Cauchy-Schwarz inequality.
The monopole approximation is a simplification of the Mullins-Sekerka model in the
case when all particles are non-overlapping
spheres and the centers of the particles do not move.
We derive the monopole approximation and prove its well-posedness
by considering a gradient flow restricted
on collections of finitely many non-overlapping spheres. After that, we prove
one-sided universal bounds on the coarsening rate for the monopole approximation.