DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ANALYTIC SOLUTIONS OF UNSTEADY CRYSTALS
AND RAYLEIGH-TAYLOR BUBBLES
XUMING XIE
We study the initial value problem for 2-dimensional dendritic crystal growth with zero
surface tension and classical Rayleigh-Taylor problems. If the initial data is analytic, it
is proved that unique analytic solution exists locally in time. The analysis is based on a
Nirenberg theorem on abstract Cauchy-Kovalevsky problem in properly chosen Banach
spaces.
Copyright © 2006 Xuming Xie. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The phenomenon of dendritic crystal growth is one of the earliest scientific problems,
tackled first by Kepler [4] in 1661 in his work on six-sided snowflake crystals. It has long
been a subject of continued interest to physicists, metallurgists as well as mathematicians.
Dendrite constitutes a good example of pattern selection and stability in nonequilibrium
systems. From mathematical point of view, dendrite formation is a free boundary problem
like the Stefan problem. Many review papers on this subject have appeared in the
literature, for example, Langer [8], Kessler et al. [5], Pelce [13], Levine [9]. For unsteady
dendritical crystal growth, Kunka et al. [6, 7] studied the linear theory of localized disturbances
and a class exact zero-surface-tension solutions if the initial conditions include
only poles. They also studied the singular behavior of unsteady dendritical crystal with
surface tension. In those situations, a zero of the conformal map that describes the crystal
gives birth to a daughter singularity that moves away from the zero and approaches the
interface.
The motion of the interface of a heavy fluid resting above a lighter fluid in the presence
of gravity (Rayleigh-Taylor flow) is very basic but important problem. When the fluids
are immiscible, the sharp interface deforms into a pattern containing rising bubbles of
lighter fluid and falling spikes of heavier fluid. Model equations for the location of the
interface have been derived (see Baker et al. [2], Moore [10], Sharp [16], and references
therein). These studies are numerical and asymptotic, but important to furthering physical
understanding of the flow dynamics. Numerical calculation ran into the traditional
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1149–1157