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Advances in Differential Equations
Volume 9, Numbers 3-4, March/April 2004, Pages 299–327
COUPLED SURFACE AND GRAIN BOUNDARY MOTION:
NONCLASSICAL TRAVELING-WAVE SOLUTIONS
Jacob Kanel and Amy Novick-Cohen
Department of Mathematics, Technion-IIT, Haifa 32000, Israel
Arkady Vilenkin
The Racah Institute of Physics, The Hebrew University of Jerusalem
Jerusalem 91904, Israel
(Submitted by: Y. Giga)
In Memory of W.W. Mullins
Abstract. We study a geometric free-boundary problem for a bicrystal
in which a grain boundary is attached at a groove root to the exterior
surface of the bicrystal. Mathematically, this geometric problem couples
motion by mean curvature of the grain boundary with surface diffusion
of the exterior surface. If the groove root effects are localized, it is
realistic to look for traveling-wave solutions. We show that travelingwave solutions can be determined via solutions of Problem PΨ in which
Ψsss = sin Ψ,
s ∈ (−∞, 0) ∪ (0, ∞),
Ψ(±∞) = 0,
and appropriate jump conditions are prescribed at s = 0. We prove
existence of solutions to Problem PΨ for all m, 0 ≤ m < 2, where m
denotes the ratio of the surface energies of the grain boundary and of
the exterior surface. We show numerically that for ≈ 1.81 < m ≤ 2, the
corresponding solutions to the original geometric problem are not singlevalued as functions of x, where x varies along the unperturbed exterior
surface of the bicrystal. We refer to these solutions as “nonclassical
traveling-wave solutions.”
1. Introduction
In the present paper, we analyze traveling-wave solutions for the motion
of a grain boundary which is attached at a grain groove to an exterior surface which evolves under the influence of surface diffusion in the context of
a “quarter loop geometry”; see Figure 1. The quarter loop setting [5, 6] in
Accepted for publication: October 2003.
AMS Subject Classifications: 34A34, 34B10, 35K55, 74N15, 74N20.
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