Advances in Differential Equations

Coupled surface and grain boundary motion: nonclassical traveling-wave solutions

Jacob Kanel, Amy Novick-Cohen, and Arkady Vilenkin

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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 traveling-wave solutions can be determined via solutions of Problem $P_{\Psi}$ in which $$\Psi_{sss}= \sin \Psi, \quad s \in (-\infty, \, 0) \cup (0, \, \infty), \quad \Psi(\pm \infty)=0,$$ and appropriate jump conditions are prescribed at $s=0$. We prove existence of solutions to Problem $P_{\Psi}$ for all $m,$ $0 \le m <2$, where $m$ denotes the ratio of the surface energies of the grain boundary and of the exterior surface. We show numerically that for $\,\approx\! 1.81 < m \le 2$, the corresponding solutions to the original geometric problem are not single-valued 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."

Article information

Adv. Differential Equations Volume 9, Number 3-4 (2004), 299-327.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 74N05: Crystals
Secondary: 34A37: Differential equations with impulses 34B40: Boundary value problems on infinite intervals 35R35: Free boundary problems 74E15: Crystalline structure 74N25: Transformations involving diffusion


Kanel, Jacob; Novick-Cohen, Amy; Vilenkin, Arkady. Coupled surface and grain boundary motion: nonclassical traveling-wave solutions. Adv. Differential Equations 9 (2004), no. 3-4, 299--327.

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