Stability of travelling wave solutions for coupled surface and grain boundary motion |
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Authors: | Margaret Beck |
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Institution: | a Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA b Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore c Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2 |
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Abstract: | We investigate the spectral stability of the travelling wave solution for the coupled motion of a free surface and grain boundary that arises in materials science. In this problem a grain boundary, which separates two materials that are identical except for their crystalline orientation, evolves according to mean curvature. At a triple junction, this boundary meets the free surfaces of the two crystals, which move according to surface diffusion. The model is known to possess a unique travelling wave solution. We study the linearization about the wave, which necessarily includes a free boundary at the location of the triple junction. This makes the analysis more complex than that of standard travelling waves, and we discuss how existing theory applies in this context. Furthermore, we compute numerically the associated point spectrum by restricting the problem to a finite computational domain with appropriate physical boundary conditions. Numerical results strongly suggest that the two-dimensional wave is stable with respect to both two- and three-dimensional perturbations. |
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Keywords: | Grain boundary Mean curvature motion Surface diffusion Linear stability Spectrum Free boundary |
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