Abstract
This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton–Cabrera–Frank-type model following the Xiang’s work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first-order convergence rate.
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Notes
Compared to Xiang (2002), we drop all the physical constants that are mathematically unimportant.
For the convenience of calculation, we set the coefficients slightly different from Dal Maso et al. (2014). Moreover, instead of taking h to be increasing as in Dal Maso et al. (2014), we take h to be decreasing corresponding to physical interpretation of h being the height of the vicinal surface, which is the same convention as Xiang (2002), Xiang and E (2004).
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Acknowledgements
We would like to thank the support by the National Science Foundation under Grants DMS-1514826 (JGL), DMS-1454939 (JL), and also through the research network KI-Net RNMS11-07444. We thank Dionisios Margetis, Jeremy Marzuola, Yang Xiang and Aaron N.K. Yip for helpful discussions.
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Communicated by Robert V. Kohn.
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Gao, Y., Liu, JG. & Lu, J. Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces. J Nonlinear Sci 27, 873–926 (2017). https://doi.org/10.1007/s00332-016-9354-1
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DOI: https://doi.org/10.1007/s00332-016-9354-1