具有弹性效应的台阶外延表面(2+1)维连续体模型的存在性、唯一性和能量标度

IF 1.2 Q2 MATHEMATICS, APPLIED
Gang-Han Fan, T. Luo, Y. Xiang
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引用次数: 0

摘要

本文研究了Xu和Xiang (SIAM J. appll)提出的长时间弹性相互作用下阶梯外延表面演化的2+1维连续体模型。数学。69,1393-1414,2009)。该模型中的远程相互作用项和两个长度尺度使得PDE分析具有挑战性。此外,与1+1维情况不同,2+1维情况下的总能量存在非凸性贡献,且在演化过程中不容易证明解总是处于适定状态。在本文中,我们提出了一个基于基础物理的改进的2+1维连续体模型。这种修正修正了由于能量泛函的非凸性而可能产生的病态问题。我们证明了静态解和动态解的存在唯一性,并推导了它们的最小能量标度律。结果表明,能量最小的表面轮廓主要是由阶梯弯曲不稳定曲面获得的。这与阶跃聚束表面轮廓在弹性效应下得到的1+1维外延表面的能量标度规律有本质区别。我们还讨论了在2+1维中从阶跃聚束不稳定性到阶跃弯曲不稳定性的转变。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Existence, Uniqueness and Energy Scaling of (2+1)-Dimensional Continuum Model for Stepped Epitaxial Surfaces with Elastic Effects
We study the 2+1 dimensional continuum model for the evolution of stepped epitaxial surface under long-range elastic interaction proposed by Xu and Xiang (SIAM J. Appl. Math. 69, 1393-1414, 2009). The long-range interaction term and the two length scales in this model makes PDE analysis challenging. Moreover, unlike in the 1+1 dimensional case, there is a nonconvexity contribution in the total energy in the 2+1 dimensional case, and it is not easy to prove that the solution is always in the well-posed regime during the evolution. In this paper, we propose a modified 2+1 dimensional continuum model based on the underlying physics. This modification fixes the problem of possible illposedness due to the nonconvexity of the energy functional. We prove the existence and uniqueness of both the static and dynamic solutions and derive a minimum energy scaling law for them. We show that the minimum energy surface profile is mainly attained by surfaces with step meandering instability. This is essentially different from the energy scaling law for the 1+1 dimensional epitaxial surfaces under elastic effects attained by step bunching surface profiles. We also discuss the transition from the step bunching instability to the step meandering instability in 2+1 dimensions.
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CiteScore
2.70
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