多晶材料中晶界演化的新连续介质模型的分析验证

中文摘要 AI 辅助

为了基于线缺陷（位错）的潜在微观机制以及多种热力学驱动力的综合效应来描述晶界的演化，Zhang等人在2017年构建了一个连续介质方程。我们证明了该模型初边值问题弱解的全局时间存在性、唯一性和正则性，还建立了稳态解的存在性与唯一性。最后，我们研究了演化问题弱解的长时间行为，证明该解在合适的意义下收敛到稳态解，并通过数值模拟验证了分析结果。主要定理证明中的核心困难源于带奇异性的非局部项、与未知量梯度相关的最高阶导数的非光滑系数，以及自由能并非从下方一致有界的特殊形式；证明的关键要素包括能量方法、希尔伯特型奇异积分的估计、希尔伯特变换的傅里叶变换，以及针对未知量时间导数的带权估计。

英文摘要

To describe the evolution of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and the integrated effects of a diverse range of thermodynamic driving forces, Zhang, et al. in 2017 formulated a continuum equation. We prove the global-in-time existence, uniqueness, and regularity of the weak solution to an initial-boundary value problem for this model. The existence, uniqueness of the stationary solution are also established. Finally, we investigate the large-time behavior of the weak solution of the evolution problem, and show that the solution converges to the stationary solution in a suitable sense. Numerical simulations are carried out to validate the analysis results. The main difficulties in the proof of main theorems are due to a non-local term with singularity, a non-smooth coefficient of the highest derivative associated with the gradient of the unknown, and the special form of free energy which is not uniformly bounded from below. The key ingredients in the proof are the energy method, an estimate for a singular integral of the Hilbert type, Fourier transform of Hilbert transformations, and an estimate with a weight, for time-derivative of the unknown.

发表机构

• Shanghai University（上海大学）
• Hong Kong University of Science and Technology（香港科技大学）

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