论文标题
晶粒边界运动连续方程的初始有限价值问题的弱解决方案
Weak solutions to an initial-boundary value problem for a continuum equation of motion of grain boundaries
论文作者
论文摘要
我们研究了连续方程的初始（周期性）边界值问题，这是基于线缺陷（断开连接）的基础显微镜机制运动的模型，并整合了热力学驱动力的各种范围的效果。在情况下，我们首先证明了弱解决方案问题的全球时间存在和弱解决方案的唯一性，并具有正均衡断开密度参数B，然后研究溶液的渐近行为，因为B占零。主要定理证明的主要困难是由于b = 0的退化造成的，一个具有奇异性的非本地项，以及与未知梯度相关的最高导数的非平滑系数。证明中的关键成分是能量法，对希尔伯特类型的奇异积分的估计以及紧凑的引理。
We investigate an initial-(periodic-)boundary value problem for a continuum equation, which is a model for motion of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and integrated the effects of a diverse range of thermodynamic driving forces. We first prove the global-in-time existence and uniqueness of weak solution to this initial-boundary value problem in the case with positive equilibrium disconnection density parameter B, and then investigate the asymptotic behavior of the solutions as B goes to zero. The main difficulties in the proof of main theorems are due to the degeneracy of B=0, a non-local term with singularity, and a non-smooth coefficient of the highest derivative associated with the gradient of the unknown. The key ingredients in the proof are the energy method, an estimate for a singular integral of the Hilbert type, and a compactness lemma.
