The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model
The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model
复制标题
DOI：
10.1016/j.jde.2011.08.007
复制
发表时间：
2010-10
期刊：
影响因子：
--
通讯作者：
A. Fino;H. Ibrahim;R. Monneau
中科院分区：
文献类型：
--
作者：
A. Fino;H. Ibrahim;R. Monneau
研究了n维完全过阻尼Frenkel-Kontorova模型的推广.该模型描述了晶体中每个原子的位置的演化，并且在数学上由耦合的一阶常微分方程的无限系统给出。我们证明了，对于一个适当的重新缩放的这个模型，解决方案收敛到Peierls-Nabarro模型，这是一个耦合系统的两个PDE（通常是一个椭圆型PDE在一个域的边界上的演化PDE）的解决方案。这种从离散模型到连续模型的过渡是在粘度解的框架内完成的。
We study a generalization of the fully overdamped Frenkel–Kontorova model in dimension n⩾1. This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls–Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions.