Multi-scale analysis of phase boundaries under the influence of fluctuations and random fields

我们考虑材料中界面的形成和演化的模型，其特征是界面能量的相互作用在很小的尺度上与时间无关的随机和/或确定性的不均匀性建模热波动。当前的项目集中在这类系统动力学的两个方面：第一个方面是大时间尺度上的动态大偏差，特别是成核现象。第二个方面涉及(已经开发的)界面的有效演化。我们感兴趣的是起源于较小尺度的非均匀和噪声对这些大时空尺度上的有效演化规律的影响。对于这两个方面，我们的目标都是严格的数学结果。我们考虑的模型是具有随机扰动的平均曲率流(MCF)和几种简化的具有时空噪声和/或具有随机或周期系数的反应扩散型梯度流结构的连续介质模型。

We consider models for the formation and evolution of interfaces in materials which are characterised by the interplay of¿ the interfacial energy¿ time-independent random and/or deterministic inhomogeneities on a very small scale¿ space-time dependent noise modelling thermal fluctuations.The current project is focused on two aspects of the dynamics of such systems: The first aspect are dynamic large deviations on large time scales and, in particular, the phenomenon of nucleation.The second aspect concerns the effective evolution of an (already developed) interface. We are interested in the influence of the inhomogeneities and the noise - which originate on a smaller scale - on these effective evolution laws on a large time-space scale. For both aspects we aim at rigorous mathematical results. The models we consider are Mean Curvature Flow (MCF) with random perturbations and several simplified continuum models of reaction-diffusion type of gradient-flow structure with space-time noise and/or with random or periodic coefficients.