Effective properties of interface evolution in a random environment

这是一项将偏微分方程(PDE)分析和应用概率分析联系起来的数学研究。该项目的主要目标是推导出非均匀随机环境中界面的宏观演化规律，并用具有随机系数的非线性PDE在小范围内描述。特别是，我们希望超越经典的同质化，以便治疗出现长期集体行为的系统。这个项目的一个关键方面是在分析和概率之间建立一种新的联系，使这两个领域都受益，通过研究由应用程序激发的问题。它的动机是以下情况：与界面相关联的标量称为其能量(例如，考虑其面积)，它试图减少该标量，即，它执行梯度流动。这种能量在很小的范围内通过障碍物或杂质受到扰动，并受到某种大规模的力量的驱动。杂质是随机的，也就是说，我们只有关于在某个地方找到某些杂质的概率的信息，而不是关于它们的确切性质和位置的信息。我们感兴趣的是大范围内的有效速度和其他定性属性，远远大于扰动变化的尺度。在这个尺度上，扰动应该是平均的，但问题是：大尺度上的有效演化规律是什么，界面的定性属性是什么，例如，由于所有的随机非均质性，界面在哪些尺度上看起来很粗糙？这一切是如何依赖于杂质定律的？这一点很重要，因为我们感兴趣的是我们日常生活中的一个系统对这个尺度上的输入的反应。例如，我们想知道一块金属在车祸中如何改变形状，我们对每个原子的位置不感兴趣，而且我们无论如何也无法计算出这些位置。但大多数逼真材质在精细比例上都有一些随机结构。

This is mathematical research connecting the analysis of partial differential equations (PDEs) and (applied) probability.The main goal of this project is to derive macroscopic evolution laws for interfaces in heterogeneous, random environments, described on a small scale by nonlinear PDEs with random coefficients. In particular, we want to go beyond classical homogenisation in order to treat systems where a long-range collective behaviour emerges. A key aspect of this project is to establish a new link between analysis and probability, benefitting both fields, by working on problems motivated by applications.It is motivated by the following situation:With an interface is associated a scalar quantity called its energy (think e.g. of its area) which it tries to decrease, i.e., it performs a gradient flow. This energy is perturbed through obstacles or impurities on a very small scale, and driven by some large-scale force. The impurities are random, i.e., we have information only on the probability of finding certain impurities in a certain place, not on their precise nature and location.We are interested in the effective velocity and other qualitative properties of the interface on a large scale, much larger than the scale on which the perturbations vary. On that scale, the perturbations should average out, but the questions is:What is the effective evolution law on a large scale, and what are the qualitative properties of the interface, e.g. on which scales does it look rough due to all the random heterogeneities? How does all this depend on the law of the impurities?This is important because we are interested in the reaction of a system on the scale of our everyday life to an input on that scale. E.g. we would like to know how a piece of metal changes shape in a car crash, we are not interested in the position of each single atom, and we wouldn't be able to compute those anyway. But most realistic materials have some random structure on a fine scale.