Makroskopische Dynamik in diskreten Gittern

这个项目的目的是研究偏微分方程解表现出几个空间尺度。特别是，我们对那些产生快速空间振荡的系统感兴趣，这些系统的宏观行为必须得到描述。我们想要处理抛物型和双曲型问题。具体地说，需要开发一些方法来刻画给定进化问题的所有杨度量解的集合。此外，还将调查这些措施的时间演变。粗略地说，有三种不同的情况：(1)在空间一维半线性波动方程中，空间振动可以在任意小尺度上传输，不需要小参数。在最多两个特征速度的情况下，这一理论可以用经典的杨氏测量很好地理解。我们希望开发使用非局部关联的方法，允许描述具有两个以上特征速度的系统。这在线性情况下甚至是一个悬而未决的问题。(2)在色散波动方程中，调制方程的理论是很好理解的。在相当一般的条件下，波包的演化可以用一个类似于非线性薛定谔方程的振幅方程来描述。我们希望将这一理论与Young-测解理论联系起来，这很可能也将给半导体理论中的微观方程提供新的关系。(3)抛物型问题通常会抑制小尺度上的空间振荡。然而，在某些情况下，较小的参数e会使这些振荡变得有利。该参数e给出了小空间尺度和大空间尺度之间的比率。在极限e&gt；0中，我们得到了一个宏观演化方程，它可以是双曲线的，也可以是抛物线的。特别地，我们想要处理通常在调制方程的Ginzburg-Landau形式中研究的情况。

The aim of this project is to study partial differential equations whose solutions display several spatial scales. In particular we are interested in systems which develop fast spatial oscillations whose macroscopic behavior has to be described. We want to treat parabolic as well as hyperbolic problems. In particular, methods are to be developed which allow to characterize the set of all Young-measure solutions to a given evolutionary problem. Moreover, the time evolution of such measures will be investigated. Roughly spoken there are three different cases.(1) In spatially one-dimensional semilinear wave equations spatial oscillations can be transported on arbitrarily small scales; no small parameter is needed. The theory is well understood with classical Young-measures in the case of at most two characteristic speeds. We want to develop methods using nonlocal correlations which allow for the description of systems with more than two characteristic speeds. This is even an open problem in the linear setting.(2) In dispersive wave equations the theory of Modulation equations is well understood. Under quite General conditions the evolution of wave packets can be described by an amplitude equation like the nonlinear Schrödinger equation. We want to connect this theory to the theory of Young-measure solutions, which very likely will also give new relations to microscopic equations in semiconductor theory.(3) Parabolic problems usually damp out spatial oscillations on small scales. However, there are situations where a small parameter e makes these oscillations favorable. This parameter e gives the ratio between the small and the large spatial scale. In the limit e>0 we obtain a macroscopic evolution equation which may be either hyperbolic or parabolic. In particular we want to treat the case which is usually studied in the Ginzburg-Landau formalism for modulation equations.