Makroskopische Dynamik in diskreten Gittern

本课题的目的是研究其解具有多个空间尺度的偏微分方程。我们特别感兴趣的是那些发展出快速空间振荡的系统，它们的宏观行为必须被描述。我们既要处理抛物线问题，也要处理双曲问题。具体地说，将发展一些方法，使之能够描述给定进化问题的所有杨测度解的集合。此外，还将研究这些措施的时间演变。大致说来，有三种不同的情况。(1)在空间一维半线性波动方程中，空间振荡可以在任意小尺度上传递；不需要小的参数。在至多两个特征速度的情况下，用经典的杨测度可以很好地理解这一理论。我们希望开发使用非局部相关的方法，允许描述具有两个以上特征速度的系统。这在线性环境中也是一个开放的问题。(2)在色散波动方程中，调制方程理论得到了很好的理解。在相当一般的条件下，波包的演化可以用振幅方程来描述，如非线性Schrödinger方程。我们想把这个理论和Young-measure解理论联系起来，Young-measure解很可能也会给半导体理论中的微观方程提供新的关系。(3)抛物型问题通常在小尺度上抑制空间振荡。然而，在某些情况下，小的参数e会使这些振荡变得有利。这个参数e给出了小空间尺度和大空间尺度之间的比率。在极限e>0下，得到了一个宏观演化方程，该方程可以是双曲型的，也可以是抛物线型的。特别地，我们要处理的情况，通常是研究在金兹堡-朗道形式的调制方程。

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.