Analysis of complex and non-smooth flows

该计划包括四个项目，它们是：（1）各向异性函数空间中的乘法和Nemytskij算子，（2）非均相催化，（3）有边和顶点区域上的Stokes方程，（4）有边和顶点区域上的非均相催化，（5）有边和顶点区域上的非均相催化，（6）有边和顶点区域上的非均相催化，（7）有边和顶点区域上的非均相催化。（4）动态接触线：方案（1）给出了处理非线性，即拟线性问题的一个重要工具。在（1）中获得的结果将在项目（2）和（4）中得到广泛应用。事实上，由于在Lp设置中处理（4）中的接触线模型缺乏光滑性和可积性，因此具有p的最佳值是至关重要的，这只能通过对（1）中给出的各向异性函数空间的乘法的详细分析来提供。尽管经典椭圆和抛物型偏微分方程的边和顶点区域的理论已经很成熟，但对于Stokes方程的相应结果却很少。特别是对于非定常Stokes方程，只有很少的结果存在。项目（3）的目标是提供此类结果。它们代表了（4）中所考虑的描述接触线运动的模型的分析方法的关键成分。结合该专题的最新发展，我们非常有信心，我们可以在这些方向上取得根本性进展。项目（1）和（3）还包括进一步复杂流动方法的基本工具。作为进一步的目标，这些工具在项目（2）中用于处理描述多相催化的模型。总而言之，该提案的主要目标是在非光滑流动，i.p.动态接触线和复杂流动（如描述多相催化的模型）的分析方面取得实质性进展。

The proposal comprises four projects addressing some fundamental problems in mathematical fluid dynamics as well as the development of functional analytic methods and tools for their treatment.The four projects are:(1) Multiplication and Nemytskij operators in anisotropic function spaces;(2) Heterogeneous catalysis;(3) Stokes equations on domains with edges and vertices;(4) Dynamic contact lines.Project (1) yields an important tool for the treatment of nonlinear, i.p. quasilinear, problems. The outcome obtained in (1) will intensively be used in projects (2) and (4). In fact, by the lack of smoothness and integrability for the treatment of the contact line models in (4) in the Lp-setting it is crucial to have the optimal value for p which can only be provided by the detailed analysis on multiplication of anisotropic function spaces given in (1). Even though theory on domains with edges and vertices for classical elliptic and parabolic PDE is well-developed, corresponding results for the Stokes equations are very rare. Particularly, for the instationary Stokes equations only very few results exist. It is the objective to provide such kind of results by Project (3). They represent the key ingredient for an analytical approach to models describing contact line movement as considered in (4). In connection with very recent developments on the topic we are quite confident that we can achieve fundamental progress in these directions.Projects (1) and (3) also include basic tools for an approach to further complex flows. As a further objective these tools are utilized in Project (2) for the treatment of models describing heterogeneous catalysis. Summarizing, the main objective of the proposal is to derive substantial progress in the analysis of non-smooth flows, i.p. dynamic contact lines, and of complex flows such as models describing heterogeneous catalysis.