Analysis of complex and non-smooth flows

复杂和非光滑流动的分析

• 批准号: 416057450
• 负责人: Professor Dr. Jürgen Saal
• 金额: --
• 依托单位: Lehrstuhl für Angewandte Analysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/416057450?language=en
• 关键词: Analysis; complex; non; smooth; flows

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.

该提案包括四个项目，涉及数学流体动力学中的一些基本问题，以及开发用于处理这些问题的泛函分析方法和工具。这四个项目是：(1)各向异性函数空间中的乘法算子和Nemytskij算子；(2)多相催化；(3)具有边和顶点的域上的Stokes方程；(4)动态接触线。项目(1)提供了一个处理非线性，特别是拟线性问题的重要工具。在(1)中获得的成果将集中用于项目(2)和(4)。事实上，由于(4)中接触线模型在lp设置下的处理缺乏平滑性和可积性，因此p的最优值是至关重要的，而p的最优值只能通过(1)中给出的各向异性函数空间乘法的详细分析来提供。尽管关于经典椭圆型和抛物型偏微分方程的边和顶点区域的理论已经很成熟，但是关于Stokes方程的相应结果却很少。特别是对于Stokes方程，只有很少的结果存在。项目(3)的目标就是提供这样的结果。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。关于这个问题最近的事态发展，我们非常有信心，我们能够在这些方向上取得根本进展。项目(1)和(3)还包括用于进一步复杂流程的方法的基本工具。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。