Existence and regularity theory and qualitative analysis

存在性规律理论与定性分析

• 批准号: 470900796
• 负责人: Professor Dr. Michael Winkler
• 金额: --
• 依托单位: Fachgebiet Differentialgleichungen
• 依托单位国家: 德国
• 项目类别: Research Units
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/470900796?language=en
• 关键词: Existence; regularity; theory; qualitative; analysis

The adequate modeling of dynamical behavior in piezoceramic materials of the considered type requires detailed knowledge about certain classes of partial differential equations which appropriately account for both the dispersive and the dissipative among the mechanisms influencing the respective real system.The project part ANA firstly intends to develop fundamental solution theories for various models relevant for the work in the project part MESS; here, a focus is to be set not only on basic questions related to solvability and well-posedness, but beyond this also on aspects of immediate relevance for the investigations in the project parts OPT and SIM, inter alia in the course of a suitably far-reaching regularity analysis. The classes of evolution equations under consideration are to be selected in such a way that within a hierarchy of models with increasing level of complexity, both simplifying systems concentrating on particular mechanisms and comprehensive models including all essential components will be studied.Apart from that, a second central part of this project aims at identifying key characteristics of the considered models by an analysis of qualitative solution behavior. Here in line with the application contexts considered in the project part MESS, this especially includes a qualitative analysis of nonlinear and particularly thermal effects arising in the examined framework of interaction between mechanical and thermal fields. A predominant focus in this direction is to be laid on analytical descriptions of either thespontaneous occurrence or a possible exclusion of thermal "hot spot" phenomena in the shape of locally concentrated temperature distributions.

对所考虑类型的压电陶瓷材料的动力学行为进行充分的建模需要对某些类别的偏微分方程有详细的了解，这些偏微分方程适当地考虑了影响各自实际系统的机制之间的色散和耗散。项目部分 ANA 首先打算为与项目部分 MESS 工作相关的各种模型开发基本解决理论；在这里，重点不仅要放在与可解性和适定性相关的基本问题上，而且除此之外，还要放在与项目部分 OPT 和 SIM 的调查直接相关的方面，特别是在适当的深远的规律性分析过程中。所考虑的演化方程类别的选择方式是，在复杂性不断增加的模型层次结构中，将研究集中于特定机制的简化系统和包括所有基本组成部分的综合模型。除此之外，该项目的第二个中心部分旨在通过分析定性解决方案行为来识别所考虑模型的关键特征。这里，根据 MESS 项目部分考虑的应用环境，这特别包括对机械场和热场之间相互作用的检查框架中产生的非线性尤其是热效应的定性分析。这个方向的主要焦点是对自发发生或可能排除局部集中温度分布形式的热“热点”现象的分析描述。