Mathematical Sciences: Problems for Two Classes of Evolutionary Partial Differential Equations

本项目致力于推导粘弹性模型材料随时间变化的模型中出现的两类偏微分方程解的性质。第一部分讨论描述具有长期记忆的材料及其解的线性积分-微分方程式。目标是根据数据详细描述解决方案的平稳性。要采用的方法涉及允许使用更简单的相关问题的已知性质的各种变换和转换技术。因此，对理论模型的更好理解和对计算方法行为的更好洞察将是可想而知的。还将研究一类描述具有内耗的模型材料的非线性偏微分方程组。其目标是构建符合物质不能穿透自身这一基本物理原理的广义解，并分析它们的结构。对于其他类型的非线性偏微分方程组已经成功的技术，例如先验估计，将被采用。因此，应该更好地理解连续介质力学中的某些类型的模型，以及更广泛地理解描述一对一映射的偏微分方程解。

This project seeks to derive properties of solutions of two classes of partial differential equations that arise in models for the time-dependent behavior of viscoelastic model materials. The first part deals with linear integrodifferential equations that describe materials with long-term memory and their solutions. A goal is to give a detailed description of the smoothness of solutions depending on the data. The methods to be employed involve various transformation and transfer techniques that allow use of known properties of simpler, related problems. As a result, a better understanding of theoretical models and improved insight into the behavior of computational methods will is expected. Work will also be done investigating a class of nonlinear partial differential equations describing model materials with internal friction. The goal is to construct generalized solutions that obey the basic physical principle that matter cannot penetrate itself and to analyze their structure. Techniques that have been successful for other classes of nonlinear partial differential equations, such as a priori estimates, are to be employed. As a result, improved understanding of certain classes of models in continuum mechanics and more generally of solutions of partial differential equations that are to describe one-to-one mappings should result.