Long Time Behavior of Multidimensional Systems

多维系统的长时间行为拟议的研究摘要托马斯·C·西德里斯该研究项目专注于分析可压缩和不可压缩材料运动中出现的多维非线性偏微分方程组。这项工作将集中研究初值问题解的存在性和长时间性态。我们的分析将利用模型的详细结构，特别是非线性相互作用的影响。我们将研究非线性粘弹性材料的运动，可压缩和不可压缩材料之间的关系，可压缩材料中激波的形成，以及平面不可压缩理想流体中涡旋运动的限制。决定论行为的经典观点通常被认为是说，物理系统的演化规律与初始构型一起决定了系统在未来所有时间的状态。从数学上讲，物理定律编码在一个偏微分方程组中，它们的解应该存在，并唯一地、连续地依赖于初始数据。因此，为给定系统创建数学框架并验证其适当性(在上面的意义上)是一个至关重要的问题。数学分析在说明应变能关系方面提供了重要的指导，超越了纯粹的现象学假设。这项建议中的大部分研究致力于为描述大类弹性材料的动力学的模型回答这些基本问题。

Long Time Behavior of Multidimensional SystemsAbstract of Proposed ResearchThomas C. Sideris The research project focuses on the analysis of multidimensional systems of nonlinear partial differential equations that arise in the motion of compressible and incompressible materials. The work will concentrate on studying the existence and long time behavior of solutions to initial value problems. Our analysis will exploit the detailed structure of the models, and in particular, the effect of nonlinear interactions. We shall study the motion of nonlinear visco-elastic materials, the relationship between compressible and incompressible materials, the formation of shock waves in compressible materials, and the confinement of vortical motion in planar incompressible ideal fluids. Classical ideas of deterministic behavior usually are regarded as saying that the evolutionary laws of a physical system together with the initial configuration determine the state of the system at all future times. Mathematically speaking, the laws of physics are encoded in a system of partial differential equations and their solutions should exist and depend uniquely, and continuously, on the initial data. Creating the mathematical framework for a given system and verifying its well-posedness (in the sense above) is thus a problem of fundamental importance. Mathematical analysis offers important guidance in the specification of strain energy relationships beyond purely phenomenological assumptions. Much of the research in this proposal is devoted to answering these basic questions for models describing the dynamics of broad classes of elastic materials.