Nonlinear Multidimensional Systems of Hyperbolic Partial Differential Equations

建议的研究将集中在涉及弹性动力学、不可压缩流体流动和电磁学中的多维非线性偏微分方程组的模型。这项工作的重点将是使用分析方法来研究解的存在性和长期行为，这些方法基于零条件，利用模型的基本结构。在非线性弹性动力学领域，将研究应变-能量关系的结构与初值问题的整体适定性之间的联系。这项研究旨在更好地了解可压缩和不可压缩材料之间的关系，冲击波的形成，以及天体物理应用中弹性物体的行为。最近发展的弹性动力学方法将应用于非线性电磁学中的Born-Infeld系统，并最近在弦理论中复兴。研究平面不可压缩理想流体中涡旋运动的约束问题。根据经典决定论原理，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，它的解应该存在，并且唯一地和连续地依赖于初始构形。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。这项建议中的大部分研究都致力于描述弹性材料动力学的模型的基本问题。材料的性质是通过应变和能量之间的关系的规范来表达的，使用现象学的论点。然而，现象学本身并不能完全描述模型的结构。描述未来所有时间的动态的愿望导致了进一步的结构性假设的强加，这些假设与问题的数学分析密切相关。从这个意义上说，数学分析为模型的选择提供了重要的指导。有了数学上易于处理的理论，就有可能考虑定性问题，例如系统的行为在材料性质变化时如何变化。特别是，将研究当材料变得不可压缩时系统的稳定性。例如，这包括橡胶类材料。这些技术还将被修改，以允许包括天体物理应用所必需的相对论效应。来自非线性电动力学的Born-Infeld模型(最近在弦理论中重新出现)也将被研究，因为它具有类似的结构特性。

The proposed research will focus on models involving multidimensional systems of nonlinear partial differential equations, arising in elastodynamics, incompressible fluid flow, and electromagnetics. The emphasis of this work will be on existence and long time behavior of solutions using analytical methods which exploit the fundamental structure of the models, based on the null condition. In the area of nonlinear elastodynamics, the connection between the structure of the strain-energy relations and global well-posedness for the initial value problem will be studied. The research aims at a better understanding of the relationship between compressible and incompressible materials, the formation of shock waves, and the behavior of elastic bodies in astrophysical applications. The recent methods developed for elastodynamics will be applied to the Born-Infeld system from nonlinear electromagnetics and recently revived in string theory. The confinement of vortical motion in planar incompressible ideal fluids will be studied. According to the classical principle of determinism, the the evolutionary laws of a physical system together with an initial configuration should determine the state of the system at all future times. Mathematically speaking, the the laws of physics are encoded in a system of partial differential equations solutions of which should exist and depend uniquely and continuously on the initial configuration. Creating the mathematical framework for a given system and verifying its well-posedness (in the sense above) is thus a problem of fundamental importance. Much of the research in this proposal is devoted to such basic questions for models describing the dynamics of elastic materials. Material properties are expressed through the specification of the relation between strain and energy, using phenomenological arguments. However, phenomenology alone does not fully characterize the structure of the model. The desire to describe dynamics for all future times leads to the imposition of further structural assumptions which are deeply connected with the mathematical analysis of the problem. In this sense, mathematical analysis offers important guidance in the selection of the model. With a mathematically tractable theory in hand, it is possible to consider qualitative questions, such as how the behavior of the system varies under changes in material properties. In particular, the stability of the system as the material becomes incompressible will be studied. This includes rubber-like materials, for example. The techniques will also be adapted to allow for the inclusion of relativistic effects, necessary for astrophysical applications. The Born-Infeld model from nonlinear electrodynamics (and recently revived in the theory of strings) will also be investigated, since it shares similar structural properties.