Problems in Hyperbolic Field Theories

双曲场论中的问题

• 批准号: 0301207
• 负责人: Abdolreza Tahvildar-Zadeh
• 金额: --
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301207&HistoricalAwards=false
• 关键词: Problems; Hyperbolic; Field; Theories

PI: Abdolreza Tahvildar-Zadeh, Rutgers UniversityDMS-0301207--------------------------------------Problems in Hyperbolic Field TheoriesThis is a three-year proposal for studying some of the hyperbolic systems of partial differential equations arising in physical theories that are derivable from a Lagrangian, focusing on questions of long-time existence and asymptotic behavior of classical solutions, mapping properties of the linear operators involved, and nonlinear dynamical stability of static and stationary solutions. Specific problems proposed are (1) Obtaining space-time Strichartz estimates for solutions of the linear wave and Schroedinger equations in presence of potentials with critical (i.e. inverse-square) decay at infinity and/or with local singularities. (2) Proving stability of vortex-like wave maps from the Minkowski space into the sphere, utilizing the above estimates. (3) Obtaining a sharp dispersive estimate for solutions of the anisotropic Maxwell equations of crystal optics, one that encodes the direction-dependence of the decay. (4) Proving global existence of small-amplitude waves for the Euler-Maxwell system describing the dynamics of plasma modeled by an electron fluid moving in a constant ion background. (5) Using the wave map formulation of symmetry-reduced Einstein equations of general relativity to obtain results on the future and past asymptotic behaviors of Gowdy metrics, on the existence of constant mean curvature hypersurfaces in symmetric spacetimes with twist, on the oscillatory approach to the initial singularity in these spacetimes, and on the global existence for the Einstein-Vlasov system in cylindrical symmetry.Field Theory is the most enduring paradigm of classical as well as modern physics. Electromagnetics, fluid and solid mechanics, weak and strong interactions of elementary particles, and Einstein's theory of gravitation are all describable in the framework of a field theory. One of the most important physical phenomena to be understood in this framework is the phenomenon of waves, their creation, propagation, interaction, and dispersion. Some examples are electromagnetic waves, material waves, and gravitational waves. Each of the problems proposed here has a direct consequence in the understanding of a specific aspect of the wave phenomenon. With the dawn of a new century, as advances in technology force scientists to address the inherently nonlinear behavior of nature in more detail than ever before, mathematical analysts are in a position to take up the challenge of doing research in those areas of physical mathematics that have long been neglected by others. This fundamental research involves going beyond numerical simulations and approximate equations, and addressing the hard problems that lie at the core of the subject, i.e. in the theory of nonlinear partial differential equations. Understanding nonlinear waves is an important step in this direction. This is very much a collaborative effort, and in particular collaborations with members of mathematical communities in other parts of the world where a tradition of caring about physical problems is well-maintained, provides us with an opportunity to play a role in preventing the erosion of the leading status of the US in these key areas of mathematical sciences.

PI: Abdolreza Tahvildar-Zadeh, Rutgers UniversityDMS-0301207--------------------------------------Problems in Hyperbolic Field TheoriesThis is a three-year proposal for studying some of the hyperbolic systems of partial differential equations arising in physical theories that are derivable from a Lagrangian, focusing on questions of long-time existence and asymptotic behavior of classical solutions, mapping涉及的线性算子的性能以及静态和固定溶液的非线性动力学稳定性。 提出的具体问题是（1）在存在无穷大和/或局部奇点的势势（即相反平方）衰减的电势的情况下，获得线性波和Schroedinger方程溶液的溶液的估计值（1）。 （2）利用上述估计值证明了从Minkowski空间进入球体的涡流样波图的稳定性。 （3）获得尖锐的分散估计，以供晶体光学的各向异性麦克斯韦方程溶液，该方程式编码衰减的方向依赖性。 （4）证明了Euler-Maxwell系统的全球振幅波的全球存在，描述了由电子流体在恒定离子背景中移动的电子流体模拟的等离子体的动力学。 （5）使用波图对称对称性的爱因斯坦等方程的一般相对性方程，以获取Gowdy指标的未来和过去的渐近行为的结果，以与对称空间中的持续平均曲率超出曲线的存在，以及在这些旋转系统中的持续曲折中的持续曲率，以及在这些空间的最初奇异系统中的存在 - 以及在EIN全局中的存在，并跨越了EIN的全球范围。圆柱形对称性。场理论是古典和现代物理学中最持久的范式。 电磁学，流体和固体力学，基本颗粒的弱和强相互作用以及爱因斯坦的引力理论都在田间理论的框架中都可以描述。 在此框架中要理解的最重要的物理现象之一是波浪现象，它们的创造，繁殖，相互作用和分散。 一些例子是电磁波，材料波和引力波。 这里提出的每个问题在理解波浪现象的特定方面都有直接的结果。 随着新世纪的曙光，随着技术的进步迫使科学家比以往任何时候都更详细地解决自然的非线性行为，数学分析师可以承担在长期以来一直被其他人忽略的物理数学领域进行研究的挑战。 这项基本研究涉及超越数值模拟和近似方程，并解决了受试者核心的严重问题，即在非线性偏微分方程理论中。 了解非线性波是朝这个方向迈出的重要一步。这是一项合作的努力，尤其是与世界其他地区数学社区成员的合作，在这里，关心身体问题的传统得到了良好的维护，这为我们提供了有机会在防止美国领先地位的侵蚀中发挥作用的机会。