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.

双曲场理论中的Abdolreza Tahvildar-Zadeh，Rutgers UniversityDMS-0301207--------------------------------------Problems这是一个为期三年的建议，用于研究从拉格朗日引出的物理理论中的一些双曲型偏微分方程组，重点是经典解的长期存在和渐近行为，所涉及的线性算子的映射性质，以及静态和定常解的非线性动力学稳定性。所提出的具体问题是：(1)求出线性波动方程和薛定谔方程的解的时空Strichartz估计，其中的势在无穷远处具有临界(即平方反比)衰减和/或具有局部奇性。(2)利用上述估计，证明了从Minkowski空间到球面的涡旋波映射的稳定性。(3)得到了晶体光学各向异性麦克斯韦方程组的解的精确色散估计，该方程编码了衰变的方向相关性。(4)证明了描述由电子流体在恒离子背景中运动的等离子体动力学的欧拉-麦克斯韦系统的整体小振幅波的存在性。(5)利用广义相对论对称约化爱因斯坦方程的波图形式，得到了Gowdy度规未来和过去的渐近行为，对称扭曲时空中常平均曲率超曲面的存在性，这些时空中初始奇性的振荡方法，以及柱对称下爱因斯坦-弗拉索夫系统的整体存在性。场论是经典和现代物理学中最持久的范例。电磁学、流体和固体力学、基本粒子的弱相互作用和强相互作用，以及爱因斯坦的引力理论，都可以在场论的框架下描述。在这个框架中要理解的最重要的物理现象之一是波的现象，波的产生、传播、相互作用和色散。例如电磁波、物质波和引力波。这里提出的每一个问题都对理解波浪现象的一个特定方面有直接的影响。随着新世纪的到来，随着技术的进步迫使科学家比以往任何时候都更详细地研究自然界固有的非线性行为，数学分析人员能够接受挑战，在那些长期被其他人忽视的物理数学领域进行研究。这项基础性研究超越了数值模拟和近似方程，并解决了这门学科的核心难题，即非线性偏微分方程组理论。了解非线性波是朝着这个方向迈出的重要一步。这在很大程度上是一种合作努力，特别是与世界其他地区数学界成员的合作，在这些地区，关心物理问题的传统得到了很好的维护，这为我们提供了一个机会，让我们有机会在防止美国在这些关键数学科学领域的领先地位受到侵蚀方面发挥作用。