Problems in Hyperbolic Field Theories

PI:Abdolreza Tahvildar-Zadeh，罗格斯大学dfs -0301207--------------------------------------双曲场理论问题这是一项为期三年的研究计划，旨在研究一些由拉格朗日导出的物理理论中出现的偏微分方程双曲系统，重点研究经典解的长期存在性和渐近行为问题，所涉及的线性算子的映射性质，以及非线性动力稳定性的静力和定常解。提出的具体问题有：(1)在无穷远处具有临界（即逆平方）衰减和/或具有局部奇点的势存在时，获得线性波和薛定谔方程解的时空Strichartz估计。(2)利用上述估计证明了从Minkowski空间到球体的涡状波映射的稳定性。(3)获得了晶体光学各向异性麦克斯韦方程解的尖锐色散估计，该估计编码了衰减的方向依赖性。(4)证明了描述恒定离子背景下电子流体运动的等离子体动力学的Euler-Maxwell系统的小振幅波的全局存在性。(5)利用广义相对论对称约简爱因斯坦方程的波映射公式，得到了高迪度量的未来和过去的渐近行为，具有扭转的对称时空中常平均曲率超曲面的存在性，这些时空中初始奇点的振荡逼近，以及柱对称Einstein- vlasov系统的全局存在性的结果。场论是古典物理学和现代物理学最经久不衰的范式。电磁学、流体力学和固体力学、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内描述。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，它们的产生、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们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，数学分析师们正处于一个接受挑战的位置，在那些长期被其他人忽视的物理数学领域进行研究。这项基础研究超越了数值模拟和近似方程，并解决了该学科核心的难题，即非线性偏微分方程理论。理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项合作努力，特别是与世界其他地区的数学界成员的合作，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，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。

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.