Problems in Nonlinear Geometric Field Theories

非线性几何场论中的问题

• 批准号: 9704430
• 负责人: Abdolreza Tahvildar-Zadeh
• 金额: $ 8.25万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704430&HistoricalAwards=false
• 关键词: Problems; Nonlinear; Geometric; Field; Theories

The proposed research is in the area of nonlinear hyperbolic systems of partial differential equations arising in mathematical physics and deals specifically with regularity, break-down, and large-time behavior of solutions to the initial value problem for wave maps. While the ultimate goal of any study in this area is the understanding of the dynamics of physically relevant field theories such as General Relativity and Yang-Mills, valuable insight can be gained by first focusing on wave maps, which is a simpler geometric field theory exhibiting many similarities with the above, and in which many of the difficulties in dealing with those physical theories are present in a more transparent way. Wave maps (known to physicists as sigma-models) are the hyperbolic analogue of harmonic maps between manifolds, where the domain manifold instead of being Riemannian is Lorentzian. They thus satisfy a system of semilinear wave equations with a nonlinearity which is quadratic in the gradient. Some of the problems to be studied are (1) existence of smooth stationary solutions to the equations and their stability under small perturbations, (2) constructing all possible self-similar wave maps of the Minkowski space, (3) investigating the genericity of singularities arising from self-similar solutions, and (4) finding examples of blowup in two space dimensions. Hyperbolic partial differential equations lie at the heart of mankind's efforts to quantify and understand the evolutionary phenomena in nature. From subatomic particles to galactic clusters, every phenomenon known to Man which involves the propagation of signals and disturbances at finite speeds is modeled by a hyperbolic differential equation. Examples of such phenomena are the production and propagation of sound waves (acoustics), water waves (hydrodynamics), seismic body and surface waves (elastodynamics), electromagnetic waves (electrodynamics), and gravitational waves (general relativity). Despite considerable progress in recent years in the study of nonlinear hyperbolic systems, the basic questions regarding regularity, break-down and large time behavior of their solutions in more than one space dimension remain largely unanswered. Progress in this area requires a good understanding of continuum physics and is only possible through rigorous mathematical analysis of a host of simpler problems, each one modeling only a few of the many difficulties present in the actual physical problem. With such a long-term plan, it is equally important to educate the next generation of scientists who will have the physical and mathematical ability, as well as the courage and enthusiasm, to continue the work being done today. This requires a serious re-evaluation of the existing mathematics curriculum and the development of new courses dealing specifically with mathematics as it relates to continuum physics at all levels of college and graduate education.

拟议的研究是在数学物理中产生的偏微分方程的非线性双曲系统的领域，具体涉及的规则性，故障，和大的时间行为的解决方案的初始值问题的波映射。 虽然在这一领域的任何研究的最终目标是理解物理相关的场论，如广义相对论和杨米尔斯的动力学，有价值的洞察力可以通过首先关注波图，这是一个更简单的几何场论，表现出与上述许多相似之处，其中许多困难，在处理这些物理理论是目前在一个更透明的方式。 波映射（物理学家称为sigma模型）是流形之间调和映射的双曲模拟，其中域流形不是黎曼流形而是洛伦兹流形。 因此，它们满足一个系统的半线性波动方程的非线性是二次的梯度。 要研究的一些问题是：（1）方程的光滑定态解的存在性及其在小扰动下的稳定性;（2）构造Minkowski空间的所有可能的自相似波映射;（3）研究自相似解产生的奇点的一般性;（4）寻找二维空间中爆破的例子。 双曲型偏微分方程是人类量化和理解自然界进化现象的核心。 从亚原子粒子到星系团，人类已知的每一种涉及信号和扰动以有限速度传播的现象都可以用双曲型微分方程来模拟。 这些现象的例子是声波（声学）、水波（流体动力学）、地震体波和表面波（弹性动力学）、电磁波（电动力学）和引力波（广义相对论）的产生和传播。 尽管近年来非线性双曲方程组的研究取得了很大的进展，但关于其解在多维空间中的正则性、崩溃性和大时间行为的基本问题仍然没有得到解答。 这一领域的进展需要对连续介质物理学有很好的理解，并且只有通过对许多简单问题进行严格的数学分析才有可能，每个问题只模拟了实际物理问题中存在的许多困难中的一小部分。 有了这样一个长期计划，教育下一代科学家同样重要，他们将拥有物理和数学能力，以及勇气和热情，继续今天正在做的工作。 这就需要对现有的数学课程进行认真的重新评估，并开发专门处理数学的新课程，因为它涉及到大学和研究生教育各个层次的连续统物理。