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.

所提出的研究属于数学物理中出现的偏微分方程的非线性双曲系统领域，具体涉及波图初值问题解的规律性、分解和大时间行为。 虽然该领域任何研究的最终目标是理解广义相对论和杨米尔斯等与物理相关的场论的动力学，但首先关注波图可以获得有价值的见解，波图是一种更简单的几何场论，与上述场论表现出许多相似之处，并且在处理这些物理理论时，许多困难都以更透明的方式呈现。 波图（物理学家称为西格玛模型）是流形之间调和图的双曲​​模拟，其中域流形不是黎曼流形，而是洛伦兹流形。 因此，它们满足具有梯度二次的非线性的半线性波动方程组。 要研究的一些问题是（1）方程平滑平稳解的存在性及其在小扰动下的稳定性，（2）构造闵可夫斯基空间的所有可能的自相似波图，（3）研究自相似解产生的奇点的通用性，以及（4）寻找二维空间中爆炸的例子。 双曲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以有限速度传播信号和扰动的现象都可以通过双曲微分方程来建模。 这种现象的例子有声波（声学）、水波（流体动力学）、地震体和表面波（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系统的研究取得了相当大的进展，但有关其解在多个空间维度上的规律性、分解和大时间行为的基本问题在很大程度上仍未得到解答。 这一领域的进展需要对连续介质物理学有很好的理解，并且只有通过对一系列更简单的问题进行严格的数学分析才有可能实现，每个问题仅对实际物理问题中存在的许多困难中的一小部分进行建模。 有了这样一个长期计划，教育下一代科学家也同样重要，他们将拥有物理和数学能力，以及继续今天正在进行的工作的勇气和热情。 这需要对现有的数学课程进行认真的重新评估，并开发专门针对数学的新课程，因为它与大学和研究生教育各级的连续介质物理相关。