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学中产生的部分微分方程的非线性双曲系统的领域，并专门涉及波浪图的初始值问题的规律性，分解和大型解决方案行为。 虽然该领域的任何研究的最终目标是了解物理相关的现场理论的动态，例如一般相对论和杨利米尔，但首先要专注于波浪图，可以获得宝贵的见解，这是一种更简单的几何学领域理论，表现出与上述许多相似之处，并且在与这些物理理论相处的许多困难都具有更为临时的方式。 波浪图（物理学家称为Sigma-Models）是歧管之间谐波图的双曲线类似物，其中域歧管而不是Riemannian是Lorentzian。 因此，他们满足了具有非线性的半线性波方程系统，该系统在梯度中是二次的。 要研究的一些问题是（1）在小扰动下存在平稳的固定溶液及其稳定性，（2）构建Minkowski空间的所有可能的自相似波图，（3）研究由自相似的解决方案引起的奇异性的一般性，以及（4）在两个太空二小剂中找到爆炸的示例。 双曲线部分微分方程在于人类为量化和理解自然界进化现象的努力的核心。 从亚原子颗粒到银河系簇，涉及在有限速度下传播信号和干扰的人已知的每个现象都是由双曲线微分方程建模的。 这种现象的例子是声波的产生和传播（声波），水波（水动力学），地震体和表面波（弹性动力学），电磁波（电动力学）和引力波（一般相对论）。 尽管近年来在非线性双曲系统的研究中取得了长足的进步，但有关其解决方案的规律性，分解和大的时间行为的基本问题在不止一个空间维度上仍未得到解答。 该领域的进展需要对连续体物理学有很好的了解，只有通过对许多简单问题的严格数学分析才有可能，每个问题只对实际物理问题中存在的许多困难进行建模。 有了这样的长期计划，教育下一代的科学家同样重要的是，他们将具有身体和数学能力以及勇气和热情，以继续今天的工作。 这需要对现有数学课程的重新评估以及专门针对数学的新课程的开发，因为它与大学和研究生教育的各个级别的连续体物理学有关。