Problems in Nonlinear Geometric Field Theories

拟议的研究是在数学物理中产生的非线性双曲型偏微分方程组的领域，具体涉及波图初值问题解的正则性、分解和大时间行为。虽然这一领域的任何研究的最终目标都是理解物理相关场论的动力学，如广义相对论和杨-米尔斯，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理论，与上面的许多相似之处，其中许多处理这些物理理论的困难以更透明的方式呈现。波图(物理学家称为西格玛模型)是流形之间调和映射的双曲模拟，其中区域流形而不是黎曼流形是洛伦兹的。因此，它们满足一个半线性波动方程组，该方程组具有在梯度上是二次的非线性。要研究的问题有：(1)方程的光滑定常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)研究由自相似解引起的奇性的一般性；(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，人类已知的每一种现象，涉及有限速度的信号和扰动的传播，都用双曲型微分方程式来模拟。这种现象的例子有声波(声学)、水波(流体动力学)、地震体波和面波(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。尽管近年来对非线性双曲方程组的研究取得了长足的进展，但关于其解在一维以上空间的正则性、分解和大时间性态等基本问题仍然很大程度上没有得到解答。在这一领域的进展需要对连续介质物理有很好的理解，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，每个问题只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。有了这样一个长远的计划，培养下一代科学家也同样重要，他们将拥有物理和数学能力，以及勇气和热情，继续今天正在进行的工作。这需要认真重新评估现有的数学课程，并开发专门针对数学的新课程，因为它涉及大学和研究生各级教育中的连续统一体物理。

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.