Shock Waves and Geometry

该奖项下的研究调查了几何学和冲击波数学理论之间的联系。这包括作者最近关于发展适用于爱因斯坦方程的冲击波数学理论的工作的继续。爱因斯坦方程描述了时空曲率的时间演变，根据广义相对论，引力场。由于可压缩欧拉方程作为爱因斯坦方程的一个子系统，在低速和弱引力场的极限下精确地出现，所以在这里是相关的。这项工作包括构建一个新的宇宙学模型，在该模型中，宇宙的膨胀(以哈勃常数衡量)被模拟为在前沿产生冲击波的大爆炸的影响，类似于进入静态背景的核爆炸，但规模很大，而不是标准的宇宙学模型，在标准宇宙模型中，假设整个宇宙正在以哈勃定律测量的速度膨胀。如果成功，这些结果将为宇宙学提供第一个也是最简单的模型，假设哈勃常数只测量局部膨胀，并与今天在宇宙中观察到的能量密度和背景辐射水平一致。这项研究还有一个有趣的哲学含义，那就是，如果存在来自大爆炸的冲击波，那么与宇宙学的标准模型相比，将会有相关的信息损失，使得无法从目前的数据重建初始事件的细节。此外，还将发展Glimm方法的协变版本，试图为爱因斯坦方程提供激波相互作用的第一个普遍存在理论。这是朝着开发可应用于广义相对论中冲击波的数值模拟的原理迈出的一步。这些研究也将对经典流体产生影响，因为爱因斯坦方程提供了一个独特的环境，在这个环境中，自然的几何结构对流体动力学产生了影响。这项工作与正在进行的关于冲击波解的大时间行为如何与给出冲击波相互作用的散射图的李代数的几何关系的研究相吻合。因此，这项研究在多个层面上将冲击波理论和几何学联系起来。爱因斯坦的广义相对论方程描述了宇宙的大规模发展。尽管它们是在80多年前首次提出的，但它们仍然构成了令人敬畏的数学挑战。这项研究将继续早期的工作，找到这些方程的精确冲击波解。将发展出一种数学理论，它将导致标准的大爆炸理论的替代，并允许在广义相对论中对冲击波进行一般理论处理。由于这些方程包含了经典流体动力学方程作为特例，因此研究也将为这些更常见的问题带来新的见解。

The research under this award investigates connections between geometry and the mathematical theory of shock waves. This includes a continuation of the authors' recent work on developing a mathematical theory of shock-waves that applies to the Einstein equations - the equations that describe the time evolution of spacetime curvature, the gravitational field according to general relativity. Shock waves are relevent here because the compressible Euler equations appear as a subsystem of the Einstein equations, and emerge exactly in the limit of low velocities and weak gravitational fields. The work includes the construction of a new cosmological model in which the expansion of the universe, (as measured by the Hubble constant), is modeled as the effect of a great explosion that generates a shock wave at the leading edge, something like a nuclear explosion into a static background, except on an enormous scale, instead of by the standard cosmological model in which it is assumed that the entire universe is expanding at a rate measured by the Hubble law. If successful, the results will provide the first and simplest model for cosmology under the assumption that the Hubble constant measures only a localized expansion, and consistent with the energy density and background radiation levels observed in the universe today. This research also has an interesting philosophical implication in that, if there were a shock wave present from the Big Bang, then, in contrast to the standard model of cosmology, there would be an associated loss of information that would make it impossible to reconstruct the details of the initial event from present data. In addition, a covariant version of Glimm's Method will be developed in an attempt to provide the first general existence theory of shock wave interactions for the Einstein equations. This is a step toward developing principles that can be applied to the numerical simulation of shock waves in general relativity. These studies will also have implications for classical fluids because the Einstein equations provide a unique setting in which natural geometrical constructs put a handle on the fluid dynamics. This work dovetails with ongoing research into how the large time behavior of shock wave solutions is related to the geometry of the Lie Algebra that gives the scattering picture forshock wave interactions. Thus this research connects the theory of shock waves and geometry on several levels. Einstein's equations of general relativity describe the large scale developmentof the universe. Although they were first proposed more than eighty years ago,they still pose formidable mathematical challenges. This research will continue earlier work in which exact shock wave solutions of these equations were found.A mathematical theory will be developed that leads to alternatives of the standard"Big Bang" theory and that allows a general theoretical treatment of shock waves within the general theory of relativity. Since these equations contain the classical fluid dynamics equations as a special case, the research will also lead to new insights for these more familiar problems.