Shock Waves and Geometry

该奖项下的研究调查了几何与激波数学理论之间的联系。这包括作者最近工作的延续，即发展冲击波的数学理论，该理论适用于爱因斯坦方程——描述时空曲率的时间演化的方程，根据广义相对论的引力场。激波在这里是相关的，因为可压缩欧拉方程作为爱因斯坦方程的一个子系统出现，并且恰好出现在低速和弱引力场的极限中。工作包括建设一个新的宇宙的膨胀宇宙模型,(以哈勃常数),是建模为一个伟大的影响爆炸产生的冲击波前沿,像核爆炸成一个静态的背景,除了一个巨大的规模,而不是由标准宇宙模型中,假定整个宇宙扩张的速度来衡量哈勃定律。如果成功，该结果将提供第一个也是最简单的宇宙学模型，该模型假设哈勃常数只测量局部膨胀，并且与今天在宇宙中观测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，即如果大爆炸中存在冲击波，那么，与宇宙学的标准模型相反，将会有相关的信息丢失，这将使从现有数据重建初始事件的细节成为不可能。此外，将开发格里姆方法的协变版本，试图为爱因斯坦方程提供冲击波相互作用的第一个一般存在理论。这是朝着发展可以应用于广义相对论中激波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，自然几何结构可以处理流体动力学。这项工作与正在进行的研究相吻合，即冲击波解的大时间行为如何与给出冲击波相互作用散射图的李代数几何有关。因此，本研究将激波理论与几何理论在若干层面上联系起来。爱因斯坦的广义相对论描述了宇宙的大尺度发展。尽管它们是在80多年前首次提出的，但它们仍然构成了巨大的数学挑战。这项研究将继续早期的工作，其中这些方程的精确激波解被发现。一种数学理论将被发展出来，它将导致标准“大爆炸”理论的替代方案，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熟悉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.