The Geometry of Shock Waves

冲击波的几何形状

• 批准号: 9500694
• 负责人: John Temple
• 金额: $ 9.2万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500694&HistoricalAwards=false
• 关键词: Geometry; Shock; Waves

DMS-9500694 Temple This proposal addresses the problem of constructing, analyzing and numerically computing shock-wave solutions of systems of nonlinear, hyperbolic conservation laws, with emphasis on the compressible Euler equations of gas dynamics. Shock-waves are modeled by discontinuous solutions in inviscid equations, and shock-waves are what introduce time-irreversibility, decay, loss of information, dissipation and increase of entropy into the dynamics of solutions. The author's recent construction of a shock wave solution of the Einstein equations takes advantage of the fact that the gravitational metric is continuous across a shock, even though the fluid variables are discontinuous. This allows for solutions that are weaker than shock wave, and the proposal explores the organizing effect that the gravitational metric has on the geometry of shock wave propagation. In the nonlinear theory of sound waves, the authors address the open problem of explaining the stability of periodic solutions. Their idea is to treat Lie bracket errors in wave interactions as virtual waves, and thereby they can identify and study a nonlinear cancellation mechanism that explains why the dissipation due to increasing entropy defeats the amplification of waves due to nonlinear interaction in the long time. The proposal involves the analysis and interpretation of the author's recent construction of a new shock wave solution of the Einstein equations of general relativity. This exact solution resolves a problem that was first posed in a famous 1939 paper by J. Robert Oppenheimer and his student H. Snyder. The new shock wave solution of the Einstein equations of general relativity provides a model in which the universe begins with a shock wave explosion instead of the well established ``Big Bang'' scenario (in which the entire universe burst from a single point). The proposal explores the possible application of this construction to modeling the dynamics of stars.

本文讨论了非线性双曲型守恒律组的激波解的构造、分析和数值计算问题，重点讨论了气体动力学的可压缩欧拉方程。激波由无粘方程中的间断解来描述，而激波在解的动力学中引入了时间的不可逆性、衰减性、信息损失、耗散性和增益性。作者最近构造了爱因斯坦方程的激波解，利用了这样一个事实，即引力度规在激波中是连续的，即使流体变量是不连续的。这允许比冲击波更弱的解决方案，该提案探索了引力度规对冲击波传播几何形状的组织效应。在声波的非线性理论中，作者讨论了解释周期解稳定性的公开问题。他们的想法是将波相互作用中的李括号误差视为虚拟波，从而他们可以识别和研究一种非线性抵消机制，该机制解释了为什么在很长一段时间内，由于熵增加而引起的耗散击败了由于非线性相互作用而导致的波的放大。该建议涉及对作者最近构造的广义相对论爱因斯坦方程的新激波解的分析和解释。这个精确的解决方案解决了J·罗伯特·奥本海默和他的学生H·斯奈德在1939年发表的一篇著名论文中首次提出的一个问题。爱因斯坦广义相对论方程的新激波解提供了一个模型，在该模型中，宇宙是从激波爆炸开始的，而不是公认的“大爆炸”情景(整个宇宙从一个点爆发)。该提案探索了将这种结构应用于恒星动力学建模的可能性。