Hyperbolic Systems of Conservation Laws

守恒定律的双曲系统

• 批准号: 0505430
• 负责人: Alberto Bressan
• 金额: $ 15.15万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505430&HistoricalAwards=false
• 关键词: Hyperbolic; Systems; Conservation; Laws

Award Abstract DMS-0505430, Alberto Bressan, Pennsylvania State UniversityTitle: HYPERBOLIC SYSTEMS OF CONSERVATION LAWSABSTRACT:This research addresses fundamental issues in the theory of hyperbolic systems of conservation laws, in one or more space dimensions. The project also includes the analysis of some new physical models of nonlinear wave propagation, and applications to the theory of differential games.For systems of equations in two or three space dimensions, the P.I. will study basic problems concerning the existence and stability of solutions, in suitable functional spaces. A major goal will be the description of singularities determined by the focusing of nonlinear waves in radially symmetric solutions, and the understanding of oscillation effects, arising from transport phenomena with highly irregular velocity fields. The P.I. also plans to develop new mathematical techniques for the analysis of non-linear waves, described by a class of integro-differential equations. The theoretical results will becomplemented by some rigorous studies on the performance of computational algorithms, for solutions containing shock waves.Phenomena related to the propagation of waves can be found nearly everywhere in nature, and are of great importance in science and engineering. Non-linear effects are responsible for changes in the shape of the waves, and allow what is mathematically described as "singularity formation". In practice, this means that waves can break, such as sea waves rolling up along a beach or shock waves forming at the passage of a supersonic airplane. In two or three space dimensions, the singularities canbe even more dramatic because of "focusing" effects, producing a high concentration of energy near a single point. This is exemplified by an optical lens, concentrating sun raysat a single spot. In the numerical computation of solutions,the loss of regularity due to wave breaking is a considerable source of difficulties, because it can greatly reduce the accuracy of computer algorithms.This research project is concerned with mathematical models describing the propagation of nonlinear waves. Its main goal is to understand the formation andthe evolution of singularities, in one or more space dimensions. Particular attention will be given at specific equations of physical significance, such as one describingwaves in a liquid crystal. The research will also address some issues related to numericalcomputation. In particular, the P.I.~will study the effectiveness of discrete approximations, in case of solutions containing shock waves. Moreover, work will begin on the design of algorithms that can automatically recognize the location of shock fronts,with the eventual goal of producing more efficient computational codes.

奖项摘要DMS-0505430，Alberto Bressan，宾夕法尼亚州立大学标题：双曲守恒律系统简介：这项研究解决了守恒律双曲系统理论中的基本问题，在一个或多个空间维度。 该项目还包括一些新的非线性波传播的物理模型的分析，并应用于微分对策理论。将研究有关的存在性和稳定性的解决方案，在适当的功能空间的基本问题。一个主要的目标将是描述的奇异性所确定的非线性波的径向对称的解决方案，并了解振荡的影响，所产生的运输现象与高度不规则的速度场。私家侦探还计划开发新的数学技术，用于分析由一类积分微分方程描述的非线性波。 对于含有激波的解，理论结果将通过对计算算法性能的严格研究来验证。与波的传播有关的现象在自然界中几乎随处可见，在科学和工程中具有重要意义。非线性效应导致波浪形状的变化，并允许数学上描述为“奇点形成”。 在实践中，这意味着波浪可以破碎，例如沿着海滩卷起的海浪或在超音速飞机通过时形成的冲击波。在二维或三维空间中，由于“聚焦”效应，奇点可能会更加引人注目，在单个点附近产生高浓度的能量。这是由一个光学透镜，集中在一个单一的点太阳光线的例子。在数值解的计算中，由于波浪破碎而导致的规则性损失是一个相当大的困难，因为它会大大降低计算机算法的精度。本研究项目涉及描述非线性波浪传播的数学模型。它的主要目标是理解奇点的形成和演化，在一个或多个空间维度。将特别注意具有物理意义的特定方程，例如描述液晶中的波的方程。研究还将解决一些与数值计算相关的问题。特别是P.I.~将研究离散近似的有效性，在包含冲击波的解决方案的情况下。 此外，还将开始设计能够自动识别冲击波阵面位置的算法，最终目标是产生更有效的计算代码。