Hyperbolic Systems of Conservation Laws

奖项摘要DMS-0505430，阿尔贝托·布雷桑，宾夕法尼亚州立大学标题：双曲型守恒定律：这项研究在一个或多个空间维度上解决了双曲型守恒律理论中的基本问题。该项目还包括一些新的非线性波传播物理模型的分析，以及在微分博弈理论中的应用。对于二维或三维空间的方程组，P.I.将在适当的函数空间中研究有关解的存在性和稳定性的基本问题。一个主要的目标将是描述由径向对称解中的非线性波聚焦所确定的奇点，以及理解由具有高度不规则速度场的传输现象引起的振荡效应。P.I.还计划开发新的数学技术来分析非线性波，用一类积分-微分方程式来描述。理论结果将通过对含有冲击波的解的计算算法的性能的严格研究来实现。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的意义。非线性效应导致了波浪形状的变化，并导致了数学上所说的“奇点形成”。在实践中，这意味着波浪可以破裂，比如沿着海滩翻滚的海浪或超音速飞机通过时形成的冲击波。在两个或三个空间维度上，奇点可能会因为“聚焦”效应而变得更加戏剧性，在单点附近产生高度集中的能量。这是一个光学透镜的例子，它将太阳光线集中在一个单一的点上。在解的数值计算中，由于波浪破碎造成的正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。它的主要目标是了解奇点在一个或多个空间维度中的形成和演化。我们将特别注意具有物理意义的特定方程，例如描述液晶中波的方程。这项研究还将解决与数值计算有关的一些问题。特别是，P.I.~将研究离散近似的有效性，在解包含冲击波的情况下。此外，将开始设计能够自动识别冲击波锋面位置的算法，最终目标是产生更有效的计算代码。

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.