Hyperbolic Systems of Conservation Laws

奖项摘要：DMS-0505430， Alberto Bressan，宾夕法尼亚州立大学题目：守恒定律的双曲系统摘要：本研究在一个或多个空间维度上解决守恒定律的双曲系统理论中的基本问题。该项目还包括分析非线性波传播的一些新的物理模型，以及在微分对策理论中的应用。对于二维或三维空间的方程组，P.I.将研究在合适的泛函空间中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非线性波的聚焦所决定的奇点，以及理解由高度不规则速度场的输运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-微分方程来描述。理论结果将通过对包含激波的解的计算算法性能的一些严格研究来完善。与波的传播有关的现象在自然界中几乎无处不在，在科学和工程中具有重要意义。非线性效应导致了引力波形状的变化，并允许在数学上被描述为“奇点形成”。实际上，这意味着波浪会破裂，比如沿着海滩卷起的海浪，或者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，奇点可能更加引人注目，在单个点附近产生高度集中的能量。光学透镜就是一个例子，它把太阳光集中在一个点上。在解的数值计算中，由于波浪破碎而导致的规则性损失是一个相当大的困难来源，因为它可以大大降低计算机算法的准确性。这个研究项目涉及描述非线性波传播的数学模型。它的主要目标是在一个或多个空间维度上理解奇点的形成和演化。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。本研究也将涉及一些与数值计算相关的问题。特别地，P.I.~将研究离散近似在含有激波的解的情况下的有效性。此外，将开始设计能够自动识别冲击前沿位置的算法，最终目标是产生更高效的计算代码。

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.