Measure-valued solutions of hyperbolic conservation laws

双曲守恒定律的测值解

• 批准号: 1108048
• 负责人: Mikhail Perepelitsa
• 金额: $ 11.88万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108048&HistoricalAwards=false
• 关键词: Measure; valued; solutions; hyperbolic; conservation

This project is aimed at the analysis of hyperbolic systems of conservation laws. The focus is on: (1) vanishing viscosity limits to the multi-dimensional Euler equations of gas dynamics; and (2) new representation formulas for solutions of conservation laws. The first project aims at the characterization of limits of families of the vanishing viscosity, radial solutions to the multidimensional Euler equations with large, discontinuous data. The methods of compensated compactness will be adopted for systems of equations that do not possess invariant regions, using only the balance of total energy. In the second project the Principal Investigator will establish a new kinetic formulation for hyperbolic conservation laws. In this approach solutions of conservation laws are represented by probability measures on the phase space which, in turn, are represented as divergences of vector fields obtained as values of a contraction semigroup on suitable Hilbert spaces.Hyperbolic systems of conservation laws are fundamental equations in physics. They model diverse phenomena in dynamics of gases, elasticity, and electromagnetism. Because of their importance in applications these equations have been extensively studied. However, there is no complete theory that allows one solving the equations for generic data and describing properties of the solutions. The project introduces several innovative analytical tools to approach these issues and creates a theoretical basis for solving a large variety of equations of this type. The research will potentially have impact on the areas of computational and applied mathematics where the hyperbolic systems are involved. Results of this research will be disseminated through presentations at national and international conferences, seminars and publications in scientific journals. Graduate student projects will be integrated into this research.

这个项目的目的是分析双曲守恒律系统。重点是：（1）消除了多维气体动力学欧拉方程的粘性极限;（2）守恒律解的新表示公式。第一个项目的目的是在家庭的消失粘度，径向解决方案的多维欧拉方程的大，不连续的数据的特点的限制。对于不具有不变区域的方程组，将采用补偿紧性方法，只利用总能量的平衡。在第二个项目中，首席研究员将建立一个新的双曲守恒定律的动力学公式。在这种方法中，守恒律的解由相空间上的概率测度表示，而相空间上的概率测度又表示为向量场的散度，而向量场的散度又表示为合适的Hilbert空间上的压缩半群的值。他们模拟气体动力学、弹性和电磁学中的各种现象。由于这些方程在实际应用中的重要性，人们对它们进行了广泛的研究。然而，还没有一个完整的理论，允许一个通用的数据求解方程和描述的解决方案的属性。该项目引入了几种创新的分析工具来解决这些问题，并为解决这种类型的各种方程奠定了理论基础。该研究将对涉及双曲系统的计算和应用数学领域产生潜在的影响。这项研究的结果将通过在国家和国际会议、研讨会上的介绍以及在科学杂志上发表文章的方式传播。研究生项目将被纳入这项研究。