Measure-valued solutions of hyperbolic conservation laws

这个项目的目的是分析双曲型守恒律系统。重点是：(1)气体动力学多维欧拉方程的粘性极限消失；(2)守恒律解的新表示式。第一个项目旨在刻画具有大而不连续的数据的多维Euler方程的消失粘性族的极限，即径向解。对于不具有不变区域的方程组，将采用补偿紧致的方法，仅使用总能量的平衡。在第二个项目中，首席调查员将为双曲守恒定律建立一个新的动力学公式。在该方法中，守恒律的解被表示为相空间上的概率度量，而相空间上的概率度量又被表示为在适当的Hilbert空间上作为压缩半群的值的向量场的发散。双曲型守恒律组是物理学中的基本方程。它们模拟了气体动力学、弹性和电磁学中的各种现象。由于它们在应用中的重要性，这些方程得到了广泛的研究。然而，没有一个完整的理论允许人们解一般数据的方程并描述解的性质。该项目引入了几种创新的分析工具来解决这些问题，并为解决大量这类方程奠定了理论基础。这项研究可能会对涉及双曲型系统的计算和应用数学领域产生影响。这项研究的结果将通过在国家和国际会议、研讨会和科学期刊上发表的报告来传播。研究生项目将被纳入这项研究。

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.