Divergence-Measure Fields and Nonlinear Conservation Laws

发散测度场和非线性守恒律拟议研究摘要Monica Torres在这个项目中，主要研究员将研究发散测度场;即，其发散是Radon测度的向量场，以及它们在非线性守恒律中的应用。尽管仅从分析的角度来看，发散测度场可能是有趣的，但研究它们的主要动机是为了提高我们对非线性守恒律熵解的理解。 Glimm证明了一维严格双曲守恒律方程组在有界变差函数空间中有一个整体熵解，假设初始数据具有足够小的总变差。然而，当初始数据很大或系统不是严格双曲的（特别是对于多维情况），则解u通常是符号Radon测度或L^p函数。 了解散度测度场的更多性质将有助于我们进一步理解非线性守恒律的熵解，Ambrosio，Anzellotti，Chen-Frid，Chen-Torres和Ziemer等作者已经研究了散度测度场.然而，关于它们的分析，特别是对于无界向量场的分析，仍然有许多悬而未决的问题。因此，主要研究者将研究发散测度场的分析性质，包括有限周长上无界发散测度场的正常迹线和高斯-格林公式。首席研究员还将应用理论的发散措施领域的发展的一般框架柯西通量在定向表面是有限周长集的边界。这个非常一般的框架将允许捕获测量值的生产密度的平衡律和熵耗散的双曲守恒律的熵解的制定。 多维守恒律的研究，无论是标量方程和系统，是目前的重大研究工作的主题。在许多维度（甚至在一维情况下）的双曲系统的调查是激励寻找新的分析工具，可以带来一些洞察这些困难的方程。主要研究者打算使用几何测度理论技术来研究某些双曲守恒律。

Divergence-measure fields and nonlinear conservation lawsAbstract of proposed researchMonica TorresIn this project, the principal investigator will study divergence-measure fields; that is, vector fields whose divergence is a Radon measure, and their applications to nonlinear conservation laws. Even though divergence-measure fields can be interesting solely from the analytical point of view, the main motivation to study them is to advance our understanding of entropy solutions for nonlinear conservation laws. It was proven by Glimm that the one-dimensional system of strictly hyperbolic conservation laws has a global entropy solution in the space of functions of bounded variation, assuming that the initial data has sufficiently small total variation. However, when the initial data is large or the system is not strictly hyperbolic (especially for the multidimensional case), then the solution u is generally a signed Radon measure or a L^p-function. Understanding more properties of divergence-measure fields will advance our understanding of entropy solutions for nonlinear conservation laws.Divergence-measure fields have been studied by authors including Ambrosio, Anzellotti, Chen-Frid, Chen-Torres and Ziemer. However, there are still many open questions concerning their analysis, especially for the case of unbounded vector fields. Thus, the principal investigator will study analytical properties of divergence-measure fields including normal traces and Gauss-Green formulas for unbounded divergence-measure fields, over sets of finite perimeter. The principal investigator will also apply the theory of divergence-measure fields to the development of a general framework for the Cauchy flux over oriented surfaces that are boundaries of sets of finite perimeter. This very general framework will allow to capture measure-valued production density in the formulation of the balance law and entropy dissipation for entropy solutions of hyperbolic conservation laws. The study of multidimensional conservation laws, both scalar equations and systems, is currently the subject of significant research effort. The investigation of hyperbolic systems in many dimensions (and even in the one-dimensional case) is motivating the search for new analytical tools that could bring some insight to these difficult equations. The principal investigator intends to use geometric measure theory techniques to study certain hyperbolic conservation laws.