Divergence-Measure Fields and Nonlinear Conservation Laws

拟研究摘要：monica torin本项目主要研究散度测量场；即散度为Radon测度的向量场及其在非线性守恒律中的应用。尽管散度测量场可能仅仅从分析的角度来看是有趣的，但研究它们的主要动机是促进我们对非线性守恒定律的熵解的理解。Glimm证明了一维严格双曲守恒律系统在有界变分函数空间中具有全局熵解，假设初始数据的总变分足够小。但是，当初始数据很大或系统不是严格的双曲型时（特别是对于多维情况），那么解u通常是带符号的Radon测度或L^p函数。了解散度测量场的更多性质将促进我们对非线性守恒定律的熵解的理解。包括Ambrosio， Anzellotti, Chen-Frid， Chen-Torres和Ziemer在内的作者已经研究了散度测量领域。但是，对于它们的分析，特别是对于无界向量场的分析，仍有许多问题有待解决。因此，主要研究者将在有限周长集合上研究发散测量场的解析性质，包括法向迹和无界发散测量场的高斯-格林公式。首席研究员还将应用散度测量场理论来开发柯西通量在有限周长集边界的定向表面上的一般框架。这个非常一般的框架将允许在双曲守恒定律的熵解的平衡定律和熵耗散的公式中捕获测量值生产密度。多维守恒定律的研究，无论是标量方程还是标量系统，都是目前重要的研究课题。在许多维度（甚至在一维情况下）对双曲系统的研究激发了对新的分析工具的探索，这些分析工具可以为这些困难的方程带来一些见解。主要研究者打算用几何测量理论的方法来研究某些双曲型守恒定律。

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.