Divergence-Measure Fields and Nonlinear Conservation Laws

散度场和非线性守恒律研究摘要在这个项目中，首席研究员将研究散度场，即散度为Radon测度的矢量场，以及它们在非线性守恒律中的应用。尽管散度度量场仅仅从分析的角度来看可能是有趣的，但研究它们的主要动机是促进我们对非线性守恒律的熵解的理解。Glimm证明了一维严格双曲型守恒律组在有界变差函数空间中存在整体熵解，假设初始数据具有足够小的总变分。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)时，解u通常是符号Radon测度或L^p-函数。更多地了解散度场的性质将促进我们对非线性守恒律的熵解的理解。Ambrosio，Anzellotti，Chen-Frid.Chen-Torres和Ziemer等人已经研究了散度场。然而，关于它们的分析，特别是对于无界向量场的情况，仍然有许多悬而未决的问题。因此，主要研究者将研究散度度量场的分析性质，包括有限周长集合上的无界散度度量场的法迹和Gauss-Green公式。主要研究人员还将应用散度度量场的理论来开发定向表面上的柯西通量的一般框架，定向表面是有限周长集合的边界。这一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的公式化中捕捉测量值生产密度和熵耗散。多维守恒定律的研究，无论是标量方程还是系统，目前都是重要的研究课题。对多维双曲系统的研究(甚至在一维情况下)正促使人们寻找新的分析工具，以便为这些困难的方程带来一些见解。主要研究人员打算利用几何测度论的技巧来研究某些双曲型守恒律。

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.