In 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.