The study of multidimensional conservation laws, for both scalar and systems of equations, is currently the subject of broad research efforts. Though significant progress has been made in the case of multidimensional scalar conservation laws, there is currently no general theory for multidimensional systems of hyperbolic conservation laws. One of the main difficulties is that solutions can develop singularities in finite time, regardless of the smoothness of the initial data. These singularities are known as shock waves. For the strictly hyperbolic one-dimensional system, it can be shown that if the initial data has sufficiently small total variation, then there exists a global entropy solution in the space of functions of bounded variation. However, solutions of conservation laws are not in general functions of bounded variation. Moreover, it has been shown that the space of functions of bounded variation is mathematically insufficient for describing solutions of multidimensional systems of conservation laws. These shortcomings in the state-of-the-art theory for systems of conservation laws have motivated the principal investigator to develop a theory for divergence-measure fields. Divergence-measure fields provide a more general framework for characterizing solutions of systems of conservation laws. The principal investigator conjectures that solutions of systems of conservation laws have a special structure, in the sense that the shock waves are supported on a codimension-one rectifiable set where the solution has strong traces. Outside the shock waves, the solution is conjectured to be approximately continuous. This project will investigate these questions by analyzing the blow-up limits of the rescalings of the solution directly on the equation given by the entropy inequality. This plan hinges on the analysis of divergence-measure fields in that the existence of strong traces of the solution are related to the existence of weak normal traces of divergence-measure fields. Moreover, the analysis of divergence-measure fields provides information on the entropy dissipation measures. The project will also explore the structure of solutions to degenerate parabolic-hyperbolic equations, for they relate to divergence-measure fields in the same fashion as the solutions of hyperbolic conservation laws.
Conservation laws and their associated vector fields govern physical processes from broad scientific disciplines, including fluid mechanics, solid mechanics, acoustics, chemistry, and electromagnetism. Shock waves are ubiquitous in physical systems, occurring in aerodynamics, biological systems, and chemical processes, yet their mathematical structure is not well understood. The analysis of the structure of solutions of systems of hyperbolic conservation laws will open new doors to the understanding of shock waves. The analysis of the space of divergence-measure fields, which is larger than the space of so-called bounded variation vector fields and is the focal point of this project, will provide new tools to research other equations where "weakly differentiable vector fields" appear. The research plans of this proposal are tightly integrated with the mentoring of graduate students and cross-collaborations. This will encourage research dissemination through collaboration among students, postdocs, and faculty who will be interacting with the principal investigator. She will also integrate her research plan with activities intended to broaden the participation of underrepresented groups, as she has done in the past.
