Divergence-measure fields and the structure of solutions of systems of hyperbolic conservation laws

对于标量和方程组的多维守恒定律的研究，是目前广泛研究的课题。虽然在多维标量守恒律方面已经取得了重大进展，但目前还没有关于双曲守恒律多维系统的一般理论。主要的困难之一是，无论初始数据是否平滑，解都可能在有限时间内产生奇点。这些奇点被称为冲击波。对于严格双曲型一维系统，可以证明，如果初始数据的总变分足够小，则在有界变分函数空间中存在全局熵解。然而，守恒律的解并不是有界变分的一般函数。此外，还证明了有界变分函数空间在数学上不足以描述具有守恒律的多维系统的解。守恒定律系统理论的这些缺陷促使首席研究员发展一种散度测量领域的理论。散度测量场为描述守恒律系统的解提供了一个更一般的框架。主要研究者推测守恒律系统的解有一个特殊的结构，在这个意义上，激波被支持在一个协维1的可整流集合上，其中的解有强迹。在激波外，解被推测为近似连续的。本项目将通过分析直接在熵不等式给出的方程上解的重标度的爆破极限来研究这些问题。该方案依赖于散度场的分析，解的强迹的存在与散度场的弱法向迹的存在是相关的。此外，对散度测度场的分析提供了熵耗散测度的信息。该项目还将探索退化抛物-双曲方程解的结构，因为它们与散度测量场的关系与双曲守恒律解的关系相同。守恒定律及其相关的向量场控制着广泛科学学科的物理过程，包括流体力学、固体力学、声学、化学和电磁学。冲击波在物理系统中无处不在，出现在空气动力学、生物系统和化学过程中，但它们的数学结构尚未被很好地理解。双曲守恒律系统解的结构分析将为理解激波打开新的大门。发散测度场的空间比所谓有界变分向量场的空间大，是本课题的重点，对发散测度场空间的分析将为研究其他出现“弱可微向量场”的方程提供新的工具。本提案的研究计划与研究生指导和交叉合作紧密结合。这将鼓励通过学生、博士后和教师之间的合作来传播研究成果，他们将与首席研究员互动。她还将把她的研究计划与旨在扩大代表性不足群体参与的活动结合起来，就像她过去所做的那样。

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.