Mathematical Aspects of Some PDE Models for Granular Matter and Fish Harvest

本研究项目对偏微分方程理论中的数学问题进行分析和数值研究，这些问题出现在两个由应用程序驱动的模型中。 第一个项目涉及基于平衡定律系统的颗粒流数学模型。 迄今为止的结果主要关注一个空间维度，PDE系统是弱线性退化的情况下。 进一步的理论问题，在一个空间维度将得到解决，包括适定性的系统在空间中的功能与有界变化和源项的影响。 在两个空间维度的模型也将进行研究，从径向对称的解决方案。 第二个项目涉及几家竞争公司的海洋资源捕捞。 由于模型的性质，最优策略被定义在一个空间的正Radon测度。 研究课题包括二维空间模型的最优控制的存在性、解的必要条件和正则性、最优控制措施的唯一性、海岸公园的最优选址等。 颗粒流慢侵蚀极限的研究结果有望为偏微分方程的分析提供新的技术。 最优渔业管理的研究将在二维空间中的最优控制问题和微分对策的测度值解方面产生新的结果。 材料科学、生态学和经济系统中的重要现象的模型自然是根据偏微分方程来制定的，这是本研究项目的主题。 了解雪崩（或滑坡）的动态演变和缓慢侵蚀的影响可以提高规划或减轻此类自然灾害的能力。 关于海洋资源最佳收获的研究将有助于了解如何在竞争性经济环境中保护海洋资源。 该项目的结果将适用于分析这两个重要过程的基本模型的方程。

This research project conducts analytical and numerical investigation of mathematical questions in the theory of partial differential equations that arise in two models motivated by applications. The first project concerns mathematical models for granular flow that are based on systems of balance laws. Results to date concern primarily one spatial dimension, for the case where the PDE system is weakly linearly degenerate. Further theoretical issues in one space dimension will be addressed, including well-posedness of the system in the space of functions with bounded variation and effects of the source term. Models in two space dimensions will also be studied, starting with radially-symmetric solutions. The second project deals with the harvest of marine resources by several competing companies. Due to the nature of the model, optimal strategies are defined in a space of positive Radon measure. Research topics include existence of optimal control, necessary conditions and regularity of solutions, uniqueness of optimal control measures, and optimal location of marine parks, for a model in two space dimensions.Both projects aim at problems that are not well understood. Results of study of the slow erosion limit for granular flow promises to provide new techniques for analysis of the underlying partial differential equations. The study of optimal fishery management will yield new results on measure-valued solutions to optimal control problems and differential games in two space dimensions. Models of important phenomena in materials science, ecology, and economic systems are naturally formulated in terms of the partial differential equations that are the subject of this research project. Understanding the dynamic evolution of a snow avalanche (or landslide) and the effects of slow erosion could enhance the ability to plan for or mitigate such natural disasters. Research on optimal harvest of marine resources will aid in understanding how marine resources can be preserved in a competitive economic environment. The results of this project will apply to analysis of the equations underlying models of both of these important processes.