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

这项研究项目对偏微分方程组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模型中，原因是应用程序。第一个项目涉及颗粒流的数学模型，这些模型基于平衡定律系统。到目前为止，结果主要涉及一个空间维度，对于PDE系统是弱线性退化的情况。进一步的理论问题将在一维空间中被处理，包括系统在具有有界变化的函数空间中的适定性和源项的影响。还将研究两个空间维度的模型，从径向对称解开始。第二个项目涉及几家相互竞争的公司收获海洋资源。由于模型的性质，最优策略被定义在正Radon测度空间中。研究主题包括最优控制的存在性、解的必要条件和正则性、最优控制措施的唯一性以及海洋公园的最优选址。这两个项目都是针对不太清楚的问题。颗粒流缓慢侵蚀极限的研究结果有望为分析基本的偏微分方程组提供新的技术。最优渔业管理的研究将在最优控制问题的度量值解和二维空间中的微分对策方面产生新的结果。材料科学、生态和经济系统中的重要现象的模型自然是根据偏微分方程组建立的，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这类自然灾害的能力。对海洋资源最佳收获的研究将有助于理解如何在竞争激烈的经济环境中保护海洋资源。该项目的结果将适用于对这两个重要过程的模型所依据的方程式的分析。

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.