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.
|Effective start/end date||8/15/09 → 7/31/12|
- National Science Foundation: $130,000.00