The major focus of this project is to develop systematic theories to understand the solution structures of nonlinear elliptic boundary value problems arising from change-of-type systems in multidimensional conservation laws, and from experiments in morphogenesis and ecological systems. It is a distinctive feature of multidimensional conservation laws in self-similar coordinates that they change type, being hyperbolic far from the origin but of mixed type near the origin. Analysis of behavior near the origin gives rise to interesting open problems in elliptic partial differential equations, including nonlinear free boundary value problems, degenerate oblique boundary problems, and quasilinear degenerate elliptic systems. Such problems arise in many different combinations in the study of two-dimensional Riemann problems, including those for the compressible Euler equations of gas dynamics. This project also investigates a class of singular non-quasimonotone elliptic systems that arise in models of biological morphogenesis experiments and certain predator-prey interactions.
Multidimensional conservation laws are mathematical models for fundamental processes in physics and engineering, such as high-speed flows and supersonic jets. While tools for numerical simulations of multidimensional conservation laws have been developed extensively, there is very little analytical theory available. It is our goal to establish parts of the needed theory and to investigate the structure of solutions for such problems. The results of the project will have application to a wide a variety of physical systems.
