This project will develop systematic theories to understand the solution structures of nonlinear free boundary problems arising from change-of-type systems in multidimensional conservation laws and from tumor growth and treatment models. Multidimensional conservation laws are mathematical models for fundamental processes in physics and engineering, such as high-speed flows and supersonic jets. A distinctive feature of multidimensional conservation laws in self-similar coordinates is that they change their type (transonic), meaning that they are hyperbolic (supersonic) far from the origin, and mixed (subsonic) near the origin. The investigator will continue her work on these nonlinear transonic problems to gain new physical insights, to develop novel analytical tools, and to find the correct mathematical frameworks in which to pose the nonlinear conservation laws. In a different area, the investigator will develop analytical and numerical theories to understand and to refine model problems arising in tumor growth and treatment.
This project will lead to a deeper understanding of multidimensional transonic problems and tumor model problems, and will provide more efficient and effective methods for the study of systems that include compressible gas dynamics, elasticity, thermodynamics, multi-phase and porous medium flow, and complex biological systems. The project will take place at a Hispanic-serving institution and will involve undergraduate students in simulations of these nonlinear problems, preparing them for further work in design, implementation, and development of algorithms.
