RUI: Nonlinear Free Boundary Problems

这个项目将开发系统的理论来理解由多维守恒定律中的类型变化系统和肿瘤生长和治疗模型引起的非线性自由边界问题的解结构。多维守恒定律是物理和工程中基本过程的数学模型，如高速流动和超音速喷流。自相似坐标下多维守恒定律的一个显著特征是它们改变了它们的类型(跨音速)，这意味着它们在远离原点的地方是双曲的(超音速的)，在原点附近是混合的(亚音速的)。研究人员将继续她在这些非线性跨音速问题上的工作，以获得新的物理见解，开发新的分析工具，并找到正确的数学框架来提出非线性守恒定律。在不同的领域，研究人员将发展分析和数值理论，以了解和完善肿瘤生长和治疗中出现的模型问题。该项目将使人们对多维跨音速问题和肿瘤模型问题有更深入的了解，并将为包括可压缩气体动力学、弹性、热力学、多相和多孔介质流动以及复杂生物系统在内的系统的研究提供更有效的方法。该项目将在一个为拉美裔服务的机构进行，并将让本科生参与这些非线性问题的模拟，为他们在设计、实现和开发算法方面的进一步工作做好准备。

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.