RUI: Nonlinear Free Boundary Problems

RUI：非线性自由边界问题

• 批准号: 0707694
• 负责人: Eun Heui Kim
• 金额: $ 12.87万
• 依托单位: California State University-Long Beach Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707694&HistoricalAwards=false
• 关键词: RUI; Nonlinear; Free; Boundary; Problems

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.

该项目将发展系统理论来理解由多维守恒定律中的类型变化系统以及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（例如高速流动和超音速喷射）的数学模型。 自相似坐标中多维守恒定律的一个显着特征是它们改变了它们的类型（跨音速），这意味着它们远离原点是双曲（超音速）的，而靠近原点是混合（亚音速）的。 研究人员将继续研究这些非线性跨音速问题，以获得新的物理见解，开发新颖的分析工具，并找到提出非线性守恒定律的正确数学框架。 在不同的领域，研究人员将发展分析和数值理论来理解和完善肿瘤生长和治疗中出现的模型问题。该项目将导致对多维跨声速问题和肿瘤模型问题的更深入理解，并将为包括可压缩气体动力学、弹性、热力学、多相和多孔介质流以及复杂生物系统在内的系统研究提供更高效和有效的方法。 该项目将在一家为西班牙裔服务的机构进行，让本科生参与这些非线性问题的模拟，为他们进一步设计、实现和开发算法做好准备。