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Hyperbolic-Parabolic Balance Laws with Applications

双曲-抛物线平衡定律及其应用

基本信息

批准号：
1908195
负责人：
Yanni Zeng
金额：
$ 15.14万
依托单位：
University of Alabama at Birmingham
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1908195&HistoricalAwards=false
关键词：
Hyperbolic Parabolic Balance Laws Applications

项目摘要

This project aims to study a general system of partial differential equations called hyperbolic-parabolic balance laws, with two specific applications in mind: one in bio-medicine and another in gas dynamics. There is a variety of gas-dynamic phenomena that can be described mathematically by hyperbolic-parabolic balance laws. The swirl of complicated gas flows surrounding a space vehicle reentering the Earth's atmosphere is probably one of the most spectacular examples for which these balance laws serve as a (simplified) mathematical description. The second application motivating this research is chemotaxis, the movement of micro-organisms in response to a chemical stimulus. The mechanism of chemotaxis is ubiquitous in biology and medicine, from migration of bacteria or leukocytes to cancer metastasis. Arising from the physical world, the parabolic-hyperbolic systems do not fit exactly into the traditional classification in the theory of partial different equations, and the solutions' behavior is governed by both hyperbolicity and parabolicity, plus a chemical reaction. Integrated into the research activities of the project there is an educational component. It includes curriculum development of hand-on modeling of real-world problems by partial differential equations, one-on-one research mentorship to undergraduate students, and direct participation of research by Ph.D. students. The project encourages the participation of students at all levels, especially students from under-represented groups.The project has two principal mathematical objectives. The first objective is to study the stability of equilibrium solutions. This is to understand how an initial perturbation, such as inaccuracy in measurement, propagates into the solution. The project team is particularly interested in the wave pattern formed by perturbation at long time. The research unifies the theories on hyperbolic balance laws and hyperbolic-parabolic conservation laws in this regard. It has direct application to gas flows in translational and vibrational non-equilibrium. The second objective is to study another model system, Keller-Segel-Fisher/KPP system, beyond the application of the above general theory. It is a chemotaxis model that has logarithmic sensitivity and logistic growth. Here the logarithmic sensitivity is to account for Fechner's law: Subjective sensation is proportional to the logarithm of the stimulus intensity. An important practical problem is what happens if the chemical signal is near void at one end of the domain. This is equivalent to the logarithmic function is near its singular point. In the project, we carry out an in-depth investigation into such a situation. The goals are existence of solution global in time, large time behavior, and possibly large data solutions or strong wave solutions.This project is jointly funded by the Applied Mathematics Program of the Division of Mathematical Sciences and by the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目旨在研究称为双曲抛物平衡定律的偏微分方程的一般系统，并考虑到两个具体应用：一个在生物医学中，另一个在气体动力学中。有许多气体动力学现象可以用双曲-抛物平衡定律来描述。围绕着重返地球大气层的航天器的复杂气流的漩涡可能是这些平衡定律作为（简化的）数学描述的最壮观的例子之一。促使这项研究的第二个应用是趋化性，即微生物对化学刺激的反应。趋化性的机制在生物学和医学中是普遍存在的，从细菌或白细胞的迁移到癌症转移。抛物-双曲方程组起源于物理世界，不完全符合偏微分方程理论中的传统分类，其解的行为受双曲性和抛物性以及化学反应的共同控制。在该项目的研究活动中，有一个教育部分。它包括通过偏微分方程对现实世界问题进行动手建模的课程开发，对本科生进行一对一的研究指导，以及博士直接参与研究。学生该项目鼓励各级学生，特别是代表性不足群体的学生参与，该项目有两个主要数学目标。第一个目标是研究平衡解的稳定性。这是为了理解初始扰动（例如测量不准确）如何传播到解中。项目组对长时间扰动形成的波型特别感兴趣。该研究统一了双曲平衡律和双曲-抛物守恒律在这方面的理论。它直接应用于气体流动的平移和振动的非平衡。第二个目标是研究另一个模型系统，Keller-Segel-Fisher/KPP系统，超越了上述一般理论的应用。这是一个具有对数灵敏度和逻辑增长的趋化性模型。这里对数灵敏度是为了解释费希纳定律：主观感觉与刺激强度的对数成正比。一个重要的实际问题是，如果化学信号在畴的一端接近空洞，会发生什么。这等价于对数函数在其奇点附近。在项目中，我们对这种情况进行了深入调查。目标是解的存在性，时间上的全局性，大时间行为，该项目由数学科学部应用数学计划和刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的评估被认为值得支持。影响审查标准。