Collaborative Research: Variational Problems and Dynamics

合作研究：变分问题和动力学

• 批准号: 0901632
• 负责人: Eric Carlen
• 金额: $ 32.05万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901632&HistoricalAwards=false
• 关键词: Collaborative; Research; Variational; Problems; Dynamics

The proposal aims to explore the interplay of dynamics and variational inequalities. Variational inequalities provide an effective means to derive properties of solutions of evolution equations and likewise, evolution equations can be used to derive variational inequalities. Exploiting this interplay has been very fruitful in the past, and the investigators plan to approach various problems using this perspective. One is to find correction terms of various examples of the Hardy-Littlewood-Sobolev inequality by exploiting a surprising connection to the porous medium equation and to the Gagliardo-Nirenberg inequality. In particular, a correction term for the logarithmic Hardy-Littlewood-Sobolev inequality will lead to an improved understanding of the solutions of the Keller-Segel model describing the chemotaxis of certain bacteria. A similar philosophy applies as well to certain problems in kinetic theory, with the plan to derive quantitative estimates on speed of approach to equilibrium for some inhomogeneous master equations of Kac type. These investigations tie in with analogous questions in quantum mechanics. Here the PI's plan to prove hypercontractivity estimates for Lindblad operators that describe dissipative quantum mechanical systems, with the aim to obtain quantitative estimates on the speed of approach to equilibrium as well. Another circle of problems is proving Lifshitz tails in the random displacement model. The aim there is to understand the conductivity properties of materials. Many phenomena in science and technology can be modeled by evolution equations. An interesting example, treated in this proposal, is the Keller Segal model, that models the aggregation, or the absence thereof, in the motion of bacteria. Understanding the behavior of solutions of these equations is both biologically and mathematically interesting. Likewise, it is widely observed thatn systems of many interacting particles, either classical or quantum mechanical, evolve toward an equilibrium, and they do this at a certain speed, often largely independent of the number of particles. Understanding this, and determining this speed is one of the objects of this research. Another question of great interest is what distinguishes a conductor from an insulator. There are simple models in quantum mechanics that are supposed to exhibit these kind of behavior. While it is impossible to understand these phenomena by exact computations, using mathematical techniques notably from analysis, the PI's aim to understand these processes better. Conversely, applied problems, e.g., the porous medium equations that describes the seepage of water in dams, can be used to find interesting mathematical facts, which in turn lead to improved understanding of other problems. It is this interplay of pure and applied mathematics that is the focus of the PI's research and it has been an excellent way to educate graduate students as well as undergraduates, and to draw them into mathematical research.

该提案旨在探讨动态和变化不平等的相互作用。变分的不平等是一种有效的手段来得出进化方程解的特性，并且同样，进化方程也可以用于导致变异不等式。过去，利用这种相互作用一直非常富有成果，调查人员计划使用这种观点来解决各种问题。一种是通过利用与多孔培养基方程和Gagliardo-Nirenberg不平等的令人惊讶的连接来找到强大的木材 - 贝布莱夫不平等的各种示例的更正术语。特别是，对数硬​​木木 - 汤生的校正项将导致对描述某些细菌趋化性的Keller-Segel模型的解决方案的理解。类似的哲学也适用于动力学理论中的某些问题，该计划的计划得出了对某些KAC类型的某些不均匀的主方程的均衡速度的定量估计。这些研究与量子力学中的类似问题相关。在这里，PI的计划证明了描述耗散量子机械系统的Lindblad操作员的超收缩率估计值，目的是获得有关平衡方法速度的定量估计。另一个问题是在随机位移模型中证明Lifshitz尾巴。目的是了解材料的电导率。科学和技术中的许多现象都可以通过进化方程来建模。在该提案中处理的一个有趣的例子是凯勒·塞加尔（Keller Segal）模型，该模型在细菌运动中模拟了聚集或缺失。了解这些方程的解决方案的行为在生物学和数学上都很有趣。同样，人们普遍观察到，许多相互作用的粒子的系统，无论是经典的还是量子机械的，它们都会向平衡发展，并且它们以一定的速度进行，通常与粒子数量无关。理解这一点，确定这一速度是这项研究的对象之一。另一个极大的兴趣问题是，是什么区别导体与绝缘体。量子力学中有一些简单的模型应该表现出这种行为。尽管不可能通过精确的计算理解这些现象，但使用数学技术特别是通过分析来理解这些现象，但PI的目的是更好地理解这些过程。相反，应用问题，例如描述大坝中水渗漏的多孔介质方程，可用于找到有趣的数学事实，这又导致对其他问题的理解得到了改善。正是这种纯粹和应用数学的相互作用是PI研究的重点，它是教育研究生和本科生并将其吸引到数学研究的绝佳方式。