Collaborative Research: Variational Problems and Dynamics

该提案旨在探索动态和变分不等式的相互作用。变分不等式为推导演化方程解的性质提供了一种有效的手段，同样，演化方程也可以用来推导变分不等式。利用这种相互作用在过去是非常富有成效的，研究人员计划用这种观点来解决各种问题。一个是通过利用与多孔介质方程和加利亚多-尼伦伯格不等式的惊人联系，找到Hardy-Littlewood-Sobolev不等式的各种例子的修正项。特别是，对数Hardy-Littlewood-Sobolev不等式的修正项将导致对描述某些细菌趋化性的Keller-Segel模型解的更好理解。类似的理念也适用于动力学理论中的某些问题，计划对一些非齐次Kac型主方程的接近平衡的速度进行定量估计。这些研究与量子力学中的类似问题密切相关。在这里，PI计划证明描述耗散量子力学系统的Lindblad算子的超收缩性估计，目的是获得接近平衡的速度的定量估计。另一个问题是在随机位移模型中证明Lifshitz尾。目的是了解材料的导电性。科学技术中的许多现象都可以用进化方程来模拟。一个有趣的例子，在这个提议中，是Keller Segal模型，它模拟了细菌运动中的聚集或缺失。理解这些方程解的行为在生物学和数学上都很有趣。同样，人们广泛观察到，许多相互作用的粒子的系统，无论是经典的还是量子力学的，都朝着平衡发展，并且它们以一定的速度发展，通常在很大程度上与粒子的数量无关。了解这一点，并确定这一速度是本研究的目标之一。另一个非常有趣的问题是导体和绝缘体的区别。在量子力学中有一些简单的模型可以展示这些行为。虽然不可能通过精确的计算来理解这些现象，使用数学技术特别是分析，PI的目标是更好地理解这些过程。相反，应用问题，例如，描述大坝中水渗流的多孔介质方程，可以用来发现有趣的数学事实，从而提高对其他问题的理解。纯粹数学和应用数学的相互作用是PI研究的重点，这是教育研究生和本科生的一个很好的方式，并吸引他们进入数学研究。

The proposal aims to explore the interplay of dynamics and variational inequalities. Variational inequalities provide an effective means toderive 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.