Collaborative Research: Variational Problems and Dynamics

该提案旨在探讨动态和变分不平等之间的相互作用。变分不等式提供了一种有效的方法来推导发展方程的解的性质，同样，发展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取得了非常丰硕的成果，调查人员计划使用这种视角来处理各种问题。其一是通过利用与多孔介质方程和Gagliardo-Nirenberg不等式的惊人联系，找到Hardy-Littlewood-Soblev不等式的各种例子的修正项。特别是，对数Hardy-Littlewood-Sobolev不等式的修正项将有助于更好地理解描述某些细菌的趋化性的Keller-Segel模型的解。类似的哲学也适用于运动论中的某些问题，计划推导出一些Kac型非齐次主方程接近平衡的速度的定量估计。这些研究与量子力学中的类似问题相吻合。在这里，PI计划证明描述耗散量子力学系统的Lindblad算子的超缩估计，目的也是获得接近平衡的速度的定量估计。另一个问题是证明随机位移模型中的Lifshitz尾。其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演化方程来模拟。这个提案中讨论的一个有趣的例子是凯勒·西格尔模型，该模型模拟了细菌运动中的聚集或不聚集。了解这些方程的解的行为在生物学和数学上都很有趣。同样，人们普遍观察到，由许多相互作用的粒子组成的系统，无论是经典的还是量子力学的，都会进化到一种平衡，并且它们以一定的速度进化，通常在很大程度上与粒子的数量无关。理解这一点，并确定这一速度是本研究的目的之一。另一个令人非常感兴趣的问题是，导体和绝缘体的区别是什么。量子力学中有一些简单的模型被认为可以展示这种行为。虽然不可能通过精确的计算来理解这些现象，但PI的目标是更好地理解这些过程。相反，应用问题，例如描述大坝中水的渗流的多孔介质方程，可以用来寻找有趣的数学事实，这反过来又能提高对其他问题的理解。正是这种纯粹数学和应用数学的相互作用是PI研究的重点，它一直是培养研究生和本科生并吸引他们参与数学研究的一种极好的方式。

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.