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.
The research project Variational Problems and Dynamics was devoted to the development of new mathematical techniques to solve problems arising in physics and biology. Though the problems cover a wide range of science, they are all united by having to do with dynamics and optimization; i.e., variational, problems. A number of results were obtained in collaboration with Michael Loss of Georgia Tech, a long-term collaborator of the PI. Two graduate students worked on research problems developing from the proposed research. One of these, Gabriel Bouch, defended his thesis while supported by this grant, and the other, Katy Craig, started her research under this grant, and has recently defended her thesis too. Both students competed the Ph.D work in four years, and the N.S.F. support was crucial for this. Especially by adding to the national pool of top quality female researchers, the research funded in this grant has had a meaningful broader impact. The fact that many of the problems dealt with in this research came from physics and biology also adds to the broader impact. Thirteen research papers and book chapters were written, mostly now published, and all submitted. Though ambitious goals were set, such a the resolution of an old conjecture of Mark Kac concerning hard-sphere molecular collisions, they were successfully accomplished. The work accomplished has been disseminated through posting online, and presentations at various international meetings. The reception of this work, which is already being further developed by the PI, his collaborators, and others, demonstrate the intellectual merit of the research.
