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.
It is well known that in a crystal a quantum mechanical particle, like an electron, moves almost freely, i.e., the system behaves like a conductor. One question addressed in this proposal is to understand how an electron moves once the atoms in the crystal are randomly displaced so as to break the periodicity. This situation is called 'structural disorder' and it is mathematically described by the 'Random Displacement Model'. This model was formulated more than 20 years ago and it was conjectured that the system at small energies of the electrons behaves like an insulator, i.e., the electron does not move. In joint work with Klopp (France), Nakamura (Japan) and Stolz (UAB) the PI was able to show precisely that, namely that the electron stays localized at sufficiently small energies. Another outcome concerns kinetic theory which is in collaboration with Eric Carlen at Rutgers University. Large systems if undisturbed always are in an equilibrium. There is no flow of energy except for local fluctuations. It is generally observed that if one disturbs the system, it will return to its equilibrium. To describe such phenomena, one needs a model with precise mathematical assumptions and one of them is due to Mark Kac from 1956, which is now called the 'Kac Master Equation'. This model describes a collection of randomly colliding particles and Kac introduced a numerical quantity, the gap, whose size measures the speed at which the system returns to equilibrium. It has been a longstanding question to find an estimate on this quantity and jointly with Carlen (Rutgers University) and Carvalho (University of Lisbon) the PI was able to achieve that. The numerical estimate for this gap is quite reasonable and provides a realistic value for the speed of approach to equilibrium. This work could be considered as the capstone of the program initiated by Kac in 1956. Although these problems come from physics, quite sophisticated mathematics has to be used in order to solve these problems, ranging from probability theory to the theory of partial differential equations to optimization theory. Especially the latter offers a good opportunity for the training of graduate students. Two PhD students were supported under this grant and both received their PhD in 2011. One of them worked on optimization problems with the goal to prove what is known as sharp functional inequalities and took a teaching position. The other worked on the kinetic theory of gases and is now a post doctoral student at Cambridge University in the UK.
