Analytical and Geometrical Problems in Nonlinear Partial Differential Equations
Abstract of Proposed Research Luis Caffarelli
This research is to study the mathematical analysis of a number of scientific phenomena that are modeled by nonlinear partial differential equations. Specific topics include the properties of solutions of nonlinear problems involving anomalous (in particular integral) diffusion processes, such as phase transitions, fluid dynamics and optimal control. Also nonlinear random homogenization of fully non linear equations or constrained problems in randomly perforated domains. Antenna design for general signal propagation laws, and other Minkowski type problems
Specific examples of the type of phenomena being studied include the analysis of equations modeling boundary control (optimizing insulation shape across a surface, or the behavior of semi permeable membranes), surface flame propagation and the pricing of options when processes are highly discontinuous ( Levi processes). Also the effective heating of a family of small, randomly distributed, heating sources, the propagation of a flame on a medium with random occlusions or the sliding of a drop on a rough surface and the design of an optimal reflecting surface in a periodic medium.
My research team, which I define as myself, associated postdocs and graduate students, plus collaborators; have studied, among others, the following issues, thanks in good part to support from the NSF. We describe three areas of research 1. Nonlinear homogenization 2. Nonlocal equations 3. Segregation models NONLINEAR HOMOGENIZATION: What is homogenization? Consider physical phenomena that take place in a complex, highly oscillatory media, for instance: 1) a flame propagating through a media composed of thin layers of different materials, arranged periodically or randomly, 2) a chemical flowing through the pores of a semipermeable membrane If we look at these phenomena, at a scale comparable to that the layers of materials for the flame, or the channels in the membrane for the chemical flow, a description of the resulting phenomena through a mathematical model will be complex, and difficult to reproduce, since each layer or pore has to be taken into consideration. However, most of the time we are really interested in the behavior of the phenomena at a larger scale where the array of layers or channels become undistinguishable, like seen from far away. We just see bulk properties: a flame front propagating in all directions, or continuous fluid flow through the membrane in some regions and not in others. But in reality, the now "micro scale complexity" of the media reflects itself at the bulk level, for instance, in the different speeds with which the flame propagates in each direction, or in the relations between chemical concentration and flow. Our research team has made contributions in this area for different models: flame propagation, semipermeable membranes, the shape and evolution of drop on an etched or composite surface and optimal control of diffusion in random media. We showed the presence of hysteresis (past dependence) for the shape of drops and propagation of fronts, estimated the flow through the membrane in relation to the density and "capacity of the pores," etc. NONLOCAL EQUATIONS AND RELATED ISSUES: Traditionally, most mathematical modeling was based on the "continuum" idea: temperature transfer, a dispersing population, an elastic body, or the price of a stock evolved by its interaction just with its "adjacent environment," with global behavior that emerged just by each particle knowing what was happening to the next ones, resulting in a description via "differential" quantities. This is the case for heat propagating in a metal, or the displacement of a packed crowd, or the price of a stock under normal market conditions, where there is considerable agreement on its value. However, more recently, attention has turned to some phenomena where interactions occur at long range. In the case of water temperature on the ocean surface, for instance, heat transfer occurs mainly through interaction with the atmosphere. This leads to models where by interactions on the temperature of the ocean surface, through the mediation of the atmosphere, occur at long range. This is known as the quasigeostrophic model. In a similar fashion, by chemical mediation, (chemotaxis) a population of organisms can create non-adjacent interactions, where the evolution of each organism is dictated by their joint action at some range. And it is well accepted that markets may change discontinuously and exhibit large oscillations. (Levy processes) We have developed a series of analytical tools, which are used to understand which types of evolutions are feasible and stable, as well as identify the possible formation of discontinuous transitions. Several groups in the US and abroad are developing new numerical simulation methods based on our results. SEGREGATION MODELS: Segregation phenomena take place, mathematically, in different contexts. For example, when there is high competition for resources among species, when particles annihilate under contact in physics and when we want to split a given space in a number of optimal regions in terms of energy consumption, radiation, etc. We have explored a series of different phenomena, the existence and stability of limiting configurations as competition grows, etc.
