Nonlinear Partial Differential Equations and Applications

University Of Chicago, Chicago IL

Investigators

Abstract

Many applications in Physical and Social Sciences and Engineering, like porous media, composite material, turbulence and combustion, traffic models, spread of crime, climate modeling and prediction, agent models and others, involve heterogeneous media described by partial differential equations, which, typically, depend on many parameters and vary randomly on a small scale. In addition, often the available information, as, for example, in weather prediction, is not exact (deterministic) but statistical (random) with large fluctuations. On macroscopic scales that are much larger than the ones of the heterogeneities, the models often show an effective deterministic behavior, which is much simpler than the original one. The process of the averaging is known as homogenization. Mathematically, this means that the original random and inhomogeneous problem is replaced by a deterministic and homogeneous one. When this averaging is not possible, which is typically the case when the fluctuations are too strong (wild), it is necessary to deal with stochastic media (stochastic partial differential equations), which have rather singular behavior in space and time. The mathematical study of both the stochastic averaging and the stochastic partial differential equations requires original ideas and the development of new methodologies, since both topics fall outside the traditional theories of averaging and partial differential equations. Another burgeoning area of research is the theory of mean field games. Applications that have been so far looked at range from complex socio-economical topics, regulatory financial issues, crowd movement, meaningful big data and advertising to engineering contexts involving "decentralized intelligence'" and machine learning. Mean field games are the ideal mathematical structures to study the quintessential problems in the social-economical sciences, which differ from physical settings because of the forward looking behavior on the part of individual agents. Concrete examples of applications in this direction include the modeling of the macroeconomy and conflicts in the modern era. In both cases, a large number of agents interact strategically in a stochastically evolving environment, all responding to partly common and partly idiosyncratic incentives, and all trying to simultaneously forecast the decision of others. Training of graduate students is an integral part of this research project. The project is about developing general methodologies to study random homogenization, nonlinear stochastic partial differential equations and applications to front propagation, phase transitions, and mean field games. Random environments are much more general than periodic ones. The latter are basically fixed translations of a certain equation while the former can be thought as all the possible equations. This leads to considerable issues of lack of compactness. It is therefore necessary to develop novel tools that combine both the differential and random structures of the media. In this setting, the equation is the random variable and the special dependence signifies the location in space where the equation is observed. The PI and his collaborators were the first to consider stochastic homogenization in stationary ergodic environments. A large part of the project is about the further development of the theory. Stochastic partial differential equations have coefficients with very singular (Brownian) behavior. In the linear context, this can be typically handled by known methods like the classical martingale approach. The latter is based on the linear character of the higher order part of the equation and thus cannot be used for nonlinear problems, where it is necessary to find appropriate notions of solutions. In the context of first- and second-order nonlinear equations, these are the stochastic viscosity and pathwise entropy solutions, introduced by the PI and his collaborators. A part of the project is about the study of the qualitative behavior/properties of these solutions. In the context of mean field games, the PI will concentrate on the role of inhomogeneities at the several level of the game up to the master equation and models with common noise. This will require the development of novel techniques to understand the behavior of the problem past singularities and the role of the averaging. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.