Partial Differential Equations and Applications
Abstract
The PI proposes to continue his program to develop novel methods and tools for the study of problemsrequiring working across and bridging many spatio-temporal scales. The focus is on the qualitative andquantitative study of nonlinear deterministic and stochastic partial differential equations (pde and spderespectively) arising in natural and social sciences and engineering. The emphasis is on (i) pde with multiplicativerough path dependence, (ii) homogenization in random media, (iii) well-posedness in domainswith singularities, and (iv) mean-field games.For many complex phenomena, most of the available information is statistical (random) and not exact(deterministic). Incorporating the fluctuations of several physical quantities leads to equations withsingular and random dependence on some of the variables, and spde become the natural mathematicalobjects. Examples are turbulence, phase transitions, front propagation in random media, nucleations,neurophysiology, chemical pollution, macroscopic limits of particle systems, mean-field games, pathwisestochastic control theory, stochastic control with partial observations, etc..The theory of pathwise solutions is important for it allows to study two completely new classes of nonlinearspde. The subject is expected to play a crucial role in applied areas by providing the tools to analyzepreviously intractable models.The bridging of the different scales in multiscale phenomena necessitates the use of random (stationaryergodic) media, as periodicity is a rather restrictive and idealized structure for many applications, andrequires the study of averaged (macroscopic) behaviors. The penalty for this abstraction is the loss ofcompactness which is the cornerstone of the periodic theory. Homogenization in periodic media is a welldeveloped and classical field. Moving to random media, especially for nonlinear problems, requires newideas, often a change of point of view, and the use in an essential way of the probabilistic structure.Modeling in applications like traffic flow, telecommunications and control theory with discontinuous dynamicsas well as the study of some dimension reduction and relaxation problems introduce equations ondomains with singularities, a network and its nodes being one example. The main mathematical issue ishow to define solutions on the singularities to guarantee the well-posedness of the problem.A burgeoning area of research is the theory of mean-field games, which model the behavior of one or severallarge groups of agents and provide rigorous justification of some of the most important questions insocio-economics modeling. Concrete examples of applications in this direction include the modeling of themacro-economy and conflicts in the modern era. In both cases, a large number of agents interact strategicallyin a stochastically evolving environment, all responding to partly common and partly idiosyncraticincentives, and all trying to simultaneously forecast the dynamic decisions of others. The study of meanfieldgames gives rise to many challenging mathematical questions in pde (systems of backward-forwardequations, mass transport, equations in infinite dimensions) and probability (stochastic analysis, interactingparticle systems), whose study requires the introduction and development of new mathematical methodologies.The research project involves five graduate students and two postdocs the PI is currently supervising andmentoring, as well as international collaborations with Claude Le Bris and Pierre-Louis Lions.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 06, 2021
- Source ID
- N000142112325
Entities
People
- Panagiotis E. Souganidis
Organizations
- Office of Naval Research
- United States Navy
- University of Chicago