Nonlinear Partial Differential Equations and Applications

Fluctuations are ubiquitous both in real world contexts and in key technological challenges like, among others, thermal fluctuations in physical systems, algorithmic stochasticity in machine learning and modeling of weather patterns in climate dynamics. At the same time, such complex systems are subject to an abundance of influences, and depend on a large variety of parameters and interactions.

In addition, for many complex phenomena most of the available information is very often "statistical" (random) and not "exact" (deterministic). A systematic understanding of the interplay of stochasticity and complex dynamical behavior aims at unveiling universal properties, irrespectively of the many details of the concrete systems at hand. Its development relies on the derivation and analysis of universal concepts for their scaling limits, capturing not only their average behavior, but also their fluctuations.

Stochastic partial differential equations are the natural mathematical object to study and understand the role of fluctuations. Another very current and important issue arising in science and technology is the analysis, both theoretically and computationally, of problems that involve several disparate length-scales at once. For example, understanding, modeling and accurately predicting the behavior of materials at the macroscopic scale necessitates to consider their microscopic structure.

Instead of considering materials in one scale and neglecting the finer scales, modern materials science increasingly explicitly and concurrently deals with models of a given material at many different scales. Homogenization and multiscale approaches are the two, respectively theoretical and computational, facets of the mathematical theory to study such problems.

As a matter of fact, it is necessary to consider homogenization in random environments since periodicity is rather restrictive for the modeling of real materials. Growth models are natural mathematical models used in probability and mathematical physics to study stochastic partial differential equations exhibiting a universal scaling limit. Mean-field games are the ideal mathematical structures to study the quintessential problems in the social-economic sciences, which differ from physical settings because of the forward looking behavior on the part of individual agents.

In this context, an agent aims to optimize certain criteria which, together with her/his dynamics, depend on the other agents and their actions. Agents react, anticipate and strategize instead of simply reacting instantaneously. Examples of applications include the modeling of the macro-economy 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 dynamic decisions of others. Some mean-field games models in, for example, telecommunications are naturally set on networks (graphs).

This raises the need of the development of the mathematical tools to study equations on graphs and to understand their behavior across nodes. The project provides research training opportunities for graduate students.

The project is a continuation of the PI's program to develop novel methodologies and techniques for the qualitative and quantitative study of nonlinear first- and second-order deterministic and stochastic partial differential equations (PDEs and SPDEs, respectively) arising in natural and social sciences and engineering. The emphasis is on (i) PDEs with multiplicative ``rough'' path dependence; (ii) homogenization in random media; (iii) well-posedness in domains with singularities; (iv) mean-field games; and (v) convergence of growth models.

Nonlinear, first- and second-order partial differential equations with rough and, in particular, stochastic time dependence arise naturally in the study of fluctuations. The further development of the theory of pathwise solutions is important, for it allows to study of new classes of nonlinear SPDEs, and is expected to play a crucial role in applied areas by providing the tools to analyze previously intractable models.

Studying qualitatively and quantitatively stochastic homogenization problems requires the development of novel arguments and methodologies to address the loss of compactness when going from periodic to stationary ergodic media. The analysis of the behavior of scaled growth models gives rise to new equations. Their well-posedness requires the refinement of some of the by now classical tools from the theory of viscosity solutions.

It is by now well understood that a very important element of the mean-field game theory is the so-called master equation, an infinite-dimensional partial differential equation which subsumes in a single equation both the individual and collective behaviors of agents. In spite of considerable progress in the study of the properties of the smooth solution of the master equation in the presence of both idiosyncratic and common noises, less is known in the absence of the former in which case smooth solutions do not exist in general.

Another important question in this context is the development of a notion of weak solution in the relevant for application nonmonotone setting. Two of the most important questions in the study of equations of graphs is the well-posedness across vertices and the identification of the correct coupling condition.

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.