CAREER: Integro-differential and Transport Problems in Partial Differential Equations

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This award supports an integrated research program on partial differential equations and an educational program promoting scientific computing and applied mathematics at the undergraduate and graduate level. The research component of the project is concerned with mathematical models from physics and engineering, specifically models used to describe long-range forces and mass transport/kinetic effects, and seeks to answer a number of fundamental questions.

Answers to these questions will aid in the creation of practical and reliable algorithms for computer simulations of complex physical systems (such as those found in plasma physics, aerodynamics, fluid mechanics, material science) and in the creation of practical and reliable autonomous systems (including but not limited to robotics and control theory). In mathematics, long-range forces and heavy-tailed distribution are often modeled via a class of equations known as integro-differential equations.

The mathematical theory for such equations is not as advanced as that of ordinary or partial differential equations, but integro-differential equations are becoming increasingly common in theory and applications. The mathematical analysis of mass-transport, understood in the broadest sense, deals with dynamics of masses of "particles" moving in reaction to both background forces and collisions between said particles.

Depending on the context, these "particles" may represent actual physical particles or intelligent agents interacting with their environment and with one another. As such, transport models cover many situations: from dynamical formulations of models in statistical mechanics to the study of optimal or stable matchings in economics, and the mathematical results of this project may be of use to such disciplines.

The project explores interconnected questions in these two fields (many of which are relevant to the modeling of plasmas) and in particular will explore the use of transport methods to analyze problems involving long-range forces. The activities in this project also provide substantial support for the recruitment, training, and mentorship of mathematically talented undergraduate and graduate students at Texas State University.

These activities include campus-wide events with guest speakers from academia and industry, specialized advanced courses at the undergraduate and graduate levels, and research mentorship of graduate students and advanced undergraduate students. A prime objective of this project's educational efforts is increasing the numbers of graduates entering careers in industry and/or academia that require state of the art mathematical and computational training.

The research activities in this project fall, thematically speaking, in one or more of the following areas: high dimensional Hamilton-Jacobi equations, kinetic equations with nonlocal terms, integro-differential methods for interface problems, and Jacobian equations/transport problems. A common thread connecting these diverse topics is the search for pointwise estimates and regularity estimates for solutions of elliptic and parabolic partial differential equations, and as such questions about estimates makes for a considerable portion of this project's research efforts.

While a few of the problems considered in this project lie squarely within one of the categories above, they are motivated by a potential application in a different category - such as the Nonlocal Jacobian equation, which is motivated by the question of obtaining Aleksandrov-type estimates adapted to integro-differential equations. Positive outcomes in the various lines of inquiry could potentially lead to a mesh-free algorithm for viscosity solutions of Hamilton-Jacobi equations, an optimal transport formulation of solutions for nonlinear kinetic equations, new regularity results for two-phase free boundary problems, and new Harnack inequalities for nonlocal equations.

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.