CAREER: Well-posedness and long-time behavior of reaction-diffusion and kinetic equations

About This Grant

This project focuses on the behavior of physical, chemical, and biological systems that can be modelled by partial differential equations (PDEs). The forces that determine the time-evolution of these systems are complex, making their analysis subtle and technical. Two fundamental questions of interest are the qualitative behavior of these systems, e.g., whether solutions have large fluctuations, and their long-time behavior, e.g., by quantifying the speed with which an invasive species overruns a new environment. These questions are interdependent, with the latter relying on an understanding of the former. Our ability to understand the long-time behavior of PDE, including identifying the key quantities on which each long-time outcome depends, allows us to predict the behavior of real-world systems in a way that cannot be captured purely by numerical simulation, which, by necessity, is restricted to finite time scales. This project will develop novel methods for these goals. Graduate and undergraduate research will be integrated into the project, training the next generation of applied mathematicians and scientists. The project also involves a summer boot camp for entering applied mathematics PhD students transitioning from adjacent, but nonmathematical, fields that shore up their mathematical reasoning (logical thinking) and technical writing skills. Their training is impactful because these students have diverse interests (mathematical biology, machine learning, data science, PDE and numerical analysis, etc.) and go on to careers in industry, academia, and national labs. This project focuses on advances in reaction-diffusion equations and collisional kinetic equations. In the former, the project will develop a novel "Stein's method" approach to PDE that is based on the observation that monotonic steady states of a given PDE satisfy first order autonomous ordinary differential equations (ODE) and that, to show convergence of a generic solution of the PDE to such a steady state, it is enough to show that the generic solution converges to a solution of the ODE. The research will leverage new functional inequalities and ideas in the calculus of variations. In the latter, the project will import techniques from parabolic theory and stochastic analysis to characterize when blow-up occurs in generic domains (both with and without boundaries). This requires the precise and quantitative understanding of the regularity of solutions near the boundary in physical space and the decay of solutions at "large" velocities. 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.