Nonlocal and Stochastic Effects in Reaction-Diffusion and Kinetic Equations.

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In many large-scale systems, given an initial configuration (of particles, individuals, temperature, etc.) there is a trend toward a predictable state that can be described qualitatively or quantitatively. These systems arise in a wide variety of contexts in the physical, biological, and social sciences as well as engineering, and this project focuses on the mathematical study of such long-time behavior in a variety of models with a specific focus on those arising in plasma physics, turbulent combustion, and population dynamics. Two major fundamental challenges in the models considered are the existence of multiple temporal and spatial scales and nonlocal interactions. The former requires developing an understanding of how small-scale oscillations "average out" over long time scales; for instance, in flame propagation, a "small" random drift produces fluctuations of the front whose statistics are given by limiting stochastic equation. The latter requires determining the impact of complex long-range interactions. A typical example considered in this project is the influence of chemotaxis (which is the phenomenon in which each individual bacterium "senses" other bacteria and moves towards the population center) on bacterial invasions. In both cases, the aim is to determine which features of the systems are predictable and under what conditions such predictions hold. Various educational activities, including the training of young researchers at the undergraduate, graduate, and post-graduate level, are planned.The goal of the project is to develop technical tools that allow to better characterize the effects of stochastic fluctuations of the environment and nonlocal interactions between individuals affects the long-time behavior of solutions to several reaction-diffusion, Hamilton-Jacobi, and kinetic equations. The project breaks down into two major portions. The first encompasses front propagation problems in which a moving interface separating two states emerges. The shape and dynamics of this interface are strongly related to the fluctuations of the media and to internal interactions. The second is the regularity and boundedness of kinetic models coming from plasma physics. New estimates of solutions to these equations continue to emerge via the application of ideas from the parabolic theory. The goal is to combine these ideas with a precise understanding of nonlocal effects in order to weaken current restrictions on the well-posedness theory and develop physically reasonable conditions under which blow-up is prevented.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.