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

在许多大系统中，给定(粒子、个体、温度等的)初始构型。有一种趋势，可以定性或定量地描述一种可预测的状态。这些系统出现在物理、生物和社会科学以及工程的各种背景下，本项目专注于在各种模型中对这种长期行为的数学研究，特别关注那些出现在等离子体物理、湍流燃烧和人口动力学中的模型。所考虑的模型中的两个主要基本挑战是多个时间和空间尺度的存在和非局部相互作用。前者需要发展对小尺度振荡如何在长时间尺度上“平均”的理解；例如，在火焰传播中，“小的”随机漂移产生锋面的波动，其统计数字由限制性随机方程给出。后者需要确定复杂的远程相互作用的影响。这个项目中考虑的一个典型例子是趋化性(即每个单独的细菌“感觉”到其他细菌并向种群中心移动的现象)对细菌入侵的影响。在这两种情况下，目标都是确定系统的哪些特征是可预测的，以及这种预测在什么条件下成立。计划了各种教育活动，包括对本科、研究生和研究生水平的年轻研究人员的培训。该项目的目标是开发技术工具，使其能够更好地表征环境的随机波动的影响，以及个体之间的非局部相互作用影响几个反应扩散方程、哈密顿-雅可比方程和动力学方程的解的长期行为。该项目分为两个主要部分。第一个问题包括前线传播问题，在这个问题中出现了一个分离两个状态的移动界面。这种界面的形状和动力学与介质的涨落和内部相互作用密切相关。二是等离子体物理动力学模型的正则性和有界性。通过应用抛物线理论的思想，对这些方程的解的新的估计不断出现。目标是将这些想法与对非局部效应的精确理解结合起来，以削弱目前对适当性理论的限制，并开发防止爆炸的物理合理条件。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.