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

反应扩散和动力学方程中的非局部和随机效应。

• 批准号: 1907853
• 负责人: Christopher Henderson
• 金额: $ 14.32万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907853&HistoricalAwards=false
• 关键词: Nonlocal; Stochastic; Effects; Reaction; Diffusion

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.

在许多大规模系统中，给定（颗粒，个体，温度等）的初始配置，可以在定性或定量上描述可预测状态的趋势。 这些系统在物理，生物学和社会科学以及工程学的各种环境中出现，该项目重点介绍了多种模型中这种长期行为的数学研究，这些模型具有特定的重点是等离子体物理学，湍流燃烧和人群动态的数学研究。 在考虑的模型中，两个主要的基本挑战是存在多个时间和空间量表以及非局部相互作用。 前者需要对长时间尺度上的小规模振荡进行了解。例如，在火焰传播中，“小”随机漂移会产生正面的波动，其统计数据是通过限制随机方程式给出的。 后者需要确定复杂的远程相互作用的影响。 该项目中考虑的一个典型例子是趋化性的影响（这是每种细菌“感官”其他细菌并朝着人口中心移动的现象）对细菌入侵的影响。 在这两种情况下，目的都是确定系统的哪些特征是可以预测的，并且在哪些条件下，这种预测的存在。 计划了各种教育活动，包括对本科，研究生和研究生层面的年轻研究人员的培训。该项目的目的是开发技术工具，以更好地表征环境的随机波动的影响以及个人之间的非局部互动的影响，对解决方案的长期行为影响了多个反应 - 反应 - 造成反应 - 障碍的行为，Hamilton-Jacobi and Kinacic and Kinacic and Kinacic and Kinacic and Kinacic and Kincications。 该项目分为两部分。 第一个涵盖了前繁殖问题，其中移动的接口将两个状态分开。 该界面的形状和动力学与媒体的波动和内部相互作用密切相关。 第二个是来自等离子体物理学的动力学模型的规律性和界限。 这些方程式解决方案解决方案的新估计值继续通过抛物线理论的思想应用而出现。 目的是将这些想法与对非本地效应的精确理解相结合，以削弱当前对拟态性理论的限制，并在预防爆炸的情况下发展物理合理的条件。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来支持的，这是值得的。

项目成果

Non-local competition slows down front acceleration during dispersal evolution

非局部竞争减缓了扩散演化过程中的前沿加速

• DOI: 10.5802/ahl.117
• 发表时间: 2022
• 期刊: Annales Henri Lebesgue
• 作者: Calvez, Vincent;Henderson, Christopher;Mirrahimi, Sepideh;Turanova, Olga;Dumont, Thierry
• 通讯作者: Dumont, Thierry