Nonlinearity in Reaction-Diffusion and Kinetic Equations

反应扩散和动力学方程中的非线性

• 批准号: 2204615
• 负责人: Christopher Henderson
• 金额: $ 16.34万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204615&HistoricalAwards=false
• 关键词: Nonlinearity; Reaction; Diffusion; Kinetic; Equations

Diverse phenomena such as the spread of an invasive species, turbulent combustion, and the evolution of a cloud of particles in a gas can be modeled using partial differential equations. These examples necessarily involve nonlinearity, which is when the rate of change of a quantity depends on the quantity itself. A simple illustration of nonlinearity is the so-called Allee effect, where the reproduction rate of certain species is negative below a minimum population size, perhaps due to factors such as cooperative defense or mate limitation. Typically, nonlinearities present serious difficulties in the analysis of the model. In the context of various scientifically relevant classes of partial differential equations, this project aims to develop an understanding of when a model can be approximated by a (simpler) linear one and, more generally, which essential features are required by a minimal model to faithfully represent the essential behavior of the original phenomenon. The intent is to aid scientists to identify and implement the most tractable model for their investigations. The project will provide training opportunities for undergraduate students.A focus of the project is to develop technical tools for understanding the fundamental nature of the long-time behavior of several reaction-diffusion equations. These model systems exhibiting growth (reaction) and spreading (diffusion) in which an interface (front) forms and propagates with a constant speed. Classically, these systems are divided into two categories based on the underlying mechanism driving the movement of the front: linear behavior far beyond the front ('pulled' fronts) or nonlinear behavior at the front ('pushed' fronts). Often, the shape and speed of these fronts can depend strongly on intrinsic properties of the system, generally represented by a parameter. As the parameter changes, the character of the fronts may change from 'pulled' to 'pushed' or vice versa. New advances stemming from the recent introduction of ideas such as relative entropy and quantitative steepness as well as a careful understanding of the regularity of equations have opened the door to a high level of precision in characterizing this pulled-pushed transition. A second focus of the project is the development of the well-posedness theory of various collisional kinetic 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.

可以使用部分微分方程对气体中的侵入性物种，湍流燃烧的传播，湍流燃烧和颗粒云的演变等各种现象进行建模。 这些示例必然涉及非线性，这是数量变化率取决于数量本身的时候。 非线性的一个简单说明是所谓的合同效应，其中某些物种的繁殖率低于最低人口规模，这可能是由于合作防御或伴侣限制等因素所致。通常，非线性在模型的分析中遇到了严重的困难。 在各种科学相关的部分偏微分方程的背景下，该项目旨在建立对何时可以通过（简单）线性的近似模型近似的理解，更一般而言，最小模型要求这些基本特征忠实地代表原始现象的基本行为。目的是帮助科学家识别和实施其调查的最可行的模型。该项目将为本科生提供培训机会。该项目的重点是开发技术工具，以理解多个反应扩散方程的长期行为的基本性质。 这些模型系统表现出生长（反应）和扩散（扩散），其中界面（前面）形成并以恒定的速度传播。 从经典上讲，这些系统根据驱动正面运动的基本机制分为两类：线性行为远远超出了正面（“拉动”前部）或非线性行为（“推”前部）。 通常，这些前端的形状和速度可以在很大程度上取决于系统的内在特性，通常以参数为代表。 随着参数的变化，前部的特征可能会从“拉动”变为“推”，反之亦然。由于最近引入的思想，例如相对熵和定量陡度以及对方程式的规律性的仔细理解，这引起了新的进步，这为表征这种拉动的过渡的高度精确打开了大门。 该项目的第二个重点是发展各种碰撞动力学方程的适合性理论。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子优点和更广泛的影响评估标准，认为值得通过评估来获得支持。

项目成果

Local Well-Posedness for the Boltzmann Equation with Very Soft Potential and Polynomially Decaying Initial Data

• DOI: 10.1137/21m1427504
• 发表时间: 2021-06
• 期刊: SIAM J. Math. Anal.
• 作者: Christopher Henderson;W. Wang

Slow and fast minimal speed traveling waves of the FKPP equation with chemotaxis

• DOI: 10.1016/j.matpur.2022.09.004
• 发表时间: 2021-02
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: Christopher Henderson

Long-time behaviour for a nonlocal model from directed polymers

定向聚合物非局部模型的长期行为

• DOI: 10.1088/1361-6544/aca9b3
• 发表时间: 2022
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Gu, Yu;Henderson, Christopher
• 通讯作者: Henderson, Christopher