Nonlinearity in Reaction-Diffusion and Kinetic Equations

各种现象，如入侵物种的扩散，湍流燃烧，以及气体中颗粒云的演化，都可以用偏微分方程组来模拟。这些例子必然涉及到非线性，即当量的变化率取决于量本身时。非线性的一个简单例子是所谓的Allee效应，即某些物种的繁殖率在最小种群规模以下为负值，可能是由于合作防御或配偶限制等因素。通常情况下，非线性会给模型的分析带来严重困难。在各种科学上相关的偏微分方程类的背景下，这个项目的目的是了解模型何时可以用(更简单的)线性模型来近似，以及更一般地，最小模型需要哪些基本特征来忠实地表示原始现象的基本行为。其目的是帮助科学家为他们的研究确定和实施最容易处理的模型。该项目将为本科生提供培训机会。该项目的一个重点是开发技术工具，以了解几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。传统上，根据驱动锋面运动的基本机制，这些系统被分为两类：远远超出锋面的线性行为(拉动的前缘)或前缘的非线性行为(推送的前缘)。通常，这些前锋的形状和速度可以强烈地依赖于系统的固有属性，通常由一个参数来表示。随着参数的改变，正面的特征可能会从“拉”变为“推”，反之亦然。由于最近引入了相对熵和数量陡度等概念，以及对方程正则性的仔细理解，新的进展为描述这种拉力推动的转变打开了高精度的大门。该项目的第二个重点是发展各种碰撞动力学方程的适定性理论。这一奖项反映了NSF的法定使命，并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。

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.