Nonlinearity in Reaction-Diffusion and Kinetic Equations

不同的现象，如入侵物种的传播，湍流燃烧，以及气体中粒子云的演变可以使用偏微分方程建模。 这些例子必然涉及非线性，即一个量的变化率取决于该量本身。 非线性的一个简单例子是所谓的Allee效应，其中某些物种的繁殖率低于最小种群规模，可能是由于合作防御或配偶限制等因素。典型地，非线性在模型的分析中呈现严重的困难。 在各种科学相关的偏微分方程类的背景下，这个项目的目的是发展一种理解，当一个模型可以近似为一个（更简单的）线性的，更一般地说，哪些基本特征是由一个最小的模型所需的忠实地代表原始现象的基本行为。其目的是帮助科学家确定和实施最易处理的模型进行调查。本计画将提供本科生实习机会，重点在于发展技术工具，以了解几个反应扩散方程的长时间行为的基本性质。 这些模型系统表现出生长（反应）和扩散（扩散），其中界面（前沿）形成并以恒定速度传播。 传统上，这些系统根据驱动前沿运动的基本机制分为两类：远远超出前沿的线性行为（“拉”前沿）或前沿的非线性行为（“推”前沿）。 通常，这些锋的形状和速度可以强烈地依赖于系统的内在属性，通常由参数表示。 当参数改变时，前沿的特征可以从“拉”变为“推”，反之亦然。最近引入的相对熵和定量陡度等概念以及对方程规律性的仔细理解所带来的新进展，为高精度地描述这种拉-推转变打开了大门。 该项目的第二个重点是发展各种碰撞动力学方程的适定性理论。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.