Topics in Kinetic Theory

动力学理论通过考虑粒子密度函数而不是单独跟踪每个粒子来描述相互作用粒子的大系统的动力学。这种平均方法首先被用来推导波尔兹曼方程，该方程模拟了具有主要二元相互作用的稀薄气体的演化，并且它已成功地应用于推导动力学方程，该动力学方程模拟了广泛的应用，从气体和等离子体动力学到高大气空气动力学和社交网络中的集体行为。本专题将着重于两类动力学方程。第一种是动力学模型，它通过合并高阶相互作用项的总和而超越二元相互作用，并因此作为非理想气体的基本模型。第二种类型的方程被认为是模型稀释热等离子体，这出现在核聚变和托卡马克。本项目的目标是推进动力学理论中产生的非线性偏微分方程的数学理解。该项目的第一部分集中在适定性问题，矩估计和描述的长期相互作用的方程，推广玻尔兹曼方程。 一个这样的推广是二元-三元玻尔兹曼方程，它模拟了允许二元和三元相互作用的致密气体的演化。第二部分考虑相对论性朗道方程的长时间行为，包括收敛到平衡点的速度，矩估计，以及在有限时间内光滑性的传播与爆破的形成。该方程反映了爱因斯坦的狭义相对论在热等离子体中的高粒子速度的影响。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。

Kinetic theory provides a description of the dynamics of a large system of interacting particles by considering the particle density function instead of tracking each particle separately. Such averaging approach was first used to derive the Boltzmann equation that models the evolution of rarefied gases with predominantly binary interactions, and it has been successfully applied to derive kinetic equations modeling a wide range of applications, from gas and plasma dynamics to high atmosphere aerodynamics and collective behavior in social networks. This project will focus on two types of kinetic equations. The first type is a kinetic model which goes beyond binary interactions via incorporating a sum of higher order interaction terms, and as such serves as a basic model for a non-ideal gas. The second type of equations to be considered model dilute hot plasmas, which appear in nuclear fusion and tokamaks.The goal of this project is to advance the mathematical understanding of nonlinear partial differential equations arising in kinetic theory. The first part of the project focuses on questions of well-posedness, moment estimates, and description of long-range interactions of equations that generalize the Boltzmann equation. One such generalization is the binary-ternary Boltzmann equation that models the evolution of denser gases which allow both binary and ternary interactions. The second part considers the long-time behavior, including the rate of convergence to the equilibrium, moment estimates, and propagation of smoothness versus formation of blow up in finite time, of the relativistic Landau equation. This equation captures the effects of Einstein’s’ theory of special relativity due to high particle velocities in hot plasmas.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.