Collisions in Plasma: The Landau Equation and Related Models

气体动力学和等离子体物理中的现象是由粒子的碰撞和扩散驱动的。对这些效应的数学建模是应用数学和统计物理的交叉点，它涉及到动力学方程的研究，动力学方程是一类非线性偏微分方程组，它根据粒子密度来捕捉大量粒子的行为。这个项目集中在两个具体的动力学方程上：第一个描述了当粒子通过可能在非常大的微观距离上发生的二元碰撞相互作用时粒子分布的时间演化，第二个产生在带电量子粒子的统计描述中。其目的是通过研究其解的定性属性来验证这些模型对物理系统的保真度，以确保模型不会发生临时故障。该项目将为研究生和博士后研究人员提供培训机会。该项目旨在通过分析两个动力学偏微分方程式来扩大对稀薄气体和等离子体中碰撞动力学的数学理解。第一个是朗道方程，它描述了当粒子通过掠夺型的双碰撞相互作用时，粒子分布的时间演化。第二个是Landau-Fermi-Dirac方程，它出现在带电量子粒子的统计描述中。由于非线性项、非局部特征和退化系数，这两个方程都提出了挑战。该项目的第一部分涉及解的定性性质，例如整体适定性和有限时间爆破，长时间行为的特征，以及与宏观流体方程的联系。第二部分讨论了Landau-Fermi-Dirac近似作为对Boltzmann方程在放牧区域的修正的有效性。所采用的技术包括经典动力学理论和为非线性非局部积分-微分方程组和退化非局部微分算子开发的最新理论的新组合。该奖项反映了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。

Phenomena in gas dynamics and plasma physics are driven by collisions and diffusion of particles. The mathematical modeling of these effects lies at the intersection of applied mathematics and statistical physics, and it involves the study of kinetic equations, a class of nonlinear partial differential equations that captures the behavior of large number of particles in terms of the particle density. This project concentrates on two specific kinetic equations: the first describes the time evolution of the particle distribution when particles interact through binary collisions that can occur at very large microscopical distances, the second arises in the statistical description of charged quantum particles. The intent is to validate the fidelity of these models to the physical systems by studying the qualitative properties of their solutions, to ensure that a temporary breakdown of the models does not occur. The project will provide training opportunities for graduate students and postdoctoral researchers. The project aims at expanding the mathematical understanding of the dynamics of collisions in dilute gases and plasma by analyzing two kinetic partial differential equations. The first is the Landau equation, which describes the time evolution of the particle distribution when particles interact through binary collisions of grazing type. The second is the Landau-Fermi-Dirac equation, which arises in the statistical description of charged quantum particles. Both equations present challenges due to nonlinear terms, nonlocal features, and degenerate coefficients. The first part of the project deals with the qualitative properties of solutions, such as global well-posedness and finite time blow-up, characterization of the long-time behavior, and connection with macroscopic fluid equations. The second part concerns the validity of the Landau-Fermi-Dirac approximation as a correction to the Boltzmann equation in the grazing regime. The techniques employed include a novel combination of classical kinetic theory and recent theories developed for nonlinear nonlocal integro-differential equations and degenerate nonlocal differential operators.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.