CAREER: Nonlocal partial differential equations in collisional kinetic theory

本研究项目是关于粒子碰撞和扩散所驱动的重要物理现象，其数学描述基于运动型偏微分方程组。它是由气体动力学和等离子体物理中的应用以及对分析、偏微分方程式和数学物理的数学兴趣驱动的。动力学方程被用来描述相互作用粒子的演化，例如等离子体中的气体分子、离子和电子。最著名的动力学方程是波尔兹曼方程：由路德维希·波尔兹曼于1872年提出，该方程描述了一大类气体的运动。后来，1936年，列夫·朗道从玻尔兹曼方程中推导出了一个新的等离子体运动数学模型。后一个方程被称为朗道方程。尽管许多数学家和物理学家一直在研究这些方程，但由于它们的数学复杂性，许多重要的问题仍然没有答案。拟议的研究将通过弥合数学分析和物理之间的差距，从根本上促进这一领域的发展，并使人们能够进一步从数学上理解物理现象。这项工作将通过国家和国际两级的合作得到加强，并将有助于加强机构间联系。研究计划将与教育和外展活动相结合，旨在(I)扩大学生对不同研究领域的理解，(Ii)为学生提供对不同职业道路至关重要的现代技能，以及(Iii)促进DC Metro地区地方机构之间的现有联系，并帮助建立新的合作和培训高素质科学劳动力的途径。所提出的研究将促进对碰撞动力学数学模型解的整体性质的认识，提高我们对非线性动力学、熵、平衡和正则化效应的理解。相应的方程包含具有高度非线性、奇异和退化系数的积分-微分算子。这类积分-微分方程近年来受到越来越多的关注：适定性和正则性理论随着保角几何、随机控制和图像处理等新应用的出现而不断发展。该项目旨在增进动力学方程、非局部积分-微分算子和退化微分算子理论的知识。

This research project is concerned with important physical phenomena driven by collision and diffusion of particles, whose mathematical description is based on partial differential equations of kinetic type. It is driven by applications in gas dynamics and plasma physics as well by mathematical interests in analysis, partial differential equations and mathematical physics. Kinetic equations are used to describe evolution of interacting particles, such as gas molecules, ions and electrons in a plasma. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936 Lev Landau derived from the Boltzmann equation a new mathematical model for motion of plasma. This latter equation was named the Landau equation. Despite the fact that many mathematicians and physicists have been working on these equations, many important questions are still unanswered due to their mathematical complexity. The proposed research will fundamentally contribute to this field by bridging the gap between mathematical analysis and physics and enable further mathematical understanding of physical phenomena. The work will be enhanced by collaborations at the national and international levels and will help strengthen inter-institutional ties. The research program will be integrated with educational and outreach activities designed to (i) broaden the students' understanding of different research areas, (ii) provide the students with a modern skill set that is essential for different career paths, and (iii) promote existing connections between local institutions in the DC Metro area and help establish new avenues for collaboration and training of high quality scientific workforce. The proposed research will advance the knowledge in global properties of solutions to collisional kinetic mathematical models, improving our understanding of non-linear dynamics, entropy, equilibrium, and regularizing effects. The corresponding equations contain integro-differential operators that are highly nonlinear, singular and with degenerating coefficients. Integro-differential equations of this type have received increased attention recently: well-posedness and regularity theory are being developed since many new applications have emerged, among which conformal geometry, stochastic control and image processing. The project aims at advancing knowledge in the theories of kinetic equations, nonlocal integro-differential operators and degenerate differential operators.