CAREER: Nonlocal partial differential equations in collisional kinetic theory

职业：碰撞动力学理论中的非局部偏微分方程

• 批准号: 2019335
• 负责人: Maria Pia Gualdani
• 金额: $ 32.26万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2019335&HistoricalAwards=false
• 关键词: CAREER; Nonlocal; partial; differential; equations

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.

该研究项目涉及由粒子碰撞和扩散驱动的重要物理现象，其数学描述基于动力学类型的偏微分方程。它是由气体动力学和等离子体物理学的应用以及分析、偏微分方程和数学物理学的数学兴趣驱动的。动力学方程用于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玻尔兹曼方程：由路德维希·玻尔兹曼于 1872 年提出，该方程描述了一大类气体的运动。后来，1936 年 Lev Landau 从玻尔兹曼方程导出了等离子体运动的新数学模型。后一个方程被命名为朗道方程。尽管许多数学家和物理学家一直在研究这些方程，但由于其数学复杂性，许多重要问题仍未得到解答。拟议的研究将从根本上为该领域做出贡献，弥合数学分析和物理学之间的差距，并促进对物理现象的进一步数学理解。这项工作将通过国家和国际层面的合作得到加强，并将有助于加强机构间的联系。该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）为学生提供不同职业道路所必需的现代技能，以及（iii）促进华盛顿都会区当地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。拟议的研究将增进对碰撞动力学数学模型解的全局性质的了解，增进我们对非线性动力学、熵、平衡和正则化效应的理解。相应的方程包含高度非线性、奇异且具有退化系数的积分微分算子。这种类型的积分微分方程最近受到越来越多的关注：由于出现了许多新的应用，其中包括共形几何、随机控制和图像处理，适定性和正则性理论正在得到发展。该项目旨在增进动力学方程、非局部积分微分算子和简并微分算子理论的知识。