CAREER: Hilberts Sixth Problem in the Boltzmann equation

该数学研究项目旨在通过将介观模型（玻尔兹曼方程）与宏观模型（流体）和微观模型（多粒子）联系起来，建立多体系统（如气体动力学）的统一理论。研究者和合作者将在各种物理系统的模型中研究这些联系，如龙卷风、冲击波、水-油混合物和托卡马克聚变反应堆中的等离子体。研究结果的目的是导致了解这些和其他重要系统的进展。在这项研究中，研究人员计划开发研究生专题课程，本科生指导学习课程，本科生研究活动，以及为实践数学家举办的讲习班。偏微分方程的一个基本问题是发展统一的气体动力学理论，将介观玻尔兹曼方程与出现在形式极限中的宏观流体模型（流体动力极限）联系起来，将玻尔兹曼方程与微观n体问题（格拉德-玻尔兹曼极限）联系起来。在这个项目中，研究者和合作者在不同的物理情况下用数学方法研究了这两个极限的严格通道。该项目的第一部分涉及自由界面自然出现时的水动力极限：(i)不可压缩欧拉方程的涡旋块解，（ii）模拟具有尖锐界面和表面张力的二元流体混合物的两种Vlasov-Boltzmann系统，以及（iii）可压缩欧拉方程中的激波形成及其动力学近似。项目的第二部分涉及热随机边界存在下的Grad-Boltzmann极限。研究了从与随机边界相互作用的n体牛顿系统向具有漫反射边界的玻尔兹曼方程的过渡。他们将研究这种边界的混合效应，目的是在大于平均自由时间的时间间隔内建立一个严格的通道。他们还将研究介观和微观水平之间的一般稳定非平衡状态的联系。该奖项反映了美国国家科学基金会的法定使命，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。

This mathematics research project aims to develop a unified theory of many body systems (such as gas dynamics) by connecting mesoscopic models (Boltzmann equation) to macroscopic models (fluid) and microscopic models (many particles). The investigator and collaborator will study these connections in models of various physical systems such as tornados, shock waves, water-oil mixtures, and plasma in tokamak fusion reactors. Results of the research are intended to lead to advances in understanding of these and other important systems. In connection with the research, the investigators plan to develop graduate topics courses, undergraduate directed study courses, research activities for undergraduate students, and workshops for practicing mathematicians. One of the fundamental questions in partial differential equations is the development of a unified theory of gas dynamics connecting the mesoscopic Boltzmann equations to the macroscopic fluid models (hydrodynamic limits) that arise in formal limits, and the Boltzmann equations to microscopic N-body problems (Grad-Boltzmann limit). In this project, the investigator and collaborator study the rigorous passage to both limits mathematically in various physical situations. The first part of the project concerns the hydrodynamic limits when free interfaces occur naturally: (i) vortex patch solutions of the incompressible Euler equation, (ii) two-species Vlasov-Boltzmann systems modeling binary fluid mixtures with sharp interface and surface tension, and (iii) shock formation in the compressible Euler equations and its kinetic approximation. The second part of the project concerns the Grad-Boltzmann limit in the presence of a thermal stochastic boundary. The investigators study the passage from the N-body Newtonian system interacting with stochastic boundary toward the Boltzmann equation with diffuse reflection boundary. They will investigate the mixing effect of such boundaries and aim to establish a rigorous passage for a time interval greater than a mean free time. They will also study the link of generic steady non-equilibrium states between the mesoscopic and microscopic levels.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.