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（许多粒子）联系起来，发展许多身体系统（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。研究结果旨在促进对这些和其他重要系统的理解。在研究方面，研究人员计划开发研究生课题课程，本科生指导学习课程，本科生的研究活动，以及实践数学家的研讨会。偏微分方程的基本问题之一是发展气体动力学的统一理论，将介观玻尔兹曼方程连接到形式极限中出现的宏观流体模型（流体动力学极限），并将玻尔兹曼方程连接到微观N体问题（格拉德-玻尔兹曼极限）。在这个项目中，研究者和合作者研究了在各种物理情况下，在数学上达到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发生时的流体动力学极限：（i）不可压缩欧拉方程的涡补丁解决方案，（ii）两种Vlasov-Boltzmann系统模拟二元流体混合物与尖锐的界面和表面张力，和（iii）可压缩欧拉方程及其动力学近似的激波形成。该项目的第二部分涉及热随机边界存在下的梯度玻尔兹曼极限。研究了从具有随机边界的N体牛顿系统到具有漫反射边界的Boltzmann方程的过渡。他们将研究这种边界的混合效应，并旨在建立一个严格的通过时间间隔大于平均自由时间。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。

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.