Several Fundamental Questions in Hyperbolic Balance Laws

在模拟流体的非线性运动时，会出现双曲线平衡定律。这类系统的极其复杂的结构是许多公开问题的来源，这些问题在广泛的科学和工程模型中具有实际意义。这项研究将集中在几个基本问题上，目的是带来新的见解和开发新的工具，以在该领域取得突破。该项目旨在加强对这些复杂但重要的现象的了解。除了科学目标，该项目还旨在促进国际合作，培养研究生和年轻研究人员。该研究项目遵循三个主要主题。第一种方法是寻找一维可压缩非等熵欧拉方程在大初始数据下整体光滑解存在的精确条件，并继续得到一般大解的一个精确的密度下界估计。二是刻画无粘流体和粘性流体等熵近似过程中奇异极限的通过，并导出此类近似中的误差估计。最后，建立了受重力作用的可压缩流体中非线性动力Rayleigh-Taylor不稳定性的严格数学正确性，特别是对于可压缩的Navier-Stokes-Fourier系统，其中换热起着至关重要的作用。这些研究活动旨在在缺乏理论的几个领域取得突破，在当前方法之外的领域铺平道路，并增进对非线性双曲平衡定律中解的行为的理解。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。

Hyperbolic balance laws arise in the modeling of the nonlinear motion of fluid flows. The extremely complicated structure of this class of systems is a source of many open questions with practical implications in a broad class of scientific and engineering models. This research will focus on several fundamental questions, with the goal of bringing fresh insights and developing new tools to break ground in the field. The project aims to enhance understanding of these complicated but important phenomena. In addition to its scientific goals, this project aims to foster international collaborations and train graduate students and young researchers.The research project follows three main themes. The first is to look for sharp conditions for the global existence of smooth solutions to compressible, non-isentropic, Euler equations in one dimension with large initial data, and to continue the effort of deriving a sharp density lower bound estimate for generic large solutions. The second is to characterize the passage of singular limits in the isentropic approximation process for both inviscid and viscous fluids, and to derive error estimates in such approximations. The last is to establish the mathematically rigorous validity of nonlinear dynamical Rayleigh-Taylor type instability in compressible fluids subject to gravity, in particular for the compressible Navier-Stokes-Fourier system, where heat transfer plays an essential role. These research activities aim to break ground in several fields where theories are lacking, pave roads in areas that are outside current methodology, and advance understanding of the behavior of solutions in nonlinear hyperbolic balance laws.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.