Several Fundamental Questions in Hyperbolic Balance Laws

双曲平衡定律出现在流体流动的非线性运动的建模中。这类系统极其复杂的结构是许多开放问题的来源，这些问题在广泛的科学和工程模型中具有实际意义。这项研究将侧重于几个基本问题，目标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理解。除了科学目标外，该项目还旨在促进国际合作，培养研究生和年轻研究人员。第一个是寻找尖锐的条件，光滑的解决方案，可压缩的，非等熵的，欧拉方程在一维大的初始数据的整体存在，并继续努力推导出一个尖锐的密度下界估计通用的大解决方案。第二个特征是通过奇异的限制，在等熵近似过程中的无粘和粘性流体，并获得在这种近似的误差估计。最后是建立数学上严格的有效性的非线性动力学Rayleigh-Taylor型不稳定性的可压缩流体受重力，特别是可压缩的Navier-Stokes-Fourier系统，其中热传递起着至关重要的作用。 这些研究活动旨在在一些缺乏理论的领域取得突破，为当前方法之外的领域铺平道路，并促进对非线性双曲平衡定律解的行为的理解。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。

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.