Fundamental Questions in Hyperbolic Balance Laws

双曲平衡定律是描述流体、气体、等离子体和波的运动的重要的非线性方程组。该项目侧重于几个基本的和具有挑战性的问题，涉及不同流体流动模型的有效性和有效性，流体的动态稳定性和大振幅解决方案的行为，以及在一个或多个空间维度中的强波建模，这些问题经常出现在现实情况中。这项研究旨在提供新的见解，并开发新的工具来表征由非线性和共振驱动的复杂但有趣的行为。对这些建模方程的解的行为的进一步理解将在可压缩流体动力学、弹性材料力学、交通控制系统、多孔介质流动和地球物理动力学中找到重要的应用。本研究项目的主要目标有四个：（1）研究一维可压缩Euler方程在大初值条件下的整体解的存在性和有限时间爆破，并对一般大解给出精确的密度下界估计;（2）在初始熵趋于常数的情况下，对可压缩流体中的等熵近似建立合理的解释，对无粘和粘性流体都是如此：（3）研究可压缩流体中的非线性Rayleigh-Taylor型不稳定性，包括可压缩Navier-Stokes-Fourier系统;（4）研究三维带摩擦阻尼的可压缩全Euler方程的大时间渐近性态，证明达西定律，并对磁场影响下可压缩流体中短波与长波的相互作用进行建模和分析。这些研究活动旨在提供坚实的数学基础，以弥合现有结果之间的差距，并为未来的研究。 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 are important nonlinear systems of equations utilized in the description of the motion of fluids, gases, plasma, and waves. This project focuses on several fundamental and challenging questions concerning the validity and effectiveness of different fluid flow models, the dynamical stability of fluids and the behavior of large amplitude solutions, and the modeling of strong waves in one or multiple spatial dimensions, which often appear in realistic situations. This research aims to provide fresh insights and develop new tools to characterize complicated but interesting behavior driven by nonlinearity and resonances. Increased understanding of the behavior of solutions to these modeling equations will find important applications to the dynamics of compressible fluids, the mechanics of elastic materials, traffic control systems, porous medium flows, and geophysical dynamics. The project also involves international collaborations and training of graduate students and young researchers in this challenging field.This research project focuses on four goals: (1) To provide a clear picture of the global existence and finite-time blow-up for compressible Euler equations in one dimension with large initial data, and to find sharp density lower bound estimates for generic large solutions; (2) To establish a reasonable justification of the isentropic approximation in compressible fluids when initial entropy tends to a constant, for both inviscid and viscous fluids; (3) To investigate nonlinear Rayleigh-Taylor type instabilities in compressible fluids, including the compressible Navier-Stokes-Fourier system; (4) To study the large-time asymptotic behavior of compressible full Euler equations with frictional damping in three dimensions to justify Darcy's law, and to undertake the modeling and analysis of interactions between short waves and long waves in compressible fluids under the influence of magnetic fields. These research activities aim to provide a solid mathematical foundation to bridge gaps between existing results and for future studies. It is anticipated that the results will significantly advance basic understanding of 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.