Large and Multi-Dimensional Solutions to Hyperbolic Balance Laws

空气中许多扰动的运动，从普通的噪音到机翼周围的冲击波（引起熟悉的音爆），都是用可压缩流体方程来描述的。即使液体在足够高的速度（接近音速）下也是可压缩的。这个项目致力于描述可压缩流体动力学的不同版本的Navier-Stokes方程的数学研究。类似的方程为相关科学领域的现象提供了数学模型，包括弹性材料力学，多孔介质流动和地球物理动力学。 为了确定特定方程作为给定现象的模型的有效性，确定该方程是否有解以及如果有解，该解如何取决于物理参数可能是有帮助的。如果在特定的状态下，这些方程要么没有解，要么有以不连续的方式依赖于数据的解，这将表明这些现象的数学描述对于特定的应用只有有限的有效性。目前的项目将专注于模拟强波的大型解决方案的挑战性问题，这些问题经常出现在现实情况中。这项研究将带来新的见解，并开发新的工具来表征由非线性和共振驱动的复杂大型解决方案。该项目的进展将促进高性能计算方法的设计。本研究课题是对可压缩流体的动力学进行解析研究的课题，在该课题中，还将开展国际合作，培养研究生和初级研究人员。该项目将在四个相互关联的研究领域产生进一步的发展。第一个目标是研究一维可压缩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，并得到一般大解的精确密度下界估计.第二，研究具有温度相关输运系数的可压缩Navier-Stokes-Fourier方程组，旨在建立一个系统的理论，包括弱解的整体存在性，绝对温度的一致下界，以及解的大时间行为。第三个目标是研究等熵Navier-Stokes方程的局部理论与密度依赖的粘度允许真空的初始数据，与应用程序的各种浅水模式，如圣维南方程浅水。最后一个目标是研究三维可压缩全欧拉方程的大时间渐近性态，以证明达西定律在这种非常复杂的情况下的正确性，并研究磁场影响下可压缩流体中短波和长波相互作用的建模和分析。

Motion of many disturbances in the air, from ordinary noise to shock waves about an airfoil (that cause the familiar sonic boom), is described by equations of compressible fluids. Even liquids at high enough speeds (close to the sound speed) are compressible. This project is devoted to the mathematical study of the different versions of the Navier-Stokes equations that describe dynamics of compressible fluids. Similar equations provide mathematical models for phenomena in related scientific areas, including elastic material mechanics, porous medium flows, and geophysical dynamics. To determine the validity of a particular equation as a model for a given phenomenon, it can be helpful to determine whether the equation has a solution, and if it does, how this solution depends upon the physical parameters. If in particular regimes these equations either lack solutions, or have solutions which depend in a discontinuous way upon the data, this will indicate that these mathematical descriptions of the phenomena have only limited validity for the particular applications. The current project will focus on challenging problems for large solutions modeling strong waves, which often appear in realistic situations. The research will bring fresh insights and develop new tools to characterize complicated large solutions driven by nonlinearity and resonances. The progress in this project will facilitate the design of high-performance computational methods. This project also involves international collaborations and training of graduate students and junior researchers in this challenging field.This research project is devoted to the analytical study of dynamics of compressible fluids. The project will produce further developments in four inter-related research areas. The first objective is to investigate the global existence and finite time blowup for compressible Euler equations in one dimension with large initial data, and to find sharp density lower-bound estimate for generic large solutions. Second, the research on compressible Navier-Stokes-Fourier system with temperature-dependent transport coefficients that is aimed at a systematic theory including the global existence of weak solutions, uniform lower bound on absolute temperature, and large time behavior of the solutions. The third objective is to study the local theory for isentropic Navier-Stokes equations with density-dependent viscosity allowing vacuum in initial data, with applications to various shallow water models such as Saint-Venant equations for shallow water. The last objective is to study the large time asymptotic behavior of compressible full Euler equations in three dimensions to justify the Darcy's law in this very complicated case, and to study the modeling and analysis for interactions between short waves and long waves in compressible fluids under the influence of magnetic field.