Large and Multi-Dimensional Solutions of Hyperbolic Balance Laws with Dissipation

该项目致力于研究大的和/或多维的解决方案，以耗散平衡定律产生的流动建模，包括气体，流体，交通，和半导体器件。研究的重点是以下具体对象：a）一维和多维可压缩Navier-Stokes方程的整体性态，B）粘性极限从可压缩Navier-Stokes方程向具有物理粘性系数的Euler方程消失，c）松弛系统的整体稳定性或不稳定性，d）一类耗散平衡律的整体BV理论，（5）二维粘性Boussinesq方程组和二维、三维可压缩Navier-Stokes方程组的大时间行为。本项目的目标是发展新的分析方法来构造大解，研究解的奇异性，识别全局稳定性准则，理解多维解的大时间行为，本研究的目的是研究耗散双曲平衡律的几个公开问题。双曲平衡定律是模拟流体、气体和波浪运动的重要非线性偏微分方程。该研究将集中在多个空间维度的大型解决方案，强波和现实模型的问题。这些情况往往涉及复杂的，但有趣的现象所造成的非线性和共振，因此欠发达。研究结果有望应用于可压缩流体动力学、弹性材料力学、交通控制系统、半导体器件建模、多孔介质流动和地球物理动力学。这些问题的数学理解的进步起着重要的作用，在这些基本领域的科学计算有效的数值方案的设计。该项目还将在这一具有挑战性的领域为学生和青年研究人员提供教育和培训。

This project is devoted to the study of large and/or multi-dimensional solutions to dissipative balance laws arising in modeling of flows, including gases, fluids, traffics, and semi-conductor devices. The research focuses on the following particular objects: a) global behavior of compressible Navier-Stokes equations in one and several space dimensions, b) vanishing viscosity limit from compressible Navier-Stokes equations toward Euler equations with physical viscosity coefficient, c) global stability or instability of relaxation system, d) global BV theory for a class of dissipative balance laws, and e) large time behavior for certain fluid dynamics systems including viscous Boussinesq system in two dimension and compressible Navier-Stokes equations in two and three dimensions. The goals of this project are developing novel analytic approaches to constructing large solutions, investigating singular behavior of the solutions, identifying global stability criterion, understanding the large time behavior of multi-dimensional solutions.The objective of this research is to investigate several open problems for dissipative hyperbolic balance laws. Hyperbolic balance laws are important nonlinear partial differential equations modeling the motion of fluids, gas, and waves. The research will concentrate on problems for large solutions, strong waves, and realistic models in multiple spatial dimensions. These situations often involve complicated but interesting phenomena resulting from nonlinearity and resonance and thus underdeveloped. The results are expected to have applications to dynamics of compressible fluids, elastic material mechanics, traffic control systems, semi-conductor devices modeling, porous medium flows, and geophysical dynamics. The advance of mathematical understanding of these problems plays important role in the design of effective numerical schemes for scientific computation in these fundamental areas. The project will also provide education and training to students and young researchers in this challenging field.