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

该项目致力于研究在流体建模中产生的耗散平衡定律的大的和/或多维的解决方案，包括气体、流体、交通和半导体设备。研究的重点是以下特定对象：a)可压缩Navier-Stokes方程在一维和多维空间中的全局行为，b)可压缩Navier-Stokes方程向具有物理粘度系数的Euler方程的消失粘度极限，c)松弛系统的全局稳定性或不稳定性，d)一类耗散平衡律的全局BV理论，e)二维粘性Boussinesq系统和二维和三维可压缩Navier-Stokes方程等流体动力学系统的大时间行为。该项目的目标是开发新的分析方法来构建大解，研究解的奇异行为，确定全局稳定性准则，理解多维解的大时间行为。本研究的目的是探讨耗散双曲平衡律的几个开放问题。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。研究将集中在大解问题、强波问题和多空间维度的现实模型问题上。这些情况往往涉及由非线性和共振引起的复杂而有趣的现象，因此不发达。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。这些问题的数学认识的进步对于设计有效的数值格式进行这些基础领域的科学计算具有重要作用。该项目还将为这一具有挑战性的领域的学生和年轻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.