几类守恒律双曲组弱解的适定性及长时间性态

The existence and uniqueness of the weak solutions to quasilinear hyperbolic systems with conservation laws is always a quite active topic. There are many application backgrounds related to well posedness and long-time behaviors of weak solutions and, from mathematical point of view, the research on this subject is a big challenge. By a change of variable of Euler-Lagrange type, this project is mainly concerned with the explicit expressions of entropy solutions to linearly degenerate hyperbolic systems with conservation laws. Moreover, based on this result, we consider its L^1 stability and the precise description of its long-time behaviors.We are going to investigate the following problems: 1. For linearly degenerate hyperbolic systems of rich type, utilizing the explicit expressions of entropy solutions to the Cauchy problem, we are concerned with its L^1 stability and the explicit long-time behaviors. Some applications are presented. 2. For 2×2 linearly degenerate hyperbolic systems, the explicit expressions of the entropy solutions to the mixed initial boundary value problem are considered. Furthermore, we consider its L^1 stability and its explicit long-time behaviors; 3. At the prescence of viscosity effects, for some kinds of hyperbolic systems of conservation laws, we discuss the well posedness and long-time behaviors of weak solutions.

拟线性守恒律双曲组的弱解的存在唯一性问题一直是一个非常活跃的前沿研究方向。对弱解的适定性及长时间性态的研究有多方面的应用背景，在数学理论上也是一个挑战。本课题主要通过Euler-Lagrange型坐标变换的方法，研究线性退化守恒律双曲组的熵解的显示表示，在此基础上讨论其L^1稳定性并给出其长时间性态的精确描述。拟解决的主要问题包括：1. 对线性退化的富有组，借助于其Cauchy问题熵解的显示表示，讨论其L^1稳定性及显示长时间性态，并给出其应用；2. 对2×2的线性退化双曲组，考虑其混合初边值问题熵解的显示表示，进而讨论其L^1稳定性及显示长时间性态；3. 对一些具体的守恒律双曲组考虑在粘性影响下其弱解的适定性及长时间性态。

拟线性守恒律双曲组的弱解的存在唯一性问题一直是一个非常活跃的前沿研究方向。对弱解的适定性及长时间性态的研究有多方面的应用背景，在数学理论上也是一个挑战。本课题主要通过Euler-Lagrange 型坐标变换的方法，研究线性退化守恒律双曲组的熵解的显示表示，在此基础上讨论其L^1稳定性并给出其长时间性态的精确描述。解决的主要问题包括：1. 对线性退化的富有组，借助于其Cauchy 问题熵解的显示表示，讨论其L^1 稳定性及显示长时间性态，并给出其应用；2. 对可压的磁流体力学方程组的初边值问题建立了弱强唯一性原理；3. 对两类简化的可压液晶模型建立了弱强唯一性原理；4. 对可压磁流体力学方程组的初边值问题讨论了其强解的局部存在性及小初值问题的强解的整体存在唯一性；5. 对稳态的热传导Navier-Stokes方程组建立了具小Mach数大外力强解的存在性，并作为副产品获得了其小Mach数极限；6. 对具量子效应的可压NS方程组的弱解严格验证了组合的不可压极限和半经典极限; 7. 对具理想传到边界条件的稳态磁流体力学方程组，在小外力的情况下证明了强解是存在唯一的。