可压缩流体力学方程组波的稳定性研究

The stability of solutions involving rarefaction waves and shock waves to compressible Navier-Stokes equations and non-Newtonian fluids equations is one of the important, longstanding problems in the theory of compressible fluid flow. In this project, we intend to study the following problems. (1) In the case that the solutions of the isentropic Euler equations are two rarefaction waves with a vacuum intermediate state in between, we construct a sequence of solutions to the 1D compressible isentropic Navier-Stokes equations which converge to the rarefaction waves with vacuum to the corresponding Euler equations as the viscosity tends to zero. (2) We investigate nonlinear stability of viscous shock wave to 1D compressible isentropic Navier-Stokes equations with density dependent viscous coefficient with the boundary effect. (3) We study the existence and limit behavior of the shock layer for a class of 1D stationary compressible non-Newtonian fluids.Moreover, we also consider the zero dissipation limit to shock waves for a class of 1D non-stationary compressible non-Newtonian fluids. In a word, the results achieved in this project further enrich the theory of stability of waves.

在流体力学方程组的数学理论中，波的稳定性问题具有重要研究价值。本项目将围绕可压缩Navier-Stokes方程组和非牛顿流体方程组稀疏波和激波的稳定性展开研究。主要研究内容有：(1)给定等熵Euler方程组的解是1-稀疏波+真空+2-稀疏波，研究黏性系数为常数的一维可压缩等熵Navier-Stokes方程组连接真空的稀疏波的零耗散极限问题。(2)研究黏性系数依赖于密度的一维可压缩等熵Navier-Stokes方程组伴有边界效应的黏性激波稳定性。(3)建立一类一维定常可压缩非牛顿流激波层的存在性，并讨论激波层的极限行为。进而再研究一类一维非定常可压缩非牛顿流激波的零耗散极限问题。本项目的研究将进一步丰富流体力学方程组波的稳定性理论。

Euler方程和Navier-Stokes方程作为流体力学方程中的两个基本模型分别反映了理想流体和黏性流体的运动规律，两者之间具有密切的联系。Euler方程有着丰富的波现象，而描述黏性可压缩流体运动中非线性波稳定性的数学研究，长期以来是非线性偏微分方程研究热点问题之一。本项目研究了一些Euler方程的黎曼问题以及一些牛顿流和非牛顿流波的稳定性。