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Asymptotic Behavior of solutions to viscous hyperbolic conservation laws

粘性双曲守恒定律解的渐近行为

基本信息

批准号：
10640216
负责人：
NISHIHARA Kenji
金额：
$ 1.09万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640216/
关键词：
p-system diffusion wave viscous shock wave rarefaction wave inflow problem foundary layer solution P-System viscous shock wave rarefaction wave Green関数 convergence rate Green function

项目摘要

In this research we have considered one-dimensional compressible viscous flows. One is in the porous media and the viscous effect comes from the friction, so that the equations become the p-system with damping. The other has a usual Newton viscosity and the equations become the p-system with viscosity.It was known that the solution to the Cauchy problem for the p-system with damping behaves likely the diffusion wave, the solution to the corresponding parabolic equation due to the Darcy law (Hsiao, Liu etc.). Its convergence rates were also known by applying the Green function for the parabolic equation (Nishihara). We have obtained the convergence rates in several situations. For more general systems the coefficients becomes variable and hence we introduced the approximate Green function and obtained the desired results (Nishihara-Wang-Yang, Nishihara-Nishikawa). For the initial-boundary value problem on the half line we have investigated the boundary effect (Nishihara-Yang). This method has been applied to the thermoelastic system with dissipation (Nishihara-Nishibata).To investigate the p-system with viscosity, it is basic to do the Burgers equation. Depending on the flux and endstates of the data, solutions to the Cauchy problem are expected to tend to the rarefaction wave, the viscous shock wave or their superposition. In this research the global stability of the viscous shock wave and the boundary effect have been obtained (Nishihara-Zhao, Nishihara). For the original p-system with viscosity we have considered the inflow problem proposed by a joint researcher, A.Matsumura. He gave all conjectures of asymptotic behaviors, in which he introduced a new wave called a boundary layer solution. The stabilities of the boundary layer solution and the superposition of that and the rarefaction waves are rigorously proved (Matsumura-Nishihara). The stability of superposition of the boundary layer solution and viscous shock wave is remained open.

在这项研究中，我们考虑了一维可压缩粘性流动。一种是在多孔介质中，由于摩擦力的粘性作用，使得方程组成为带阻尼的p-系统。另一个方程组具有通常的牛顿粘性，方程组成为具有粘性的p-方程组，已知具有阻尼的p-方程组的Cauchy问题的解表现为扩散波，而相应的抛物方程的解由于Darcy定律（Hsiao，Liu等）而表现为扩散波。它的收敛速度也是已知的应用绿色函数的抛物方程（西原）。我们得到了几种情况下的收敛速度。对于更一般的系统，系数变得可变，因此我们引入了近似的绿色函数，并获得了所需的结果（Nishihara-Wang-Yang，Nishihara-Nishikawa）。对于半直线上的初边值问题，我们研究了边界效应（Nishihara-Yang）。将该方法应用于具有耗散的热弹性系统（Nishihara-Nishibata），研究具有粘性的p-系统，首先要求解Burgers方程。根据数据的通量和端态，柯西问题的解预计将倾向于稀疏波、粘性激波或它们的叠加。在这项研究中，粘性激波的整体稳定性和边界效应已经得到（西原赵，西原）。对于原始的p-系统的粘性，我们已经考虑了流入问题提出的联合研究员，A. Matsumura。他给出了所有的渐近行为，其中他介绍了一个新的波称为边界层解决方案。严格证明了边界层解的稳定性以及边界层解与稀疏波的叠加（Matsumura-Nishihara）。边界层解与粘性激波叠加的稳定性仍然是开放的。