Asymptotic Behavior of solutions to viscous hyperbolic conservation laws

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，粘性效应来自于摩擦，使方程成为带阻尼项的p-方程组。另一类方程具有通常的牛顿粘性，方程变成有粘性的p-系统。由于达西定律(Hsiao，Liu等)，已知具有阻尼项的p-系统的柯西问题的解类似于扩散波，即相应抛物型方程的解(肖，刘等)。通过应用抛物型方程(西原)的格林函数，我们也知道了它的收敛速度。我们得到了几种情况下的收敛速度。对于更一般的系统，系数是可变的，因此我们引入了近似格林函数，并得到了期望的结果(西原-汪洋，西原-西川)。对于半直线上的初边值问题，我们研究了边界效应(西原-杨)。该方法已应用于具有耗散项的热弹性系统(Nishihara-Nishibata)。要研究具有粘性的p-系统，首先要解Burgers方程。根据数据的通量和端态，柯西问题的解有望趋向于稀疏波、粘性激波或它们的叠加。在这项研究中，得到了粘性激波的全局稳定性和边界效应(西原-赵，西原)。对于具有粘性的原始p-系统，我们考虑了松村联合研究员提出的流入问题。他给出了所有关于渐近行为的猜想，在这些猜想中，他引入了一种称为边界层解的新波。严格证明了边界层解及其与稀疏波的叠加的稳定性(MatSumura-Nishihara)。边界层解和粘性激波叠加的稳定性仍然是开放的。

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.