Asymptotic Behavior of solutions to viscous hyperbolic conservation laws

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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.