Asymptotic Behavior of solutions to viscous hyperbolic conservation laws
Project/Area Number  10640216 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  Waseda University 
Principal Investigator 
NISHIHARA Kenji Waseda University, School of Political Science and Economics, Professor, 政治経済学部, 教授 (60141876)

CoInvestigator(Kenkyūbuntansha) 
MATSUMARA Akitaka Osaka University, Graduate School of Science Department of Mathematics, Professor, 大学院・理学研究科, 教授 (60115938)

Project Period (FY) 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)

Keywords  psystem / diffusion wave / viscous shock wave / rarefaction wave / inflow problem / foundary layer solution / PSystem / viscous shock wave / rarefaction wave / Green関数 / convergence rate / Green function 
Research Abstract 
In this research we have considered onedimensional compressible viscous flows. One is in the porous media and the viscous effect comes from the friction, so that the equations become the psystem with damping. The other has a usual Newton viscosity and the equations become the psystem with viscosity. It was known that the solution to the Cauchy problem for the psystem 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 (NishiharaWangYang, NishiharaNishikawa). For the initialboundary value problem on the half line we have investigated the boundary effect (NishiharaYang). This method has been applied to the thermoelastic system with dissipation (NishiharaNishibata). To investigate the psystem 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 (NishiharaZhao, Nishihara). For the original psystem 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 (MatsumuraNishihara). The stability of superposition of the boundary layer solution and viscous shock wave is remained open.

Report
(4results)
Research Output
(20results)