2003 Fiscal Year Final Research Report Summary
Stability of nonlinear waves for hyperbolic conservation laws will viscosity
Project/Area Number 
13640223

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 > 早稲田大学, 政治経済学部, 教授 (60141876)

CoInvestigator(Kenkyūbuntansha) 
MATSUMURA Akitaka Waseda University, Graduate School of Information Science and Technology, 大学院・情報科学研究科, 教授 (60115938)

Project Period (FY) 
2001 – 2003

Keywords  viscous shock wave / rarefaction wave / diffusion wave / stability / damped wave equation / critical exponent 
Research Abstract 
Our aim in this research is to investigate the asymptoic behavior of timeglobal solutions for one dimensional compressible flow with viscosity due to the Newton viscosity or the friction. The system is written by the hyperbolic conservation laws with viscosity, which has the nonlinear waves like viscous shock wave, rare faction wave, diffusion wave and the wave corresponding to contact discontinuity. For the compressible NavierStokes equation the global stability of strong rare faction wave is shown, whose method is applied to the JinXin relaxation model for psystem. Also, in the inflow problem on halfline the solution is shown to tend the superposition of viscous shock wave and boundary layer under some conditions, in which case the problem was open. On the other hand, the psystem with friction is modeled by the compressible flow in porous media. The solution was shown by HsiaoLiu to approach to the solution of the corresponding parabolic system due to the Darcy law. Through the precise consideration of the approach we have reached to the fact that the damped wave equation of second order is closely related to the corresponding heat equation in one and three dimensional space, which is applied to show the existence of timeglobal solution or the blowup of solution in a finite time for the semilinear damped wave equation. The critical exponent is same as that in the semilinear heat equation, which is reasonably understood by the fact obtained. It is also seen in the abstract setting. So, our result may give some suggestions in the investigation on the damped wave equation and related problem.

Research Products
(13 results)