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Stability of nonlinear waves in viscous conservation system together with diffusion phenomena of solutions of damped wave equation

粘性守恒系统中非线性波的稳定性及阻尼波动方程解的扩散现象

基本信息

批准号：
16540206
负责人：
NISHIHARA Kenji
金额：
$ 2.1万
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：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

The system of conservation laws has the shock wave, rarefaction wave and contact discontinuity as nonlinear waves. In real physics, it may become the system of viscous conservation laws by some viscous effect, which yields the viscous shock wave, rarefaction wave and viscous contact wave with diffusion wave. Our aim of this research is to observe the stability of the waves.In our research the viscous effect is by the usual Newton viscosity or friction in porous media flow. The flow approaches to the solution of corresponding parabolic system by Darcy's law, which implies that the damped wave equation behaves as the corresponding diffusion equation as time tends to infinity, what we call the diffusion phenomena. The observation of this phenomena is another aim of this research. The stability of viscous contact wave in the viscous conservation laws with Newton's viscosity has been mainly developed by the investigator. The diffusion phenomena of solutions to the damped wave equation has been investigated by the head investigator, based on the fact that the solution of the Cauchy problem far the linear damped wave equation is decomposed to the sum of the wave part exponentially decaying and the diffusion part For the corresponding diffusion equation, rather precise results are obtained thanks to the smoothing effects and the maximum principle, but these key properties do not hold for the wave equation, and further studies are necessary.

守恒律系中的激波、稀疏波和接触间断都是非线性波。在实际物理中，由于某些粘性效应，可以形成粘性守恒律系，从而产生粘性激波、稀疏波和粘性接触波与扩散波。我们研究的目的是观察波动的稳定性。在我们的研究中，粘性效应是由多孔介质流动中常见的牛顿粘性或摩擦力引起的。流动用达西定律逼近相应抛物型方程的解，这意味着当时间趋于无穷大时，阻尼波方程表现为相应的扩散方程，即我们所说的扩散现象。对这一现象的观察是本研究的另一个目的。具有牛顿粘性的粘性守恒律中的粘性接触波的稳定性主要是由研究人员发展起来的。基于线性阻尼波动方程的柯西问题的解被分解为指数衰减的波动部分和相应扩散方程的扩散部分之和，由于光滑化效应和极大值原理，得到了相当精确的结果，这是带头研究者研究的结果，但这些关键性质对波动方程不成立，需要进一步的研究。