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

守恒律系统以激波、稀薄波和接触不连续为非线性波。在实际物理中，由于某种粘滞效应，它可能成为粘滞守恒定律的系统，产生粘滞激波、稀薄波和粘滞接触波与扩散波。我们这项研究的目的是观察波浪的稳定性。在我们的研究中，粘性效应是通过通常的牛顿粘性或多孔介质流动中的摩擦来实现的。根据达西定律，流动趋向于相应抛物方程组的解，这意味着当时间趋于无穷时，阻尼波动方程表现为相应的扩散方程，我们称之为扩散现象。观察这一现象是本研究的另一个目的。在牛顿粘性守恒定律中，粘性接触波的稳定性主要是由研究者提出的。在将线性阻尼波动方程的Cauchy问题解分解为波动部分指数衰减部分和扩散部分的和的基础上，研究了阻尼波动方程解的扩散现象。对于相应的扩散方程，由于平滑效应和极大值原理，得到了相当精确的结果，但这些关键性质不适用于波动方程。进一步研究是必要的。

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.