Asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law with partially linearly degenerate flux
Asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law with partially linearly degenerate flux
复制标题
具有部分线性简并通量的标量粘性守恒定律柯西问题解的渐近行为
DOI：
10.1137/110839448
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发表时间：
2012
影响因子：
2
通讯作者：
A. Matsumura and N. Yoshida
中科院分区：
文献类型：
--
作者：
Lee;Ho-Gyu;A. Matsumura and N. Yoshida
本文研究了一维粘性守恒律方程柯西问题解的时间渐近性态，其中远场状态是给定的。特别是，我们研究的情况下，通量函数是凸的，但线性退化的一些区间。当相应的Riemann问题存在由稀疏波和接触间断组成的Riemann解时，证明了Cauchy问题的解随着时间的推移趋于稀疏波和粘性接触波的线性组合.这是关于标量粘性守恒律Cauchy问题多波型渐近性的第一个结果。本文用能量法并考虑非线性波之间的相互作用给出了证明。我们还表明，类似的论点是适用于半空间上的初边值问题。
In this paper, we investigate the asymptotic behavior in time of solutions to the Cauchy problem for a one-dimensional viscous conservation law where the far field states are prescribed. In particular, we study the case in which the flux function is convex but linearly degenerate on some intervals. When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward the linear combination of the rarefaction waves and viscous contact wave as the time goes to infinity. This is the first result concerning the asymptotics toward multiwave patterns for the Cauchy problem to the scalar viscous conservation law. The proof is given by an-energy method and a careful consideration of the interactions between the nonlinear waves. We also show that similar arguments are applicable to the initial-boundary value problem on the half space.