Asymptotic Stability of Rarefaction Wave for the Navier–Stokes Equations for a Compressible Fluid in the Half Space
Asymptotic Stability of Rarefaction Wave for the Navier–Stokes Equations for a Compressible Fluid in the Half Space
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DOI：
10.1007/s00205-008-0191-8
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发表时间：
2009-08
影响因子：
2.5
通讯作者：
S. Kawashima;P. Zhu
中科院分区：
文献类型：
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作者：
S. Kawashima;P. Zhu
本文研究了半空间中欧拉坐标系下可压缩流体中Navier-Stokes方程的出流问题解的稀疏波渐近稳定性。这是我们关于这个主题的系列论文中的第二篇。本文首先对这类初边值问题的时间渐近状态进行了完全分类，根据一些参数，即空间渐近状态和边界条件，在状态空间中画出了时间渐近状态分类的一些图。为了证明稀疏波的稳定性，我们利用Burgers方程的解构造了稀疏波的适当光滑的逼近，并建立了光滑稀疏波的Lp-模时间衰减估计。然后，我们采用的L2-能量方法证明，稀疏波是非线性稳定的小扰动下，随着时间的推移到无穷大。
This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to anoutflowproblem for the Navier–Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers’ equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates inLp-norm for the smoothed rarefaction wave. We then employ theL2-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity.