Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line
Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line
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DOI：
10.1016/j.jde.2007.06.016
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发表时间：
2007-10
影响因子：
2.4
通讯作者：
Tohru Nakamura;S. Nishibata;T. Yuge
中科院分区：
文献类型：
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作者：
Tohru Nakamura;S. Nishibata;T. Yuge
研究了一维半空间中等熵可压缩粘性流体初边值问题解的大时间性态。S.Kawashima，S.Nishibata和P.朱证明了流体从边界喷出的流出问题定常解的唯一存在性和渐近稳定性。本文的主要目的是研究解向定常解的收敛速度。对于空间无穷远的超音速流动，我们得到了一个代数或指数衰减率。准确地说，如果初始摄动在空间渐近点以代数或指数速率衰减，则当时间趋于无穷大时，解以相同的时间速率收敛到相应的定常解。给出了跨音速流动的一个代数收敛速度。用加权能量法证明了这些结果。
We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.