Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier-Stokes Equations
Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier-Stokes Equations
复制标题
DOI：
10.1137/s003614100342735x
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
K. Nishihara;Tong Yang;Huijiang Zhao
中科院分区：
文献类型：
--
作者：
K. Nishihara;Tong Yang;Huijiang Zhao
本文关注一维可压缩纳维-斯托克斯方程解的强稀疏波的时间渐近行为。假设可压缩欧拉方程对应的黎曼问题可以通过稀疏波(V-R, U-R, S-R)(t, x)求解。如果非等熵可压缩纳维-斯托克斯方程的初始数据 (v(0), u(0), s(0))(x) 是近似稀疏波的小扰动，如 [ S. Kawashima, A. Matsumura, and K. Nishihara, Proc.日本科学院.序列。 A, 62 (1986), pp. 249-252]，然后我们证明，对于一般气体，柯西问题承认一个唯一的全局平滑解 (v, u, s)(t, x)，当 t 趋于无穷大时，该解趋于 (V-R, U-R, S-R)(t, x)。假设绝热指数 gamma 接近 1，也可以建立非等熵理想多变气体的全局稳定性结果。此外，我们还表明，对于等熵可压缩纳维-斯托克斯方程，只要所得可压缩欧拉方程是严格双曲的并且两个特征场都是真正非线性的，相应的全局稳定性结果成立。在这里，全局稳定性意味着初始扰动可能很大。由于我们不要求稀疏波的强度很小，因此这些结果给出了一维可压缩纳维-斯托克斯方程的强稀疏波的非线性稳定性。
This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (V-R, U-R, S-R)(t, x). If the initial data (v(0), u(0), s(0))(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [ S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (V-R, U-R, S-R)(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent gamma is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.