TY - JOUR

T1 - Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space

AU - Kawashima, Shuichi

AU - Zhu, Peicheng

N1 - Funding Information:
* Corresponding author. Present address: Department of Mathematics, Tech. Univ. Darmstadt, 64289 Darmstadt, Germany. E-mail addresses: kawashim@math.kyushu-u.ac.jp (S. Kawashima), zhu@mathematik.tu-darmstadt.de (P. Zhu). 1 The author was supported by JSPS postdoctoral fellowship under P99217.

PY - 2008/6/15

Y1 - 2008/6/15

N2 - In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L2-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are "good" and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.

AB - In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L2-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are "good" and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.

KW - Asymptotic stability

KW - Compressible Navier-Stokes equations

KW - Eulerian coordinate

KW - Superposition of a rarefaction wave and a stationary solution

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U2 - 10.1016/j.jde.2008.01.020

DO - 10.1016/j.jde.2008.01.020

M3 - Article

AN - SCOPUS:42649117067

SN - 0022-0396

VL - 244

SP - 3151

EP - 3179

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 12

ER -