Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier-Stokes/Allen-Cahn system

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Yin, Haiyan¹  Zhu, Changjiang² 

[1] Huaqiao Univ, Sch Math Sci, Quanzhou 362021, Peoples R China

[2] South China Univ Technol, Sch Math, Guangzhou 510641, Guangdong, Peoples R China

关键词: Navier-Stokes/Allen-Cahn system;    Inflow problem;    Rarefaction wave;    Stationary solution;    Stability;   

DOI  :  10.1016/j.jde.2018.11.034

来源: Elsevier

【 摘 要 】

In this paper, we investigate the large time behavior of the solutions to the inflow problem for the one-dimensional Navier-Stokes/Allen-Cahn system in the half space. First, we assume that the space-asymptotic states (rho(+), u(+), chi(+)) and the boundary data (rho(b), u(b), chi(b)) satisfy some conditions so that the time-asymptotic state of solutions for the inflow problem is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Then, we show the existence of the stationary solution by the center manifold theorem. Finally, we prove that the nonlinear wave is asymptotically stable when the initial data is a small perturbation of the nonlinear wave. The proof is mainly based on the energy method by taking into account the effect of the concentration chi and the complexity of nonlinear wave. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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