Stability of stationary for inflow problem on the coupled compressible Navier–Stokes/Allen–Cahn system

Abstract

In this paper, we study the large-time behavior of solutions to the Navier–Stokes/Allen–Cahn system of compressible viscous gases in half space. We prove that the corresponding stationary solutions of the small amplitude to the inflow problem for the Navier–Stokes/Allen–Cahn system are time-asymptotically stable. The proof of asymptotic stability is based on the basic energy method.

Keywords:

• Navier–Stokes/Allen–Cahn system
• stationary solutions
• inflow problem
• stability

AMS subject classifications:

• 35M10
• 35B40
• 35B35

Acknowledgements

Ting Luo would like to thank Professors Changjiang Zhu and Haiyan Yin for many fruitful discussions on the topic of the paper. Ting Luo was supported by the National Natural Science Foundation of China $\mathrm{\#}$12001249, the Educational Commission Science Programm of Jiangxi Province $\mathrm{\#}$GJJ190280 and the Scientific Research Funds of Jiangxi University of Finance and Economics $\mathrm{\#}$012270624.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Ting Luo would like to thank Professors Changjiang Zhu and Haiyan Yin for many fruitful discussions on the topic of the paper. Ting Luo was supported by the National Natural Science Foundation of China $\mathrm{\#}$12001249, the Educational Commission Science Programm of Jiangxi Province $\mathrm{\#}$GJJ190280 and the Scientific Research Funds of Jiangxi University of Finance and Economics $\mathrm{\#}$012270624.