GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY*

doi:10.1007/s10473-024-0111-5

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (1): 195-214.doi: 10.1007/s10473-024-0111-5

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*

Shijin Ding, Yinghua Li^†, Yu Wang

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

收稿日期:2022-09-23 修回日期:2023-06-16 出版日期:2024-02-25 发布日期:2024-02-27
通讯作者: † Yinghua Li,E-mail:yinghua@scnu.edu.cn
作者简介:Shijin Ding, E-mail: dingsj@scnu.edu.cn; Yu Wang, E-mail: yuwang@m.scnu.edu.cn
基金资助:
Ding's research was supported by the Key Project of the NSFC (12131010), the NSFC (11771155, 12271032) and the NSF of Guangdong Province (2021A1515010249, 2021A1515010303). Li's research was supported by the NSFC (11971179, 12371205).

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*

Shijin Ding, Yinghua Li^†, Yu Wang

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2022-09-23 Revised:2023-06-16 Online:2024-02-25 Published:2024-02-27
Contact: † Yinghua Li,E-mail:yinghua@scnu.edu.cn
About author:Shijin Ding, E-mail: dingsj@scnu.edu.cn; Yu Wang, E-mail: yuwang@m.scnu.edu.cn
Supported by:
Ding's research was supported by the Key Project of the NSFC (12131010), the NSFC (11771155, 12271032) and the NSF of Guangdong Province (2021A1515010249, 2021A1515010303). Li's research was supported by the NSFC (11971179, 12371205).

摘要/Abstract

摘要： This paper is concerned with the Navier-Stokes/Allen-Cahn system, which is used to model the dynamics of immiscible two-phase flows. We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form of $\eta(\rho)=\rho^\alpha$. The existence of unique global $H^{2m}$-solutions $(m\in\mathbb N)$ to the free boundary problem is proven for when $0<\alpha<\frac14$. Furthermore, we obtain the global $C^\infty$-solutions if the initial data is smooth.

关键词: Navier-Stokes/Allen-Cahn system, density-dependent viscosity, free boundary, global solutions

Abstract: This paper is concerned with the Navier-Stokes/Allen-Cahn system, which is used to model the dynamics of immiscible two-phase flows. We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form of $\eta(\rho)=\rho^\alpha$. The existence of unique global $H^{2m}$-solutions $(m\in\mathbb N)$ to the free boundary problem is proven for when $0<\alpha<\frac14$. Furthermore, we obtain the global $C^\infty$-solutions if the initial data is smooth.

Key words: Navier-Stokes/Allen-Cahn system, density-dependent viscosity, free boundary, global solutions

中图分类号:

35A02

Shijin Ding, Yinghua Li, Yu Wang. GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*[J]. 数学物理学报(英文版), 2024, 44(1): 195-214.

Shijin Ding, Yinghua Li, Yu Wang. GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 195-214.

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GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*

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