STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

doi:10.1007/s10473-024-0417-3

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (4): 1507-1523.doi: 10.1007/s10473-024-0417-3

STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

Yunkun Chen¹, Bin Huang², Xiaoding Shi^2,*

1. School of Mathematics and Computer Science, Anshun University, Anshun 561000, China;
2. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

收稿日期:2022-07-25 修回日期:2022-12-14 出版日期:2024-08-25 发布日期:2024-08-30

STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

Yunkun Chen¹, Bin Huang², Xiaoding Shi^2,*

1. School of Mathematics and Computer Science, Anshun University, Anshun 561000, China;
2. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

Received:2022-07-25 Revised:2022-12-14 Online:2024-08-25 Published:2024-08-30
Contact: *E-mail: shixd@mail.buct.edu.cn
About author:E-mail: cyk2013@gznu.edu.cn; abinhuang@gmail.com
Supported by:
fyn:Chen's work was supported by the National Natural Science Foundation of China (12361044); Shi's work was supported by the National Natural Science Foundation of China (12171024, 11971217, 11971020). This paper was also supported by the Academic and Technical Leaders Training Plan of Jiangxi Province (20212BCJ23027).

摘要/Abstract

摘要： This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space. For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation, and where the strength of the initial phase field could be arbitrarily large, we prove that the solution of the Cauchy problem exists for all time, and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero. The proof is mainly based on a scaling argument and a basic energy method.

关键词: compressible Navier-Stokes equations, Allen-Cahn equation, rarefaction wave, sharp interface limit, stability

Abstract: This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space. For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation, and where the strength of the initial phase field could be arbitrarily large, we prove that the solution of the Cauchy problem exists for all time, and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero. The proof is mainly based on a scaling argument and a basic energy method.

Key words: compressible Navier-Stokes equations, Allen-Cahn equation, rarefaction wave, sharp interface limit, stability

中图分类号:

35M10

Yunkun Chen, Bin Huang, Xiaoding Shi. STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM[J]. 数学物理学报(英文版), 2024, 44(4): 1507-1523.

Yunkun Chen, Bin Huang, Xiaoding Shi. STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM[J]. Acta mathematica scientia,Series B, 2024, 44(4): 1507-1523.

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STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

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