INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM*

doi:10.1007/s10473-024-0114-2

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (1): 247-274.doi: 10.1007/s10473-024-0114-2

INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^*

Zhenhua Guo^1,2, Xueyao Zhang^1,†

1. School of Mathematics and CNS, Northwest University, Xi'an 710127, China;
2. School of Mathematics and Information Science, Guangxi University, Nanning 530004, China

收稿日期:2022-09-07 修回日期:2023-07-22 出版日期:2024-02-25 发布日期:2024-02-27
通讯作者: † Xueyao Zhang, E-mail: xyzhang05@163.com
作者简介:Zhenhua Guo, E-mail: zhguo@gxu.edu.cn
基金资助:
NSFC (11931013) and the GXNSF (2022GXNSFDA035078).

INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^*

Zhenhua Guo^1,2, Xueyao Zhang^1,†

1. School of Mathematics and CNS, Northwest University, Xi'an 710127, China;
2. School of Mathematics and Information Science, Guangxi University, Nanning 530004, China

Received:2022-09-07 Revised:2023-07-22 Online:2024-02-25 Published:2024-02-27
Contact: † Xueyao Zhang, E-mail: xyzhang05@163.com
About author:Zhenhua Guo, E-mail: zhguo@gxu.edu.cn
Supported by:
NSFC (11931013) and the GXNSF (2022GXNSFDA035078).

摘要/Abstract

摘要： In this paper, we study the one-dimensional motion of viscous gas near a vacuum, with the gas connecting to a vacuum state with a jump in density. The interface behavior, the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficient $\mu(\rho)=\rho^{\alpha}$ for any $0<\alpha<1$; this includes the time-weighted boundedness from below and above. The smoothness of the solution is discussed. Moreover, we construct a class of self-similar classical solutions which exhibit some interesting properties, such as optimal estimates. The present paper extends the results in [Luo T, Xin Z P, Yang T. SIAM J Math Anal, 2000, 31(6): 1175-1191] to the jump boundary conditions case with density-dependent viscosity

关键词: decay rates, interface, Navier-Stokes equations, vacuum

Abstract: In this paper, we study the one-dimensional motion of viscous gas near a vacuum, with the gas connecting to a vacuum state with a jump in density. The interface behavior, the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficient $\mu(\rho)=\rho^{\alpha}$ for any $0<\alpha<1$; this includes the time-weighted boundedness from below and above. The smoothness of the solution is discussed. Moreover, we construct a class of self-similar classical solutions which exhibit some interesting properties, such as optimal estimates. The present paper extends the results in [Luo T, Xin Z P, Yang T. SIAM J Math Anal, 2000, 31(6): 1175-1191] to the jump boundary conditions case with density-dependent viscosity

Key words: decay rates, interface, Navier-Stokes equations, vacuum

中图分类号:

35Q30

Zhenhua Guo, Xueyao Zhang. INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^*[J]. 数学物理学报(英文版), 2024, 44(1): 247-274.

Zhenhua Guo, Xueyao Zhang. INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 247-274.

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INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^*

INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^*

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