ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY

doi:10.1007/s10473-021-0514-5

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (5): 1635-1658.doi: 10.1007/s10473-021-0514-5

ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY

苏奕帆, 郭真华

School of Mathematics, CNS, Northwest University, Xi'an 710127, China

收稿日期:2019-11-20 修回日期:2021-04-12 出版日期:2021-10-25 发布日期:2021-10-21
通讯作者: Zhenhua GUO E-mail:zhguo@nwu.edu.cn
作者简介:Yifan SU,E-mail:mayifansu@163.com
基金资助:
The second author was supported by the National Natural Science Foundation of China (11671319, 11931013).

ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY

Yifan SU, Zhenhua GUO

School of Mathematics, CNS, Northwest University, Xi'an 710127, China

Received:2019-11-20 Revised:2021-04-12 Online:2021-10-25 Published:2021-10-21
Contact: Zhenhua GUO E-mail:zhguo@nwu.edu.cn
Supported by:
The second author was supported by the National Natural Science Foundation of China (11671319, 11931013).

摘要/Abstract

摘要： This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity. In particular, we focus our attention on the viscosity taking the form $\mu(\rho)=\rho^\epsilon (\epsilon > 0).$ For the selected density-dependent viscosity, it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates, in addition to the scaling argument and elementary energy analysis. Moreover, our results extend the studies in[Xin Z P. Comm Pure Appl Math, 1993, 46(5):621-665].

关键词: compressible Navier-Stokes equations, density-dependent viscosity, rarefaction wave, zero dissipation limit

Abstract: This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity. In particular, we focus our attention on the viscosity taking the form $\mu(\rho)=\rho^\epsilon (\epsilon > 0).$ For the selected density-dependent viscosity, it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates, in addition to the scaling argument and elementary energy analysis. Moreover, our results extend the studies in[Xin Z P. Comm Pure Appl Math, 1993, 46(5):621-665].

Key words: compressible Navier-Stokes equations, density-dependent viscosity, rarefaction wave, zero dissipation limit

中图分类号:

76N15

苏奕帆, 郭真华. ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2021, 41(5): 1635-1658.

Yifan SU, Zhenhua GUO. ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY[J]. Acta mathematica scientia,Series B, 2021, 41(5): 1635-1658.

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ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY

ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY

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