The uniqueness of entropy solution for the compressible Euler equations is a fundamental and challenging problem. In this paper, the uniqueness of a composite wave of shock and rarefaction of one-dimensional compressible Euler equations is proved in the inviscid limit of compressible Navier-Stokes equations. Moreover, the relative entropy around the original Riemann solution consisting of shock and rarefaction under the large perturbation is shown to be uniformly bounded by the framework developed in [M. J. Kang and A. F. Vasseur, Invent. Math., 224 (2021), pp. 55--146]. The proof contains two new ingredients: (1) a cut-off technique and the expanding property of rarefaction are used to overcome the errors generated by the viscosity related to inviscid rarefaction; (2) the error terms concerning the interactions between shock and rarefaction are controlled by the compressibility of shock, the decay of derivative of rarefaction, and the separation of shock and rarefaction as time increases.

Publication:

SIAM Journal on Mathematical Analysis Volume 56, Issue 3 Jun 2024

https://doi.org/10.1137/23M156584X

Author:

FEIMIN HUANG

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing,100049, China

Email: fhuang@amt.ac.cn

WEIQIANG WANG

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing,100049, China

Email: wangweiqiang@amss.ac.cn

YI WANG

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing,100049, China

Email: wangyi@amss.ac.cn

YONG WANG

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing,100049, China

Email: yongwang@amss.ac.cn