Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier--Stokes Equations

Kenji Nishihara Tong Yang Huijiang Zhao

Analysis of PDEs mathscidoc:1912.43932

SIAM journal on mathematical analysis, 35, (6), 1561-1597, 2004
This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier--Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (<i>V<sup>R</sup></i>, <i>U<sup>R</sup></i>, <i>S<sup>R</sup></i>)(<i>t</i>,<i>x</i>). If the initial data (<i>v</i><sub>0</sub> , <i>u</i><sub>0</sub> ,<i>s</i><sub>0</sub> )(<i>x</i>) to the nonisentropic compressible Navier--Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [S. Kawashima, A. Matsumura, and K. Nishihara, <i>Proc. Japan Acad. Ser. A</i>, 62 (1986), pp. 249--252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (<i>v</i>, <i>u</i>, <i>s</i>)(<i>t</i>,<i>x</i>) which tends to (<i>V<sup>R</sup></i>, <i>U<sup>R</sup></i>, <i>S<sup>R</sup></i>)(<i>t</i>,<i>x</i>) as <i>t</i> tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent is close to 1. Furthermore, we show that for the
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@inproceedings{kenji2004nonlinear,
  title={Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier--Stokes Equations},
  author={Kenji Nishihara, Tong Yang, and Huijiang Zhao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210735161508496},
  booktitle={SIAM journal on mathematical analysis},
  volume={35},
  number={6},
  pages={1561-1597},
  year={2004},
}
Kenji Nishihara, Tong Yang, and Huijiang Zhao. Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier--Stokes Equations. 2004. Vol. 35. In SIAM journal on mathematical analysis. pp.1561-1597. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210735161508496.
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