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Stability of rarefaction waves to the 1D compressible NavierStokes equations with density-dependent viscosity

Quansen Jiu Yi Wang Zhouping Xin

Analysis of PDEs mathscidoc:1912.43751

Communications in Partial Differential Equations, 36, (4), 602-634, 2011.1
In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic NavierStokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.
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@inproceedings{quansen2011stability,
  title={Stability of rarefaction waves to the 1D compressible NavierStokes equations with density-dependent viscosity},
  author={Quansen Jiu, Yi Wang, and Zhouping Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205522204880315},
  booktitle={Communications in Partial Differential Equations},
  volume={36},
  number={4},
  pages={602-634},
  year={2011},
}
Quansen Jiu, Yi Wang, and Zhouping Xin. Stability of rarefaction waves to the 1D compressible NavierStokes equations with density-dependent viscosity. 2011. Vol. 36. In Communications in Partial Differential Equations. pp.602-634. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205522204880315.
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