Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier--Stokes equations with far field vacuum

Jinkai Li South China Normal University Zhouping Xin The Chinese University of Hong Kong

Analysis of PDEs mathscidoc:2002.03001

In the presence of vacuum, the physical entropy for polytropic gases behaves singularly and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the onedimensional heat conductive compressible Navier--Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than $O(\frac{1}{x^2})$. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier--Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.
heat conductive compressible Navier--Stokes equations; global existence and uniqueness; uniformly bounded entropy; far field vacuum; De Giorgi iteration; singular estimates.
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@inproceedings{jinkaientropy-bounded,
  title={Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier--Stokes equations with far field vacuum},
  author={Jinkai Li, and Zhouping Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200211201711940197621},
}
Jinkai Li, and Zhouping Xin. Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier--Stokes equations with far field vacuum. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200211201711940197621.
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