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Global well-posedness of the one-dimensional compressible navier-stokes equations with constant heat conductivity and nonnegative density

Jinkai Li South Chine Normal University

Analysis of PDEs mathscidoc:2002.03003

SIAM J. MATH. ANAL., 51, (5), 366-{3693, 2019
In this paper we consider the initial-boundary value problem to the one-dimensional compressible Navier{Stokes equations for ideal gases. Both the viscous and heat conductive coeffcients are assumed to be positive constants, and the initial density is allowed to have vacuum. Global existence and uniqueness of strong solutions is established for any H2 initial data, which generalizes the well-known result of Kazhikhov and Shelukhin [J. Appl. Math. Mech., 41 (1977), pp. 273{282] to the case that with nonnegative initial density. An observation to overcome the diculty caused by the lack of the positive lower bound of the density is that the ratio of the density to its initial value is inversely proportional to the time integral of the upper bound of the temperature along the trajectory.
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@inproceedings{jinkai2019global,
  title={GLOBAL WELL-POSEDNESS OF THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH CONSTANT HEAT CONDUCTIVITY AND NONNEGATIVE DENSITY},
  author={Jinkai Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200211202747552703623},
  booktitle={SIAM J. MATH. ANAL.},
  volume={51},
  number={5},
  pages={366-{3693},
  year={2019},
}
Jinkai Li. GLOBAL WELL-POSEDNESS OF THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH CONSTANT HEAT CONDUCTIVITY AND NONNEGATIVE DENSITY. 2019. Vol. 51. In SIAM J. MATH. ANAL.. pp.366-{3693. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200211202747552703623.
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