Abstract
In this paper, we consider the Cauchy problem to the heat conductive compressible Navier–Stokes equations in the presence a of vacuum and with a vacuum far field. Global well-posedness of strong solutions is established under the assumption, among some other regularity and compatibility conditions: that the scaling invariant quantity \(\Vert \rho _0\Vert _\infty (\Vert \rho _0\Vert _3+\Vert \rho _0\Vert _\infty ^2\Vert \sqrt{\rho _0}u_0\Vert _2^2)(\Vert \nabla u_0\Vert _2^2+ \Vert \rho _0\Vert _\infty \Vert \sqrt{\rho _0}E_0\Vert _2^2)\) is sufficiently small, with the smallness depending only on the parameters \(R, \gamma , \mu , \lambda ,\) and \(\kappa \) in the system. Notably, the smallness assumption is imposed on the above scaling invariant quantity exclusively, and it is independent of any norms of the initial data, which is different from the existing papers. The total mass can be either finite or infinite. An equation for the density-more precisely for its cubic, derived from combining the continuity and momentum equations-is employed to get the \(L^\infty _t(L^3)\) type estimate of the density.
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Acknowledgements
The author is grateful to the anonymous referees for the kind suggestions that improved this paper. This work was supported in part by the National Natural Science Foundation of China Grants 11971009, 11871005, and 11771156, by the Natural Science Foundation of Guangdong Province Grant 2019A1515011621, by the South China Normal University start-up Grant 550-8S0315, and by the Hong Kong RGC Grant CUHK 14302917.
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Li, J. Global Small Solutions of Heat Conductive Compressible Navier–Stokes Equations with Vacuum: Smallness on Scaling Invariant Quantity. Arch Rational Mech Anal 237, 899–919 (2020). https://doi.org/10.1007/s00205-020-01521-7
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DOI: https://doi.org/10.1007/s00205-020-01521-7