Global small solutions of heat conductive compressible Navier-Stokes equations with vacuum: smallness on scaling invariant quantity

Jinkai Li South China Normal University

Analysis of PDEs mathscidoc:1906.03001

In this paper, we consider the Cauchy problem to the heat conductive compressible Navier-Stokes equations in the presence of vacuum and with 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 $\|\rho_0\|_\infty(\|\rho_0\|_3+\|\rho_0\|_\infty^2\|\sqrt{\rho_0}u_0\|_2^2)(\|\nabla u_0\|_2^2+ \|\rho_0\|_\infty\|\sqrt{\rho_0}E_0\|_2^2)$ 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, see, e.g., \cite{HLX12,HUANGLI11,WENZHU17}. The total mass can be either finite or infinite. A new equation for the density, more precisely for the cubic of the density, derived from combing the continuity and momentum equations, is employed to get the $L^\infty_t(L^3)$ type estimate of the density.
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@inproceedings{jinkaiglobal,
  title={Global small solutions of heat conductive compressible Navier-Stokes equations with vacuum: smallness on scaling invariant quantity},
  author={Jinkai Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190621225608771015353},
}
Jinkai Li. Global small solutions of heat conductive compressible Navier-Stokes equations with vacuum: smallness on scaling invariant quantity. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190621225608771015353.
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