Global Small Solutions of Heat Conductive Compressible Navier–Stokes Equations with Vacuum: Smallness on Scaling Invariant Quantity

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.