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 $$\Vert \rho _0\Vert _\infty (\Vert \rho _0\Vert _3+\Vert \rho _0\Vert _\infty ^2\Vert \sqrt{\rho _0}u_0\Vert _2^2)(\Vert \nabla u_0\Vert _2^2+ \Vert \rho _0\Vert _\infty \Vert \sqrt{\rho _0}E_0\Vert _2^2)$$ ‖ ρ 0 ‖ ∞ ( ‖ ρ 0 ‖ 3 + ‖ ρ 0 ‖ ∞ 2 ‖ ρ 0 u 0 ‖ 2 2 ) ( ‖ ∇ u 0 ‖ 2 2 + ‖ ρ 0 ‖ ∞ ‖ ρ 0 E 0 ‖ 2 2 ) is sufficiently small, with the smallness depending only on the parameters $$R, \gamma , \mu , \lambda ,$$ R , γ , μ , λ , and $$\kappa $$ κ 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 $$L^\infty _t(L^3)$$ L t ∞ ( L 3 ) type estimate of the density.

中文翻译：

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具有真空的热传导可压缩纳维-斯托克斯方程的全局小解：缩放不变量的小性

在本文中，我们考虑了存在真空和真空远场的导热可压缩 Navier-Stokes 方程的柯西问题。强解的全局适定性是在以下假设下建立的，其中包括一些其他规律性和兼容性条件：缩放不变量 $$\Vert \rho _0\Vert _\infty (\Vert \rho _0\Vert _3+\Vert \ rho _0\Vert _\infty ^2\Vert \sqrt{\rho _0}u_0\Vert _2^2)(\Vert \nabla u_0\Vert _2^2+ \Vert \rho _0\Vert _\infty \Vert \ sqrt{\rho _0}E_0\Vert _2^2)$$ ‖ ρ 0 ‖ ∞ ( ‖ ρ 0 ‖ 3 + ‖ ρ 0 ‖ ∞ 2 ‖ ρ 0 u 0 ‖ 2 2 ) ( ‖ ∇ u 0 ‖ 2 2 + ‖ ρ 0 ‖ ∞ ‖ ρ 0 E 0 ‖ 2 2 ) 足够小，其大小仅取决于参数 $$R, \gamma , \mu , \lambda ,$$ R , γ , μ , λ ,和 $$\kappa $$ κ 在系统中。尤其，小假设是专门对上述标度不变量强加的，它独立于初始数据的任何范数，这与现有论文不同。总质量可以是有限的或无限的。密度方程——更准确地说是它的三次方程，从连续性和动量方程的组合中推导出来——被用来得到 $$L^\infty _t(L^3)$$ L t ∞ ( L 3 ) 类型估计密度。