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

## TL;DR

This paper proves the global existence of strong solutions to the heat conductive compressible Navier-Stokes equations with vacuum, under smallness conditions on a scaling invariant quantity related to initial data.

## Contribution

It establishes the first global well-posedness result for these equations with vacuum, based on a novel smallness condition on a scaling invariant initial data quantity.

## Key findings

- Global strong solutions exist under small initial data conditions.
- The smallness condition depends only on physical parameters, not on initial data size.
- Total mass can be finite or infinite in the solutions.

## Abstract

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. The total mass can be either finite or infinite.