Global Well-Posedness to the 3D Cauchy Problem of Nonhomogeneous Heat Conducting Navier–Stokes Equations with Vacuum and Large Oscillations

Abstract

We study global well-posedness of strong solutions to the Cauchy problem of nonhomogeneous heat conducting Navier–Stokes equations with vacuum on the whole space ${\mathbb {R}}^3$. We derive the global existence and uniqueness of strong solutions provided that $\Vert \rho _0\Vert _{L^\infty }\Vert \sqrt{\rho _0}{\mathbf {u}}_0\Vert _{L^2}^2\Vert \nabla {\mathbf {u}}_0\Vert _{L^2}^2$ is suitably small, with the smallness depending only on the viscosity coefficient $\mu$ of the system under consideration. Moreover, we also obtain the large time decay rates of the solution. In particular, the smallness condition is independent of any norms of the initial data and allows the solution to have large oscillations. Furthermore, there is no need to require compatibility conditions for the initial data via time weighted techniques.