Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

Abstract

We consider the Cauchy problem for the full compressible Navier--Stokes equations with vanishing of density at infinity in $\mathbb{R}^3$. Our main purpose is to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time. Our main results show that the strong solution exists globally in time if the initial mass is small for the fixed coefficients of viscosity and heat conduction, and can be large for the large coefficients of viscosity and heat conduction. Moreover, large-time behavior and a surprisingly exponential decay rate of the strong solution are obtained. Finally, we show that the global strong solution can become classical if the initial data are more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of $\mathbb{R}^3$ and that it can be of a nontrivially compact support. To our knowledge, this paper contains the first resul...