Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data

TL;DR

This paper proves the global existence and uniqueness of strong and classical solutions to the 3D full compressible Navier-Stokes equations with vacuum at infinity, including decay rates and large data scenarios.

Contribution

It provides the first known results on global solutions with vacuum at infinity for the full compressible Navier-Stokes equations, including exponential decay rates.

Findings

Global strong solutions exist with small initial mass or large viscosity and heat conduction.

Solutions exhibit exponential decay over time.

Solutions can become classical with more regular initial data.

Abstract

We consider the Cauchy problem for the full compressible Navier-Stokes equations with vanishing of density at infinity in R3. 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 is more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of R3 and that it can be of a non trivially compact support.To our knowledge, this paper contains the first result so far for the global existence of solutions to the full compressible Navier-Stokes equations when density vanishes at infinity (in space). In addition, the exponential decay rate of the strong solution is of independent interest.