Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum

Abstract

This paper is concerned with the Cauchy problem of Navier–Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in [Formula: see text]. For less regular data and weaker compatibility condition than those proposed by Cho–Kim [Existence results for viscous polytropic fluids with vacuum, J. Differ. Equ. 228 (2006) 377–411], we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is more regular. We also establish the global well-posedness of classical solutions with large oscillations and vacuum in the case when the initial total energy is suitably small. The exponential decay estimates of the global solutions are obtained.