Global Classical Large Solutions to Navier--Stokes Equations for Viscous Compressible and Heat-Conducting Fluids with Vacuum

Abstract

We consider the one-dimensional Navier--Stokes equations for viscous compressible and heat-conducting fluids (i.e., the full Navier--Stokes equations). Because of the stronger nonlinearity and degeneracies of the equations due to the temperature equation and vanishing of density (i.e., appearance of vacuum), respectively, there are few results until now about global existence of regular solutions to the full Navier--Stokes equations. In the paper, we get a unique global classical solution to the equations with large initial data and vacuum. Our analysis is based on some delicate energy estimates together with some new ideas which were used in our previous paper [Ding, Wen, and Zhu, J. Differential Equations, 251 (2011), pp. 1696--1725] to get the high-order estimates of the solutions. This result could be viewed as the first one on the global well-posedness of regular (classical) solutions to the Navier--Stokes equations for viscous compressible and heat-conducting fluids when initial data may be large and vacuum could appear.