Global symmetric classical solutions of the full compressible Navier–Stokes equations with vacuum and large initial data

Abstract

In this paper, we get a result on global existence of classical and strong solutions of the full compressible Navier–Stokes equations in three space dimensions with spherically or cylindrically symmetric initial data which may be large. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate a priori estimates globally in time which depend on the assumption κ=O(1+θq) where q>r (r can be zero), which relaxes the condition q⩾2+2r in [12,27,39]. This could be viewed as an extensive work of [16] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension.