GLOBAL CLASSICAL SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH VACUUM IN PERIODIC DOMAIN

This paper concerns the global well-posedness of classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in three spatial dimensions with periodic initial data with density allowed to vanish initially. We introduce the so-called the effective viscous flux which is the key for time-uniform upper bound of density. Based on these key ingredients, we are able to obtain the global solvability of classical solutions in three spatial dimensions, provided the smooth initial data are of small total energy. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum.