Global existence of regular solutions with large oscillations and vacuum for compressible flows

Abstract

The global existence of smooth solutions to the compressible Navier-Stokes equations is investigated. In particular, results are reviewed concerning the global existence and uniqueness of classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. The initial density is allowed to vanish and the spatial measure of the vacuum set can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous ones on classical solutions for initial densities being strictly away from vacuum, and are the first for global classical solutions that may have large oscillations and can contain vacuum states.