On local strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum

We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially. We first prove the local existence and uniqueness of the strong solutions, where the initial compatibility condition proposed in \cite{cho1,K2,coi1} is removed under suitable sense. Then, the continuous of strong solutions on the initial data is derived under an additional compatibility condition. Moreover, for the initial data satisfying some additional regularity and compatibility condition, the strong solution is proved to be a classical one.