In this mini lecture, we will investigate the smooth solutions to the multi-dimensional compressible flows in the presence of vacuum, including the insentropic Navier-Stokes equations and viscous heat-conduting Navier-Stokes-Fourier system.

Generally speaking, the smooth solutions will lose their regularities in finite time when the density is allowed to vanish in some region. Therefore, we will reveal the possible blowup mechanism which should be responsible for the formation of singularity. Based on these new observations, global in time existence of classical solution with large oscillations and vacuum will be established, which is a sharp constrast to the classical theroy with small perturbation around a non-vacuum resting state.  The method and ideas established so far were proved to be efficient tools in the investigation of compressible framework and will continue to have their value in the future research.