On Classical Solutions for Viscous Polytropic Fluids with Degenerate Viscosities and Vacuum

In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids allowing initial vacuum when viscosities depend on density in a superlinear power law. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and a vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by a vacuum. Moreover, for certain classes of initial data with a local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.