  In this paper, we consider the three-dimensional isentropic Navier-Stokes
equations for compressible fluids with viscosities depending on density in a
power law and allowing initial vacuum. We introduce the notion of regular
solutions and prove the local-in-time well-posedness of solutions with
arbitrarily large initial data and vacuum in this class, which is a
long-standing open problem due to the very high degeneracy caused by vacuum.
Moreover, for certain classes of initial data with 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.
