  Here we prove the all-time propagation of the Sobolev regularity for the
velocity field solution of the two-dimensional compressible Navier-Stokes
equations, provided the volume (bulk) viscosity coefficient is large enough.
The initial velocity can be arbitrarily large and the initial density is just
required to be bounded. In particular, one can consider a characteristic
function of a set as an initial density. Uniqueness of the solutions to the
equations is shown, in the case of a perfect gas. As a by-product of our
results, we give a rigorous justification of the convergence to the
inhomogeneous incompressible Navier-Stokes equations when the volume viscosity
tends to infinity. Similar results are proved in the three-dimensional case,
under some scaling invariant smallness condition on the velocity field.
