  Here we investigate the Cauchy problem for the barotropic Navier-Stokes
equations in R^n, in the critical Besov spaces setting. We improve recent
results as regards the uniqueness condition: initial velocities in critical
Besov spaces with (not too) \emph{negative} indices generate a unique local
solution. Apart from (critical) regularity, the initial density just has to be
bounded away from 0 and to tend to some positive constant at infinity.
Density-dependent viscosity coefficients may be considered. Using Lagrangian
coordinates is the key to our statements as it enables us to solve the system
by means of the basic contraction mapping theorem. As a consequence, conditions
for uniqueness are the same as for existence, and Lipschitz continuity of the
flow map (in Lagrangian coordinates) is established.
