  Here we prove the existence of global in time regular solutions to the
two-dimensional compressible Navier-Stokes equations supplemented with
arbitrary large initial velocity $v\_0$ and almost constantdensity
$\varrho\_0$, for large volume (bulk) viscosity. The result is generalized to
the higher dimensional case under the additional assumption that the strong
solution of the classical incompressible Navier-Stokes equations supplemented
with the divergence-freeprojection of $v\_0,$ is global. The systems are
examined in $R^d$ with $d \geq 2$, in the critical $\dot B^s\_{2,1}$ Besov
spaces framework.
