Compressible Navier-Stokes system : large solutions and incompressible limit

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 $v\mathrm{\_}0$ and almost constantdensity $\varrho \mathrm{\_}0$, 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 $v\mathrm{\_}0,$ is global. The systems are examined in $R^d$ with $d\ge 2$, in the critical ${\dot{B}}^s\mathrm{\_}2,1$ Besov spaces framework.