Compressible Navier-Stokes system : large solutions and incompressible limit

Abstract: 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.