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 v0 and almost constant density ϱ0, 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-free projection of v0, is global. The systems are examined in Rd with d≥2, in the critical B˙2,1s Besov spaces framework.