In this paper we prove the existence of global strong solution for the Navier-Stokes equations with general degenerate viscosity coefficients. The cornerstone of the proof is the introduction of a new effective pressure which allows to obtain an Oleinik-type estimate for the so called effective velocity. In our proof we make use of additional regularizing effects on the velocity which requires to extend the technics developed by Hoff for the constant viscosity case.

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In this paper we obtain new well-possedness results concerning a linear inhomogenous Stokes-like system. These results are used to establish local well-posedness in the critical spaces for initial density ρ0 and velocity, 4 , for the inhomogeneous incompressible Navier-Stokes system with variable viscosity. To the best of our knowledge, regarding the 3D case, this is the first result in a truly critical framework for which one does not assume any smallness condition on the density.

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