  This paper addresses the issue of global existence of so-called
$\kappa$-entropy solutions to the Navier-Stokes equations for viscous
compressible and barotropic fluids with degenerate viscosities. We consider the
three dimensional space domain with periodic boundary conditions. Our solutions
satisfy the weak formulation of the mass and momentum conservation equations
and also a generalization of the BD-entropy identity called: $\kappa$-entropy.
This new entropy involves a mixture parameter $\kappa \in (0,1)$ between the
two velocities $\bf{u}$ and $\bf{u} + 2\nabla\varphi(\varrho)$ (the latter was
introduced by the first two authors in [C. R. Acad. Sci. Paris 2004]), where
$\bf{u}$ is the velocity field and $\varphi$ is a function of the density
$\varrho$ defined by $\varphi'(s)=\mu'(s)/s$. As a byproduct of the existence
proof, we show that two-velocity hydrodynamics (in the spirit of S.C. Shugrin
1994) is a possible formulation of a model of barotropic compressible flow with
degenerate viscosities. It is used in construction of approximate solutions
based on a conservative augmented approximate scheme.
