Two-velocity hydrodynamics in fluid mechanics: Part II Existence of   global $\kappa$-entropy solutions to compressible Navier-Stokes systems with   degenerate viscosities
  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. 
Didier Bresch, Benoit Desjardins, Ewelina Zatorska
math.AP