Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global κ-entropy solutions to the compressible Navier-Stokes systems with degenerate viscosities

This paper addresses the issue of global existence of the 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 is kappa is an element of (0,1) between the two velocities u and u + 2 del phi(rho) (the latter was introduced by the first two authors in Bresch and Desjardins (2005) [5]), where u is the velocity field and phi is a function of the density rho defined by phi'(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. (C) 2015 Elsevier Masson SAS. All rights reserved.