On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models

The purpose of this paper is to build sequences of suitably smooth approximate solutions to the Saint-Venant model that preserve the mathematical structure discovered in [D. Bresch, B. Desjardins, Comm. Math. Phys. 238 (1-2) (2003) 211-223]. The stability arguments in this paper then apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model. Extension of this mollifying procedure to the case of compressible Navier-Stokes equations is also provided. Using the recent paper written by the authors, this provides global existence results of weak solutions for the barotropic Navier-Stokes equations and for compressible Navier-Stokes equations with heat conduction using a particular cold pressure term close to vacuum. (c) 2006 Elsevier SAS. All rights reserved.