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.

[1]  D. Bresch,et al.  Quelques modèles diffusifs capillaires de type Korteweg , 2004 .

[2]  Didier Bresch,et al.  On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids , 2007 .

[3]  D. Bresch,et al.  On Some Compressible Fluid Models: Korteweg, Lubrication, and Shallow Water Systems , 2003 .

[4]  P. Orenga.  Un théorème d'existence de solutions d'un problème de shallow water , 1995 .

[5]  D. Bresch,et al.  Existence of Global Weak Solutions for a 2D Viscous Shallow Water Equations and Convergence to the Quasi-Geostrophic Model , 2003 .

[6]  Antoine Mellet,et al.  On the isentropic compressible Navier-Stokes equation , 2005 .

[7]  B. Nicolaenko,et al.  Global splitting and regularity of rotating shallow-water equations , 1997 .

[8]  Jean-Frédéric Gerbeau,et al.  Derivation of viscous Saint-Venant system for laminar shallow water , 2001 .

[9]  J. Holton.  Geophysical fluid dynamics. , 1983, Science.

[10]  Didier Bresch,et al.  Recent mathematical results and open problems about shallow water equations. Analysis and simulation of fluid dynamic , 2006 .

[11]  Linda Sundbye.  Global Existence for the Dirichlet Problem for the Viscous Shallow Water Equations , 1996 .

[12]  F. Marche.  Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects , 2007 .

[13]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[14]  Bernardi Christine,et al.  On the Shallow Water Equations at Low Reynolds Number , 1991 .