American Institute of Mathematical Sciences

March  2008, 9(2): 281-308. doi: 10.3934/dcdsb.2008.9.281

A nonlinear degenerate system modelling water-gas flows in porous media

 1 Université du Sud Toulon Var, MC2 (Inria Futurs) and CPT/Imath, Av. de l'université, 83957 La Garde, France 2 Ecole Centrale de nantes, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 1, rue de la Noé, 44321 Nantes, France

Received  February 2007 Revised  October 2007 Published  December 2007

The aim of this paper is to study a system modelling the flow of an incompressible phase (water) and a compressible phase (gas) in porous media. Two kinds of degeneracy appear for this problem: a dissipative term and an evolution term degenerate with respect to the saturation. Global weak solutions are established for the system by introducing several approximate models. The first one consists in obtaining a non-degenerate dissipative system. The second one is a time discretization method in order to overcome the degeneracy in the evolution term. At this step, the subproblem is a non- degenerate elliptic system which is strongly coupled and highly nonlinear. Then the Leray-Schauder fixed point theorem instead of a classical Schauder fixed point theorem is the key point to solve such a problem.
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