A multiscale colloid transport model with anisotropic degenerate diffusion

Article type: Research Article

Authors: Belhadj, M. | Gerbeau, J.‐F. | Perthame, B.^;

Affiliations: INRIA Rocquencourt, BP 105, 78153 Le Chesnay cedex, France E‐mail: Mohamed.Belhadj@inria.fr, Jean‐Frederic.Gerbeau@inria.fr | DMA, École Normale Supérieure, 45, rue d'Ulm, 75230 Paris cedex 05, France E‐mail: Benoit.Perthame@ens.fr

Abstract: We consider a weakly coupled semilinear parabolic‐hyperbolic system with a degenerate and anisotropic diffusion. It arises to model the evolution of a chemical or biological tracer in a porous medium. We study the well‐posedness of the system using a L1 theory. Then, we establish the relaxation limit as the reaction constant becomes large. We prove that the system converges to a nonlinear parabolic‐hyperbolic equation that generalizes the Stefan problem. Two specificities of this paper are (i) to deal with ill‐prepared initial data and (ii) with unique entropy solutions based on a precise entropy inequality.

Keywords: transport and reaction in porous media, degenerate parabolic‐hyperbolic system, generalized Stefan problem

Journal: Asymptotic Analysis, vol. 34, no. 1, pp. 41-54, 2003