Global existence of a weak solution for a reaction–diffusion system in a porous medium with membrane conditions and mass control

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Résumé

Abstract

In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of
m
species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in



$L^1$


. This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021,
Journal of Evolution Equations
21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form



$\partial _t v_i - \Delta v_i^{r_i} = R_i$


, for an exponent



$r_i < 2$


. The case



$r_i \geq 2$


remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021,
Journal of Evolution Equations
21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.

Mots-clés

global existence, weak solutions, reaction-diffusion systems, porous medium equation, mass control, nonlinear diffusion, degenerate diffusion, multiple species