Cross-Diffusion Limit for a Reaction-Diffusion System with Fast Reversible Reaction

Dieter Bothe (1) , Michel Pierre (2) , Guillaume Rolland (2)

1 Mathematical Modeling and Analysis
2 IRMAR - Institut de Recherche Mathématique de Rennes

We consider a reaction-diffusion system which models a fast reversible reaction of type $C_1 + C_2 = C_3$ between mobile reactants inside an isolated vessel. Assuming mass action kinetics, we study the limit when the reaction speed tends to infinity in case of unequal diffusion coefficients and prove convergence of a subsequence of solutions to a weak solution of an appropriate limiting pde-system, where the limiting problem turns out to be of cross-diffusion type. The proof combines the $L^2$-approach to reaction-diffusion systems having at most quadratic reaction terms with a thorough exploitation of the entropy functional for mass action systems. The limiting cross-diffusion system has unique local strong solutions for sufficiently regular initial data, while uniqueness of weak solutions is in general open but is shown to be valid under restrictions on the diffusivities.

Mots clés: instantaneous limit; pseudo-steady-state approximation; QSSA; reaction diffusion; weak global solution; Cross-diffusion; fast reversible reaction

Domaines: Equations aux dérivées partielles [math.AP]

Dieter Bothe, Michel Pierre, Guillaume Rolland. Cross-Diffusion Limit for a Reaction-Diffusion System with Fast Reversible Reaction. Communications in Partial Differential Equations, 2012, 37 (11), pp.1940-1966. ⟨10.1080/03605302.2012.715706⟩. ⟨hal-00906640⟩