Communications in Mathematical Sciences

Volume 17 (2019)

Number 2

From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell–Stefan type

Pages: 507 – 538



Benjamin Anwasia (Centro de Matemática, Universidade do Minho, Braga, Portugal)

Patrícia Gonçalves (Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Ana Jacinta Soares (Centro de Matemática, Universidade do Minho, Braga, Portugal)


In this paper we perform a formal asymptotic analysis on a kinetic model for reactive mixtures in order to derive a reaction-diffusion system of Maxwell–Stefan type. More specifically, we start from the kinetic model of simple reacting spheres for a quaternary mixture of monatomic ideal gases that undergoes a reversible chemical reaction of bimolecular type. Then, we consider a scaling describing a physical situation in which mechanical collisions play a dominant role in the evolution process, while chemical reactions are slow, and compute explicitly the production terms associated with the concentration and momentum balance equations for each species in the reactive mixture. Finally, we prove that, under isothermal assumption, the limit equations for the scaled kinetic model is the reaction diffusion system of Maxwell–Stefan type.


Boltzmann-type equations, chemically reactive mixtures, diffusion limit, kinetic theory of gases, Maxwell–Stefan equations

2010 Mathematics Subject Classification

35Q20, 76P05, 80A32, 82C40

B.A. and A.J.S. thank Centro de Matemática da Universidade do Minho, Portugal, and the FCT/Portugal Project UID/MAT/00013/2013. B.A. thanks the FCT/Portugal for the support through the PhD grant PD/BD/128188/2016. P.G. thanks FCT/Portugal for the support through the project UID/MAT/04459/2013 and the French Ministry of Education through the grant ANR (EDNHS). The authors thank the Program Pessoa of Cooperation between Portugal and France with reference 406/4/4/2017/S.

Received 29 June 2018

Received revised 27 December 2018

Accepted 27 December 2018

Published 8 July 2019