Communications in Mathematical Sciences

Volume 20 (2022)

Number 3

Reaction-diffusion equations derived from kinetic models and their Turing instability

Pages: 763 – 801



Marzia Bisi (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Italy)

Romina Travaglini (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Italy)


We consider a binary mixture composed by a polyatomic (diatomic) and a monatomic gas, diffusing in a gaseous background (typically, the atmosphere), and undergoing reversible and irreversible chemical reactions. We show the derivation of proper reaction-diffusion equations for the number densities of the constituents, starting from suitably rescaled kinetic Boltzmann equations. The dominant process is assumed to be the elastic scattering with the host medium, while we present two different scalings for the various chemical reactions: the first option leads to a system of three reaction-diffusion equations, while the second regime leads to two reaction-diffusion equations similar to the classical Brusselator system. Then, we study the Turing instability properties of such macroscopic systems, showing their dependence on particle masses, on collision frequencies of the Boltzmann operators, and, above all, on particle internal energies.


kinetic equations, diffusive limits, reaction-diffusion equations, Turing instability

2010 Mathematics Subject Classification

35K57, 76P05, 82C40

Received 9 April 2021

Received revised 23 August 2021

Accepted 5 September 2021

Published 21 March 2022