REACTION–DIFFUSION EQUATIONS DERIVED FROM KINETIC MODELS AND THEIR TURING INSTABILITY

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