From reactive Boltzmann equations to reaction-diffusion systems

We consider the reactive Boltzmann equations for a mixture of different species of molecules, including a fixed background. We propose a scaling in which the collisions involving this background are predominant, while the inelastic (reactive) binary collisions are very rare. We show that, at the formal level, the solutions of the Boltzmann equations converge toward the solutions of a reaction-diffusion system. The coefficients of this system can be expressed in terms of the cross sections of the Boltzmann kernels. We discuss various possible physical settings (gases having internal energy, presence of a boundary, etc.), and present one rigorous mathematical proof in a simplified situation (for which the existence of strong solutions to the Boltzmann equation is known).