Propagation and quenching in a reactive Burgers–Boussinesq system

Abstract

We investigate the qualitative behaviour of solutions of a Burgers–Boussinesq system—a reaction–diffusion equation coupled via gravity to a Burgers equation—by a combination of numerical, asymptotic and mathematical techniques. Numerical simulations suggest that when the gravity ρ is small the solutions decompose into a travelling wave and an accelerated shock wave moving in opposite directions. There exists ρcr1 so that, when ρ > ρcr1, this structure changes drastically, and the solutions become more complicated. The solutions are composed of three elementary pieces: a wave fan, a combustion travelling wave and an accelerating shock, the whole structure travelling in the same direction. There exists ρcr2 so that when ρ > ρcr2, the wave fan catches up with the accelerating shock wave and the solution is quenched, no matter how large the support of the initial temperature. We prove that the three building blocks (wave fans, combustion travelling waves and shocks) exist and we construct asymptotic solutions made up of these three elementary pieces. We finally prove, in a mathematically rigorous way, a quenching result irrespective of the size of the region where the temperature was above ignition—a major difference from what happens in advection–reaction–diffusion equations where an incompressible flow is imposed.