Existence of Chapman--Jouguet Detonation and Deflagration Waves

Abstract

We study the existence of profiles for Chapman--Jouguet detonation and deflagration waves in the Navier--Stokes equations for a reacting gas. In the limit of small viscosity, heat conductivity, and diffusion, the profiles correspond to heteroclinic orbits of a system of singularly perturbed ordinary differential equations. The burned end state of the waves, however, is a nonhyperbolic equilibrium of the associated, purely gas dynamic layer problem, and hence standard methods from geometric singular perturbation theory fail. We show how to resolve this degeneracy by combining a center manifold reduction with the blow-up method. The main result is the existence of viscous profiles for various types of Chapman--Jouguet processes. In addition, we obtain results on the spatial decay rates of these waves which are expected to be relevant for the stability analysis of the waves.