Two Kinds Of Typical Initial Value Problems For The Simplest Chapman-Jouguet Combustion Model

This paper is concerned with two typical kinds of initial value problem for the simplest Chapman-Jouguet combustion model: the ignition problem and the generalized Riemann problem.By the study of ignition problem, the simplest Chapman-Jouguet model is able to exihibit instability for combustible gas due to changing of combustion waves. Under the assumption that the flux function f(u) is strictly convex, this problem was studied in [39]. However, the well-known phenomenon in combution theory that is the transition from deflagration to detonaion, cannot occur in [39]. It is because that detonation and deflagration waves cannot propagate in the same direction when f(u) is convex. In chapter 3, the ignition problem for the scalar Chapman-Jouguet combustion model without convexity is considered. With the method of characteristic analysis, we constructively obtain the existence and uniqueness of the solution under the entropy conditions in [44]. We can observe the ignition and termination of combustion waves, which shows the combustible gas is stable (unstable) when the binding energy is small (large). In addition, the transitions between deflagration and detonation appears.In chapter 4, the generalized Riemann problem for scalar convex and nonconvex Chapman-Jouguet combustion model are studied, respectively. Under the entropy conditions in [44], we obtain the perturbed Riemann solution constructively in a neighborhood of the origin (t > 0) on the (x, t) plane. Compared with the corresponding Riemann solution, it can be found that for most of the cases, the combustion waves in the corresponding Riemann solutions are able to retain their forms after perturbation, in the neighborhood of the origin. While for some other cases, the perturbation brings essential changes to the combustion waves. For instance, the perturbation may transform a strong detonation into a weak deflagration followed by a shock wave. This interesting phenomenon also appears in the numerical solutions in [58]. Our results give a reasonable explanation to this phenomenon.