Abstract
We consider the Riemann problem for a Chapman–Jouguet combustion model which comes from Majda's model with a modified, bump-type ignition function proposed in [G. Lyng and K. Zumbrun, Arch. Rational Mech. Anal. 173 (2004) 213–277; Physica D 194 (2004) 1–29]. The unique Riemann solutions are obtained constructively under the pointwise and global entropy conditions. Furthermore, we prove rigorously that these solutions are the limits of the Riemann solutions for the corresponding self-similar Zeldovich–von Neumann–Döring model as the reaction rate goes to infinity. Finally we analyze the ignition problem for this Chapman–Jouguet combustion model, and the solutions show that the unburnt state is stable (respectively unstable) when the binding energy is small (respectively large), which is the desired property for a combustion model. We can also observe the phenomenon of the transition from a weak deflagration to a strong detonation which cannot occur for the Chapman–Jouguet combustion model corresponding to Majda's model with a step-type ignition function.
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