Majda and ZND models for detonation: nonlinear stability vs. formation of singularities (Q6598483)

This paper investigates the limits of damping estimates by comparing two closely related combustion models: the Majda model and the ZND model. The focus is on the behavior of perturbations to discontinuous waves. First, it is demonstrated that singularities develop in the unweighted Lipschitz norm on both sides of the shock for both models. This result extends the classical work of \textit{F. John} [Commun. Pure Appl. Math. 27, 377--405 (1974; Zbl 0302.35064)] and \textit{T.-P. Liu} [J. Differ. Equations 33, 92--111 (1979; Zbl 0379.35048)] to variable coefficient systems by adapting John's approach to perturbations of nonconstant waves rather than constant states.\N\NSecond, the study reveals a contrast in stability between the models in exponentially weighted Sobolev spaces: the ZND model exhibits instability, while the Majda model remains stable in similar weighted settings. This analysis involves deriving high-order energy estimates that account for convective effects and the partial decay induced by a damping term, with careful attention to boundary contributions. The findings highlight that the convective effects are the primary source of instability in the weighted norms for the ZND model.