Majda and ZND Models for Detonation: Nonlinear Stability vs. Formation of Singularities

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6137-6191, October 2024.
Abstract. In this paper we explore the boundaries of damping estimates by comparing and contrasting two closely related models of combustion, the Majda and ZND models. We are especially concerned with studying the behavior of perturbations of discontinuous waves. On the one hand, we show that singularities form in the unweighted Lipschitz norm on both sides of the shock for both models, extending classical results of John in [Comm. Pure Appl. Math., 27 (1974), pp. 377–405] and Liu in [J. Differential Equations, 33 (1979), pp. 92–111] to suitable variable coefficient systems This involves adapting John’s argument to perturbations of nonconstant waves instead of perturbations of constants. On the other hand, we show instability in exponentially weighted Sobolev spaces for ZND and stability for the Majda model in similarly weighted spaces. This involves proving high order energy estimates, using convective effects and the partial decay coming from a damping term, while being careful with the boundary terms. We note that the convective effects are the origin of the instability in the weighted norm in the ZND model.