Global-in-time existence and wave-interaction structure of the generalized Riemann problem for the scalar Chapman–Jouguet combustion model

The generalized (perturbed) Riemann problem for the scalar Chapman–Jouguet combustion model — the infinite-reaction-rate limit of the scalar reduction of Majda's model of dynamic combustion — is currently solved only locally. For convex fluxes, Sun and Sheng (2009) constructed the perturbed solution in a neighborhood of the origin, showing that small perturbations of the initial data can change the combustion-wave type (a Chapman–Jouguet detonation becoming a strong detonation) or even trigger combustion where the unperturbed Riemann solution contains no combustion wave, an instability of unburnt states. The open problem is the corresponding global-in-time theory: construct the entropy solution for all $t>0$ for piecewise-smooth perturbed Riemann data, with a complete catalogue of wave interactions and type changes and the large-time behavior. Adjacent questions — continuous dependence on the perturbation, and justification of the entropy selection rule as the vanishing-reaction-rate limit of perturbed Zeldovich–von Neumann–Döring solutions — are discussed in the literature reviewed below but are recorded here as background context rather than as components of the target. Adjacent partial results exist — global weak uniqueness under a discontinuous-flux reduction (Ying 2003), global multi-D solutions for special data (Hu–Cao–Yang), and stability results for neighboring models — but the global constructive theory for perturbed Riemann data of the Chapman–Jouguet model itself is not found anywhere in the retrieved literature.

Background

The scalar combustion model — a one-equation reduction of Majda's qualitative model of dynamic combustion (Majda 1981), introduced and analyzed by Liu and Zhang (1991) — compresses reacting gas dynamics into two coupled one-dimensional equations (in the formulation used throughout this literature, as stated in Ying 2003): $\partial_t(u+qz)+\partial_x f(u)=0,\qquad \partial_t z=-K\varphi(u)z,$ where $u$ is a lumped variable standing for density, velocity, and temperature, $z\in[0,1]$ is the unburnt fraction of the gas, $q>0$ is the binding energy released by burning, $K$ is the reaction rate, and $\varphi$ is a step-type ignition function equal to 1 above the ignition temperature and 0 below; the flux $f$ satisfies $f'>0$ and strict convexity $f''\ge a_0>0$ unless explicitly stated otherwise. The model has been a standard testbed for Chorin's random-projection methodology, a numerical scheme in which the stiff reaction step is replaced by a random projection of the unburnt fraction $z$ onto the values 0 or 1. The Chapman–Jouguet (CJ) model is the infinite-reaction-rate limit $K\to\infty$: burning completes instantaneously, and the reaction zone collapses to a single discontinuity. In the Zeldovich–von Neumann–Döring (ZND) model, by contrast, $K$ is finite and the reaction zone has positive width; the CJ theory is the expected limit of the ZND theory as $K\to\infty$.

The classical (self-similar) Riemann problem — data constant on each side of the origin — was settled for the underlying gas-dynamics combustion system by Zhang and Zheng (1989), and for the scalar model itself by Liu and Zhang (1991). The admissible combustion waves, i.e. discontinuities separating burnt from unburnt gas, are the deflagrations (which travel subsonically relative to the unburnt gas) and the detonations (which travel supersonically), the latter subdivided into weak and strong detonations; the Chapman–Jouguet detonation is the strong detonation of minimal speed, whose state behind the wave lies exactly on the sonic locus (the characteristic speed of the burnt gas relative to the wave equals the wave speed). Entropy conditions select the physically relevant waves, and the CJ detonation plays the role of the unique admissible strong detonation in the standard Riemann solution. Liu and Zhang also observed that unburnt states are unstable when the binding energy is sufficiently large. Zhang and Zhang (1995) extended the classification to nonconvex fluxes, and Tan and Zhang (1992) treated the self-similar ZND model.

The generalized Riemann problem (GRP), introduced as a systematic object of study by LeFloch and Raviart (1988), replaces the two constant states by piecewise-smooth data agreeing with Riemann data to first order at the origin. The solution is then no longer self-similar, and the analysis — characteristic expansions, wave-front tracking through interactions — proceeds constructively near $t=0$. Sun and Sheng (2009) carried out this program for the scalar CJ model with convex flux: they constructed the perturbed solution in a neighborhood of the origin in the $(x,t)$ plane, classified the parametric cases through algebraic thresholds ($q^*$, $q^{**}$, $u^- - u^+$) and sign conditions on the data's derivatives, and used a global entropy condition — minimal propagation speed of the burnt–unburnt interface — to select among multiple pointwise-admissible solutions. Two of their findings drive the present problem: perturbation can change the combustion-wave type (a CJ detonation becoming a strong detonation), and perturbation can ignite combustion even when the unperturbed Riemann solution has no combustion wave at all, so that unburnt Riemann states are in a genuine sense structurally unstable. Liu and Sheng (2011) treat the analogous problem for gas-dynamic (system) combustion; Sheng, Sun and Zhang (2007) treat the nonconvex scalar CJ model, again locally near the origin.

For the global question, the relevant weak framework is Ying's (2003): under a "no cooling down" assumption on the flux, the $K=\infty$ problem reduces to a scalar conservation law for $v=u+qz$ with a flux that is discontinuous in $v$ at the ignition threshold, and Ying proved existence and uniqueness of entropy solutions for bounded-variation (BV) data constant at large $|x|$ — a class that includes perturbed Riemann data — and proved convergence of implicit and random-projection schemes. This gives global uniqueness at the weak level, but no wave-interaction structure and no counterpart for the bump-ignition variant (in which $\varphi$ is a bump-type function rather than a step, a variant proposed by Lyng and Zumbrun (2004)), for which the discontinuous-flux reduction is unavailable. The bump-ignition variant lies outside the step-ignition model to which the Problem Statement below is restricted and is mentioned only as background.

Problem Statement

For the scalar Chapman–Jouguet combustion model (the $K\to\infty$ limit of the scalar combustion model $\partial_t(u+qz)+\partial_x f(u)=0$, $\partial_t z=-K\varphi(u)z$ with strictly convex flux $f$ and step ignition function $\varphi$), construct the global-in-time theory of the generalized Riemann problem: for piecewise-smooth initial data agreeing with Riemann data to first order at the origin, construct the entropy solution — under the entropy conditions used by Sun and Sheng (2009) — for all $t>0$, and complete the wave-interaction catalogue begun locally in that paper, including all changes of combustion-wave type (CJ detonation to strong detonation, ignition of combustion in unburnt solutions, extinction of combustion) and the large-time asymptotic wave pattern.

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