The regular-versus-Mach reflection transition criterion for self-similar shock reflection: rigorous theory beyond the potential-flow counterexamples (full Euler constructions, validity of the weak-type/angle criterion, and dynamic stability)

When a plane shock reflects from a wedge or wall, the physically realized inviscid flow is either a regular reflection (RR), in which incident and reflected shocks meet at a point on the wall, or a Mach reflection (MR), in which they meet a third shock (Mach stem) at a triple point off the wall. Predicting precisely which pattern occurs for given Mach number, wedge angle, and incident-shock angle has been open since Ernst Mach and von Neumann. Elling (2009) refuted the sonic criterion in polytropic potential flow by proving structural stability of weak-type transonic RR, and his subsequent work refuted the detachment criterion and proposed a new criterion ('RR if and only if local RR exists and the angle condition holds') that remains a conjecture verified only in patches. The objective is the rigorous completion of that program: prove or disprove the proposed transition criterion across the full parameter space including the classical head-on configuration; construct global self-similar RR solutions for the full compressible Euler system, where vorticity and entropy destroy the potential-flow reduction, and determine which transition criteria apply there; and settle the dynamic (time-dependent) stability of weak- versus strong-type regular reflection, where linear analyses and numerics conflict.

Background

Shock reflection is a canonical problem of inviscid compressible gas dynamics. In regular reflection (RR) an incident shock meets a wall at a reflection point and continues as a single reflected shock; in Mach reflection (MR) the two shocks are connected away from the wall by a third, near-normal shock, the Mach stem, with a contact discontinuity (slip line) emanating from the triple point. For a wedge hit by a plane shock, the flow behind the reflection pattern is self-similar (also called pseudo-steady): it depends only on the scaled variables $\xi=x/t$, $\eta=y/t$, because the data contain no length or time scale. Two inviscid models are in play: polytropic potential flow — an ideal gas with equation of state $p=\kappa\rho^\gamma$ and constant adiabatic exponent $\gamma$, in which the flow is irrotational (the velocity is the gradient of a potential) so that the self-similar equations reduce to a single second-order equation of mixed elliptic–hyperbolic type (elliptic where the pseudo-Mach number $L=|v-x/t|/c<1$) — and the full compressible Euler system, in which vorticity and entropy are independent unknowns. Locally at the reflection point, the two-shock configuration is determined by algebraic Rankine–Hugoniot jump conditions (the statements of conservation of mass, momentum, and energy across a discontinuity); the set of downstream states reachable from a fixed upstream state across a shock forms the shock polar. For a range of deflection angles the polar gives two admissible reflected shocks: the weak-type and the strong-type (the strong-type produces a larger pressure jump and, in the transonic case, subsonic downstream flow). A local RR (two shocks meeting on the wall, satisfying the wall slip condition) may or may not extend to a global solution satisfying all far-field conditions. Three classical proposals — the detachment criterion (global RR appears generically whenever local RR is possible), the sonic criterion (global RR appears only when the local RR is supersonic behind the reflection point, the idea being that perturbations from the wall could otherwise travel back to the reflection point), and the von Neumann mechanical-equilibrium criterion (formulated through coexistence of RR with MR and meaningful only where triple configurations exist) — have been studied experimentally, numerically, and theoretically for decades without a clear result; in potential flow the von Neumann criterion does not apply at all. Elling (Arch. Ration. Mech. Anal. 194 (2009) 987–1010) proved that the sonic criterion is not universally correct: for polytropic potential flow with incident-shock angle $\alpha<90^\circ$ and wall angle $\theta>90^\circ$, there is a nonempty open set of parameters admitting global RR with a transonic reflected shock. The mechanism is that for weak-type transonic reflections, downstream perturbations do reach the reflection point but decay in space, because gradient regularity near the reflection point is a local property of the elliptic corner problem. In a companion paper (SIAM J. Appl. Math. 70 (2009) 1330–1340) Elling also refuted the detachment criterion in the same regime and proved structural instability of strong-type RR, and in J. Differential Equations 252 (2012) 2085–2103 he proved non-existence of global strong-type analogues in a natural class. He then proposed a new transition criterion: the global flow is RR if and only if local RR exists and a certain angle condition holds (the strong-type reflected-shock tangent meets the opposite wall at a sharp or right downstream side angle). This criterion is a conjecture: it is verified near trivial RR configurations (explicit exact solutions in which both shocks are straight and the flow is constant in each sector, occurring when the wall is perpendicular to the reflection-point tangent of the strong-type reflected shock), along parts of the corresponding weak-type branch, and for large-angle wedges via Chen–Feldman's global construction (Ann. of Math. 171 (2010) 1067–1182), but neither direction is known in the full parameter space. The situation is delicate because the natural class of possible competitors includes global Mach reflections whose fine structure is itself not rigorously understood: the classical von Neumann triple-point paradox (three shocks alone cannot satisfy the jump conditions for weak shocks) is resolved numerically by Guderley-reflection sequences of shocks and contacts (Tesdall–Sanders–Keyfitz, SIAM J. Appl. Math. 68 (2008)), but no rigorous construction of such configurations exists. Moreover, essentially all rigorous results concern potential flow; for the full Euler system even the global existence of self-similar regular reflection is open, and the relationship between the time-dependent (dynamic) stability problem and the steady/self-similar (structural) stability treated by Elling remains unresolved — linear analyses suggest weak-type RR is stable and strong-type unstable, while Elling's own numerical observations indicate both types are dynamically stable.

Problem Statement

Overarching question (Elling's own formulation of the long-standing problem): predict precisely for which parameters — incident-shock Mach number $M_1$, incident-shock angle $\alpha$, wall/wedge angle $\theta$, and adiabatic exponent $\gamma$ — the physically admissible global solution of the self-similar reflection problem is a regular reflection rather than a Mach reflection, and prove the correct transition criterion rigorously. After the sonic and detachment counterexamples in polytropic potential flow and Elling's proposed replacement criterion ('the global flow is RR if and only if local RR exists and the angle condition is satisfied'), the surviving open core has three linked components. (i) Potential flow, full parameter space: prove or disprove the proposed criterion, in both directions — global existence of RR wherever local RR and the angle condition hold, including the classical head-on configuration $\alpha=90^\circ$ with $\theta$ near and below the sonic angle $\theta_s$ (the wall angle below which the weak-type local reflected shock becomes transonic) where a large-data (non-perturbative) analysis is required, and non-existence of global RR in the complementary regimes, which demands ruling out all Mach-reflection alternatives whose fine structure (von Neumann triple-point paradox, Guderley reflection) is itself not rigorously constructed; establish uniqueness/stability of the global configurations. (ii) Full Euler system: construct global self-similar regular-reflection solutions for the isentropic and non-isentropic compressible Euler equations, where vorticity and entropy variations destroy the irrotational reduction, and determine which of the classical criteria — detachment, sonic, or von Neumann — if any, correctly describes the transition in Euler flow. (iii) Dynamic stability: determine whether weak-type and strong-type regular reflections are stable or unstable as solutions of the genuinely time-dependent problem, resolving the conflict between linearized analyses (weak stable, strong unstable) and numerical calculations suggesting both are dynamically stable, and thereby clarifying the precise role of stability in the selection mechanism underlying the transition criterion. These three components are conjunctive parts of the single objective of establishing the rigorous transition criterion for shock reflection: the objective is resolved only when the correct transition criterion is established — component (i) is completed either by a rigorous proof of the proposed criterion (both of its implications settled) or by a refutation of it together with the correct replacement criterion, equivalently the precise RR-versus-MR parameter partition the overarching question demands. A bare counterexample to the proposed criterion, though decisive about that conjecture, is not by itself a resolution; likewise, resolving a single component, only one implication in (i), or the nested uniqueness/stability clause is a partial result. The detachment criterion in polytropic potential flow is settled by Elling's 2009 counterexamples and lies outside the scope of this task.

References

1. Volker Elling, Counterexamples to the sonic criterion, Archive for Rational Mechanics and Analysis 194 (2009) 987–1010, DOI 10.1007/s00205-008-0196-3, arXiv:0810.2137, https://arxiv.org/abs/0810.2137
2. Volker Elling, Instability of Strong Regular Reflection and Counterexamples to the Detachment Criterion, SIAM Journal on Applied Mathematics 70 (2009) 1330–1340, DOI 10.1137/080724769, arXiv:0908.0106, https://arxiv.org/abs/0908.0106
3. Volker Elling, Non-existence of strong regular reflections in self-similar potential flow, Journal of Differential Equations 252 (2012) 2085–2103, DOI 10.1016/j.jde.2011.10.017, arXiv:1104.0330, https://arxiv.org/abs/1104.0330
4. Gui-Qiang Chen and Mikhail Feldman, Global solutions of shock reflection by large-angle wedges for potential flow, Annals of Mathematics (2) 171 (2010) 1067–1182, DOI 10.4007/annals.2010.171.1067, arXiv:0708.2540, https://arxiv.org/abs/0708.2540
5. Volker Elling and Joseph Roberts, Steady and Self-Similar Inviscid Flow, SIAM Journal on Mathematical Analysis 44 (2012) 2344–2371, DOI 10.1137/110829465, arXiv:1104.0331, https://arxiv.org/abs/1104.0331
6. Joseph Roberts, Steady and self-similar full Euler flow, Journal of Hyperbolic Differential Equations 10 (2013) 799–840, DOI 10.1142/S021989161350029X, arXiv:1211.3514, https://arxiv.org/abs/1211.3514
7. Volker W. Elling, Shock polars for ideal and non-ideal gases, Journal of Fluid Mechanics 916 (2021) A51, DOI 10.1017/jfm.2021.147, https://doi.org/10.1017/jfm.2021.147
8. Volker Elling, Barotropic Euler shock polars, Zeitschrift für angewandte Mathematik und Physik 73 (2022), article 67, DOI 10.1007/s00033-022-01685-0, https://doi.org/10.1007/s00033-022-01685-0
9. Allen M. Tesdall, Richard Sanders, and Barbara L. Keyfitz, Self-Similar Solutions for the Triple Point Paradox in Gasdynamics, SIAM Journal on Applied Mathematics 68 (2008) 1360–1377, DOI 10.1137/070698567, https://doi.org/10.1137/070698567
10. Gui-Qiang G. Chen, Mikhail Feldman, and Wei Xiang, Uniqueness and Stability for the Shock Reflection-Diffraction Problem for Potential Flow, arXiv:1904.00114 (2019), https://arxiv.org/abs/1904.00114
11. Volker Elling, Triple points and sign of circulation, Physics of Fluids 31 (2019) 126106, DOI 10.1063/1.5122679, arXiv:1910.10229, https://arxiv.org/abs/1910.10229