Global shocks for steady supersonic flow past a curved wedge: the full Euler system, large nondecaying wedge perturbations, and the detached regime

Yin (Acta Math. Sin. 2006) proved that for two-dimensional steady supersonic potential (isentropic, irrotational) flow past a curved wedge that is a small, decaying perturbation of a straight wedge, with vertex angle strictly below a critical value and the downstream state on the weak-shock branch, an attached shock exists globally. Every subsequent classical (piecewise-smooth shock-front) existence theorem for a curved wedge remains inside the potential-flow reduction and inside a smallness, degenerate-background, or hypersonic (large-Mach) regime, while the full non-isentropic Euler system is known only in the small-total-variation entropy-solution class. The open core is therefore: a global shock theory for steady supersonic flow past a curved wedge in the full Euler system without smallness or decay assumptions, together with the detached (bow-shock) regime above the critical angle, and the three-dimensional extensions.

Background

When a uniform stream of an inviscid compressible polytropic gas with speed $q_0$ and sound speed $c_0$, hence Mach number $M_\infty=q_0/c_0>1$, meets a rigid wedge, the steady (time-independent) flow in the plane is governed by the full Euler system for density $\rho$, velocity $(u,v)$, pressure $p=(\gamma-1)\rho e$ (with internal energy $e$ and adiabatic exponent $\gamma$), and specific total energy $E=e+\tfrac12(u^2+v^2)$:

$$\begin{aligned} &\partial_x(\rho u)+\partial_y(\rho v)=0,\\ &\partial_x(\rho u^2+p)+\partial_y(\rho uv)=0,\\ &\partial_x(\rho uv)+\partial_y(\rho v^2+p)=0,\\ &\partial_x\big(\rho u E+up\big)+\partial_y\big(\rho v E+vp\big)=0. \end{aligned}$$

Across a shock front the Rankine–Hugoniot jump conditions (the integrated conservation laws on a discontinuity) hold, together with an entropy inequality selecting physical shocks. For a straight wedge of half-angle $\theta$, the shock polar — the curve of downstream states compatible with the jump conditions — admits two oblique-shock branches: a weak branch, behind which the flow remains supersonic, and a strong branch, behind which it is subsonic (transonic). An attached shock exists at the vertex only for $\theta$ below a critical (detachment) angle; above it, experiments and classical theory predict a detached bow shock enclosing a subsonic pocket (Courant and Friedrichs). Schaeffer (1976) first treated a nearly straight (slightly perturbed) wedge; Chen, Xin and Yin (2002) proved global shock existence past a perturbed cone. Yin (2006) considered the isentropic irrotational (potential) reduction, in which a constant Bernoulli function (the flow-invariant total of enthalpy and kinetic energy) and vanishing vorticity reduce the steady Euler system to a $2\times2$ first-order hyperbolic system for $(u,v)$. He proved: if the wedge profile $y=b(x)$, $b(0)=0$, is a small global $C^2$ perturbation of a straight wedge with quantitative decay at infinity ($|(1+x)^k\partial_x^k(b(x)-b_0x)|\le\varepsilon_0$ for $0\le k\le2$), the vertex angle is strictly below a critical value, and the downstream state is the weak-shock state, then an attached shock exists globally, with continuous flow between shock and wedge. The open boundary of the classical theory consists of: (i) large or nondecaying perturbations of the wedge profile; (ii) angles strictly above the critical angle, where the shock detaches, and the strong (non-weak) shock branch; (iii) the extension from the potential reduction to the full non-isentropic or rotational steady Euler equations; (iv) parameter regimes beyond the weak-shock regime used in existing proofs; and (v) three-dimensional wings and finite-length wedges. Later work partialized these directions but closed none of them. For Lipschitz wedges of small total variation — no decay at infinity required — Chen, Zhang and Zhu (2006) and Chen and Li (2008) established existence and stability of global entropy solutions of the full (nonpotential) Euler system including a strong shock front, via Glimm-type schemes (random-choice difference schemes for hyperbolic conservation laws); Chen, Pu and Zhang (2025) extended this BV framework to curved wedges with strong shock fronts in an inverse-problem setting. These are entropy solutions of bounded variation (weak solutions satisfying the entropy inequality), not classical piecewise-smooth shock fronts. Within the classical potential framework, Hu (2013) re-established global existence under smallness in a weighted Sobolev norm; Hu (2018) reached boundaries of large variation by perturbing a degenerate limit-speed background solution; Hu and Qu (2025) removed smallness of the wedge height entirely, but only in the hypersonic regime where $M_\infty$ exceeds a large threshold. The strong transonic branch at straight sharp wedges was stabilized by Chen and Fang (2007) and Yin and Zhou (2009). For the detached regime, Bae and Xiang (2019, published 2023) constructed a detached shock past a convex rounded body, but for potential flow with sufficiently large Mach number and strong-shock far field. Three-dimensional results exist for cones (Chen–Xin–Yin 2002; Xin–Yin 2006; Li–Witt–Yin 2014; Hu–Zhang 2019) and for a concave cornered wedge with a jet (Qu–Xiang 2018); local theory for three-dimensional wings dates to Chen (1992).

Problem Statement

Let a uniform stream of a polytropic gas with incoming state $(\rho_0,u_0,0)$, $u_0>c_0$, hit an infinite curved wedge $y=b(x)$, $b(0)=0$, $b\in C^2$, and let the flow be governed by the two-dimensional steady full Euler system displayed in the Background, with the slip (tangency) condition on the wedge and Rankine–Hugoniot and entropy conditions on any shock front. Resolve the restrictions of the classical attached-shock theory of Yin (2006), which together form one objective, a global shock theory for steady supersonic flow past non-flat bodies in the physical full model:

(a) Attached regime, full system, no smallness: for each fixed compact interval of subcritical slopes, with upper endpoint strictly below the sonic slope of the incoming state (the largest slope for which the corresponding straight-wedge weak oblique shock leaves the downstream flow supersonic), and for each bound $A>0$ with no smallness restriction on $A$ (constants and estimates may depend on the interval and on $A$), consider wedge profiles $b\in C^2$ with $b(0)=0$ whose slope $b'(x)$ lies for all $x\ge0$ in that interval and such that $b-b_0x$ is a bounded perturbation with $\|b-b_0x\|_\infty\le A$ — without any smallness assumption and without any decay assumption at infinity. Prove or disprove that there exists a global piecewise-$C^1$ solution consisting of an attached shock front issuing from the wedge vertex together with a supersonic flow between shock and wedge. (When existence holds and $b'(x)$ has a limit as $x\to\infty$, the asymptotic behavior of the shock — for example, whether it converges to the self-similar straight-wedge shock — is a natural further question, but it is not part of the goal stated here.) A refutation must exhibit a specific admissible profile and incoming state together with a rigorous proof that no such global attached-shock solution exists (for example, because gradient catastrophe or detachment is unavoidable).

(b) Detached regime: establish existence and structure (shock curve, subsonic pocket, and sonic line — the curve where the flow speed equals the sound speed, bounding the subsonic pocket behind the shock) of a detached bow-shock solution of the steady full Euler system for general Mach number $M_\infty>1$, in at least one of the two geometry classes that admit no attached oblique shock: (i) rounded blunt bodies obtained by smoothing out the tip of a wedge, and (ii) wedges whose vertex angle is strictly above the critical (detachment) angle, that is, strictly above the largest angle for which the straight-wedge shock polar still admits an attached oblique shock. This goes beyond the currently known potential-flow constructions, which require sufficiently large $M_\infty$ and a convex body. Since weak and strong far-field states can both satisfy the entropy condition, any uniqueness statement is sought only within an explicitly specified far-field branch or solution class.

Parts (a) and (b) are the two components of the objective. The multidimensional extension — global shock fronts for three-dimensional wings and finite-length wedges with general end conditions — is a further, secondary open direction, logically downstream of (a)–(b).

References

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