Charakhch'yan (2009) computed self-similar solutions of the inviscid compressible Euler equations for a plane shock interacting with a wedge (iron plate with an aluminum-filled wedge; air with a rigid wedge), closed by the two-term equation of state, and found that the wave reflected from the wedge is a continuous compression wave whose spatial gradient decreases over time — a two-dimensional reversal of the one-dimensional gradient catastrophe. He conjectured that in real flows with viscosity and heat conduction an initial shock wave can be transformed over time into a continuous compression wave of the same amplitude. The problem asks whether, in an actual flow with dissipative processes such as viscosity and heat conduction, an initial shock wave in this two-dimensional shock-wedge setting is transformed over time into a continuous compression wave of the same amplitude — the conjectured reversal of the gradient catastrophe — or whether dissipation instead sustains a shock-like, entropy-producing structure.

Background

A Riemann problem for a system of conservation laws asks for the flow generated by piecewise-constant initial data. In one space dimension its solution is fully understood as a combination of shocks and centered rarefaction or compression fans; in two dimensions — for example when a plane shock wave strikes the inclined face of a wedge — no comparable theory exists, and such scale-free wedge-interaction flows are customarily called two-dimensional Riemann problems. Because the data contain no length or time scale, the flow is expected to be self-similar: all fields depend only on the variables ξ=x/t\xi=x/t and η=y/t\eta=y/t. The governing model is the inviscid compressible Euler system (conservation of mass, momentum, and energy of a gas or solid medium); for solids at high pressure Charakhch'yan closes it with the two-term equation of state p=ρε(γ−1)+c02(ρ−ρ0)p=\rho\varepsilon(\gamma-1)+c_0^{2}(\rho-\rho_0), where ρ\rho is density, ε\varepsilon specific internal energy, γ\gamma a Grüneisen-type exponent, and c0c_0 and ρ0\rho_0 the bulk sound speed and reference density of the material. In one dimension, a continuous compression wave — a smooth fan in which characteristics carrying higher pressure travel faster — inevitably steepens until characteristics intersect and a shock forms; this is the gradient catastrophe, and afterwards compression is discontinuous and entropy-increasing. A Lagrangian particle trajectory is the path followed by an individual material element; entropy is preserved along it across a smooth (continuous) compression but jumps across a shock. Charakhch'yan (2009) numerically obtained self-similar solutions for a plane shock in an iron plate reaching a wedge filled with aluminum (wedge angle 60∘60^{\circ}) and for a shock in air over a rigid wedge, and found the wave reflected from the wedge (or adjacent to the reflected shock) to be a two-dimensional continuous compression wave: pressure rises smoothly, particles crossing it stay isentropic, and — unlike in one dimension — the spatial gradient decreases with time because self-similar spreading in two dimensions dilutes gradients. This suggests a genuinely two-dimensional mechanism opposing gradient catastrophe, and motivated the conjecture that dissipation (viscosity, heat conduction) could convert an initial shock into such a continuous compression wave of the same amplitude. On the rigorous side, the mathematics of self-similar wedge problems is developed mainly for potential flow (the irrotational, isentropic reduction of the Euler equations): Chen and Feldman proved global existence of regular shock reflection by large-angle wedges for potential flow and built the general theory of shock reflection-diffraction, where the reflected wave is a shock, not a continuous compression. Global self-similar solutions have also been constructed for special corner-type two-dimensional Riemann problems with general or van der Waals equations of state. For the full Euler equations the theory is far less developed: Chen, Feldman, and Xiang proved that for two-dimensional Riemann problems with shocks under the isentropic Euler system the velocity need not belong to H1H^1 (the Sobolev space of square-integrable functions with square-integrable first derivatives) in subsonic regions, so even the regularity available to self-similar solutions is delicate; and Buckmaster, Shkoller, and Vicol established that smooth two-dimensional compressible flow generically forms shocks (the steepening mechanism).

Problem Statement

The problem, as left open by Charakhch'yan (2009), concerns the two-dimensional self-similar shock-wedge interaction (Riemann) problem in the source's configurations: a plane shock in an iron plate reaching a wedge filled with aluminum (wedge angle 60∘60^{\circ}), and a shock in air over a rigid wedge, with the solid medium described by the two-term equation of state p=ρε(γ−1)+c02(ρ−ρ0)p=\rho\varepsilon(\gamma-1)+c_0^{2}(\rho-\rho_0), in which the inviscid computations show the wave reflected from the wedge (or adjacent to the reflected shock) to be a two-dimensional continuous compression wave whose spatial gradient decreases over time due to the self-similarity of the solution. Determine whether the source's conjecture holds for an actual flow with dissipative processes such as viscosity and heat conduction: is an initial shock wave transformed over time into a continuous compression wave of the same amplitude — a phenomenon opposite to the one-dimensional gradient catastrophe, with entropy preserved along Lagrangian particle trajectories across the compression — or does dissipation instead sustain a shock-like, entropy-producing structure? A genuine continuous compression wave must be distinguished from a shock artificially smeared by numerical discretization.

References

  1. A. A. Charakhch'yan, "Continuous compression waves in the two-dimensional Riemann problem", Computational Mathematics and Mathematical Physics 49(10), 1774–1780 (2009). DOI: 10.1134/S096554250910011X, https://doi.org/10.1134/S096554250910011X.
  2. A. A. Charakhch'yan, K. V. Khishchenko, V. E. Fortov, A. A. Frolova, V. V. Milyavskiy, L. V. Shurshalov, "Shock compression of some porous media in conical targets: numerical study", Shock Waves 21(1), 35–42 (2011). DOI: 10.1007/s00193-010-0274-y, https://doi.org/10.1007/s00193-010-0274-y.
  3. Gui-Qiang Chen, Mikhail Feldman, "Global solutions of shock reflection by large-angle wedges for potential flow", Annals of Mathematics (2) 171(2), 1067–1182 (2010). DOI: 10.4007/annals.2010.171.1067, https://doi.org/10.4007/annals.2010.171.1067.
  4. Gui-Qiang Chen, Mikhail Feldman, "The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures", Princeton University Press (2018). DOI: 10.1515/9781400885435, https://doi.org/10.1515/9781400885435.
  5. Gui-Qiang G. Chen, Mikhail Feldman, Wei Xiang, "Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system", arXiv:2504.07034 (2025, revised 2026). https://arxiv.org/abs/2504.07034.
  6. Geng Lai, "Global solutions to a class of two-dimensional Riemann problems for the Euler equations with a general equation of state", Indiana University Mathematics Journal 68(5), 1409–1464 (2019). DOI: 10.1512/iumj.2019.68.7782, https://doi.org/10.1512/iumj.2019.68.7782.
  7. Shuangrong Li, Wancheng Sheng, "Two-dimensional Riemann problem of the Euler equations to the Van der Waals gas around a sharp corner", Studies in Applied Mathematics 152(2), 696–733 (2024). DOI: 10.1111/sapm.12658, https://doi.org/10.1111/sapm.12658.
  8. Tristan Buckmaster, Steve Shkoller, Vlad Vicol, "Formation of Shocks for 2D Isentropic Compressible Euler", Communications on Pure and Applied Mathematics 75(9), 2069–2120 (2020). DOI: 10.1002/cpa.21956, https://doi.org/10.1002/cpa.21956.