Self-Similar Shock Diffraction by Wedges

Definitions and mathematical statement

Fix arbitrary parameters $\gamma>1$ and $\rho_0>0$. Let $\Lambda\subset{\mathbb{R}}^2$ be the domain described in the supplied record as the upper half-plane outside a wedge. The wedge angle, orientation, vertex, and precise domain are not specified. Write $\xi=(\xi_1,\xi_2)$ for the self-similar spatial variables, and let $D$ and $\operatorname{div}$ denote differentiation with respect to $\xi$. Let $\Gamma_{\mathrm w}$ denote the wedge portion of $\partial\Lambda$, and let $\nu$ be a unit normal there wherever defined.

For a real-valued potential $\phi$, set \[ H_\phi=\rho_0-(\gamma-1)\left(\phi+\frac12|D\phi|^2\right), \qquad \rho_\phi=\rho(|D\phi|^2,\phi)=H_\phi^{1/(\gamma-1)}. \] The physical, real, nonnegative-density interpretation requires $H_\phi\geq0$ wherever this expression is used. Whether the intended problem requires strictly positive density or permits vacuum is not specified.

Requested existence problem. Seek a self-similar weak solution $\phi$ on $\Lambda$ satisfying \[ \operatorname{div}(\rho_\phi D\phi)+2\rho_\phi=0 \quad\text{in }\Lambda, \qquad D\phi\cdot\nu=0 \quad\text{on }\Gamma_{\mathrm w}, \] and approaching at infinity the two incident constant-state potentials separated by the incident shock. Those potentials, their parameters, the incident shock, and the meaning of this far-field convergence are not specified in the supplied record.

At the level of the interior equation alone, its distributional interpretation is \[ \int_\Lambda\left(\rho_\phi D\phi\cdot D\eta-2\rho_\phi\eta\right)\,{\,\mathrm{d}}\xi=0 \qquad\text{for every }\eta\in C_c^\infty(\Lambda), \] provided $\rho_\phi$ and $\rho_\phi D\phi$ are locally integrable. This identity does not specify the intended solution regularity, boundary traces, or shock admissibility. The record also does not state whether a boundary condition is required on any remaining portion of the boundary of the upper half-plane. Consequently, the supplied formulation describes an existence problem but does not determine a fully specified boundary-value problem or its admissible parameter range.

Short English statement

For given $\gamma>1$ and $\rho_0>0$, seek a self-similar weak potential-flow solution outside a wedge in the upper half-plane. It must satisfy the stated density law and equation, obey slip on the wedge, and approach two incident constant states separated by a shock at infinity. The wedge geometry, incident data, admissibility conditions, and precise far-field requirement remain unspecified and require mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the additional-record registry; supplied permanent ID $30001121$, exact source problem code catalogue entry, title Self-Similar Shock Diffraction by Wedges. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2008$. No individual proposer is named.
• Supplied report citation: Hyperbolic Conservation Laws (2009), Oberwolfach Reports, pp. 3139--3200, DOI 10.4171/owr/2008/56. Source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2008/56. The report is described as documenting workshop discussions and open problems. The supplied year label $2009$, proposed year $2008$, and DOI containing $2008$ are retained without resolving their bibliographic relationship.
• Supplied literature-source link labeled "Chen--Feldman shock diffraction": https://doi.org/10.4007/annals.2010.171.1067. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem partially solved and reports global self-similar shock-diffraction solutions for potential flow for large-angle wedges, including existence and stability, while describing the full parameter range and general formulation as not completely settled. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note says that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This is a supplied verification assertion, not an independent verification performed for this draft.