Shock formation after derivative blowup in two-dimensional quasilinear wave equations

Determine whether the characteristic surfaces of the two-dimensional quasilinear wave equation described in Remark Yin.5 can squeeze and consequently generate shocks after the solution’s second derivatives blow up while the solution and its first derivatives remain small.

Background

Remark Yin.5 discusses small-data solutions of a general two-dimensional quasilinear wave equation. When either of the first or second null conditions fails, prior results establish finite-time blowup of the second derivatives while the solution and its first derivatives remain small. The unresolved issue is whether this derivative blowup is accompanied by geometric squeezing of characteristic surfaces and the actual formation of shocks.

References

More precisely, at the blowup time, $|\phi|$ and $|\partial\phi|$ remain small but $|\partial{2}\phi|$ becomes infinite at the unique blowup point, while it is unknown whether the characteristic surfaces can squeeze and further the shocks are formed (see ).

— Shock formation for 3D steady supersonic flows with general short pulse data  (2608.24599 - Ding et al., 25 Aug 2026) in Remark Yin.5, Section 1