This paper proves that in three spatial dimensions, any smooth, sufficiently small disturbance of a non‑vacuum constant fluid state will develop a shock in finite time. The result requires only that the initial perturbation is compactly supported (zero outside a ball) and irrotational (no initial vorticity). No symmetry or extra compression assumptions are needed. The authors also show that the shock appears at the edge of the largest region where the standard initial‑value problem makes sense (the maximal Cauchy development). In the small‑data limit, the time when the shock appears matches the time predicted earlier by a radiation‑field analysis.

The work studies the compressible Euler equations for a perfect barotropic fluid, taking the pressure as p(ρ)=Aρ^γ with γ>1. By assuming the flow is irrotational, the fluid velocity can be written as the gradient of a potential ϕ. The evolution then reduces to a single quasilinear wave equation for ϕ whose coefficients depend on the first derivatives of ϕ. The authors use a geometric viewpoint that describes sound waves by an “acoustic metric” — a way of tracking how sound travels in the moving fluid — and they analyse how the geometry of these sound rays changes over time.

At a high level the mechanism of breakdown is the collapse of characteristic surfaces, that is, neighboring sound rays that start apart are driven together by the dynamics. This collapse makes second derivatives of the solution blow up while the solution and its first derivatives remain bounded. The paper gives a detailed geometric account of this behaviour and proves that it must occur for all small, compactly supported, irrotational perturbations of a constant non‑vacuum background state. The authors also relate their rigorous blowup time to earlier lower‑bound results (by John and Hörmander) coming from radiation‑field calculations.