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New Self-similar Euler Flows: gradient catastrophe without shock formation

We consider self-similar solutions to the full compressible Euler system for an ideal gas in two and three space dimensions. The system admits a 2-parameter family of similarity solutions depending on parameters $λ$ and $κ$. Requiring locally finite amounts of mass, momentum, and energy imply certain constraints on $λ$ and $κ$. Further constraints are imposed for particular types of flows. E.g., Guderley's pioneering construction of an unbounded converging shock wave invading a quiescent fluid, requires $κ=0$ and $λ>1$. In this work we analyze the regime $0<λ<1$, which does not appear to have been addressed previously. Our findings include: (i) non-existence of Guderley shock solutions; (ii) existence of bounded and continuous incoming similarity flows in 3-d provided $κ$ takes the value $\hatκ=\frac{2(1-λ)}{γ-1}$, $λ$ is sufficiently small, and $γ$ is sufficiently large; (iii) continuation of the latter flows beyond collapse as globally defined and continuous similarity solutions. A key feature of these solutions is that they, in contrast to Guderley solutions, remain bounded at time of collapse, while the density, velocity, and sound speed all suffer gradient blowup. It is noteworthy that, notwithstanding infinite gradients at collapse, no shock wave appears. The analysis is based on a combination of analytical and numerical calculations.