Radially Symmetric Non-isentropic Euler flows: continuous blowup with positive pressure

TL;DR

This paper demonstrates that radially symmetric, self-similar solutions to the full Euler equations can exhibit continuous blowup with positive pressure, and can be extended beyond blowup to form shock waves, challenging previous zero-pressure assumptions.

Contribution

It extends the understanding of blowup phenomena in Euler flows to include positive pressure cases for the full system, beyond simplified models.

Findings

Blowup solutions with positive pressure exist in the full Euler system.

Solutions can be continued beyond blowup, forming shock waves.

Flow remains isentropic in separated regions, satisfying weak solution criteria.

Abstract

Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its immediate wake become unbounded at time of collapse. However, these solutions are of border-line physicality: the pressure vanishes within the quiescent region due to vanishing temperature there. It is reasonable that the lack of upstream counter-pressure is conducive to large speeds, with concomitant large amplitudes. Based on Guderley's original solutions it is therefore unclear if it is the zero-pressure region that is responsible for blowup. The same applies to self-similar Euler flows describing radial cavity flow, first analyzed by Hunter (1960). Recent works have shown that the simplified isothermal and isentropic models admit continuous blowup solutions in the presence of a strictly positive pressure field. In this work we extend this conclusion to the case of the full Euler system. The solutions under consideration are radial self-similar flows in which a continuous wave focuses and blows up. We propagate the solutions beyond blowup and observe numerically that there are cases where an expanding spherical shock wave is generated at collapse. The resulting solution has the unusual property that the flow is isentropic in each of the two regions separated by the shock. We finally verify that these are admissible global weak solutions to the full, multi-d compressible Euler system.