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We establish the existence of radial self-similar Euler flows in which a continuous incoming wave generates a blowup of primary (undifferentiated) flow variables. A key point is that the solutions have a strictly positive pressure field, in contrast to Guderley's classic construction of converging shock waves. In Guderley's solutions, a converging shock invades a quiescent region at zero pressure (due to vanishing temperature), and the velocity and pressure in its immediate wake become unbounded at the time of collapse. 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. Our results demonstrate that the geometric mechanism of wave focusing is sufficiently strong on its own to drive unbounded growth. We propagate the solution beyond blowup and observe numerically that there are two distinct possibilities depending on the incoming flow: either an expanding spherical shock wave is generated, or the flow propagates in a continuous manner. Focusing on the former case, we show that the resulting flows define global admissible weak solutions to the full, multi-d compressible Euler system. These solutions have the unusual property that the flow is isentropic in each of the two regions separated by the shock. [ABSTRACT FROM AUTHOR]

Radial similarity flow offers a rare instance where concrete inviscid, multi-dimensional, compressible flows can be studied in detail. In particular, there are flows of this type that exhibit imploding shocks and cavities. In such flows, the primary flow variables (density, velocity, pressure, and temperature) become unbounded at the time of collapse. In both cases, the solution can be propagated beyond collapse by having an expanding shock wave reflect off the center of motion. These types of flows are of relevance in bomb-making and inertial confinement fusion and also as benchmarks for computational codes; they have been investigated extensively in the applied literature. However, despite their obvious theoretical interest as examples of unbounded solutions to the multi-dimensional Euler system, the existing literature does not address to what extent such solutions are bona fide weak solutions. In this work, we review the construction of globally defined radial similarity shock and cavity flows and give a detailed description of their behavior following collapse. We then prove that similarity shock solutions provide genuine weak solutions, of unbounded amplitude, to the multi-dimensional Euler system. However, both types of similarity flows involve regions of vanishing pressure prior to collapse (due to vanishing temperature and vacuum, respectively)—raising the possibility that Euler flows may remain bounded in the absence of such regions. [ABSTRACT FROM AUTHOR]