On similarity flows for the compressible Euler system

TL;DR

This paper studies radial similarity flows in the compressible Euler system, demonstrating that certain unbounded solutions involving shocks and cavities are genuine weak solutions, with implications for theoretical understanding and computational benchmarks.

Contribution

It proves that similarity shock solutions are valid weak solutions to the multi-dimensional Euler system, addressing a gap in the theoretical understanding of such flows.

Findings

Similarity shock solutions are genuine weak solutions.

Flows exhibit unbounded behavior at collapse.

Pre-collapse regions of vanishing pressure raise questions about boundedness.

Abstract

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, temperature) become unbounded at 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.