TY - JOUR

T1 - On similarity flows for the compressible Euler system

AU - Jenssen, Helge Kristian

AU - Tsikkou, Charis

N1 - Funding Information:
This material is based in part upon work supported by the National Science Foundation under Grant Nos. DMS-1311353 (Jenssen) and DMS-1714912 (Tsikkou). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Publisher Copyright:
© 2018 Author(s).

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

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U2 - 10.1063/1.5049093

DO - 10.1063/1.5049093

M3 - Article

AN - SCOPUS:85059614466

VL - 59

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

M1 - 121507

ER -