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, 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.
| Original language | English (US) |
|---|---|
| Article number | 121507 |
| Journal | Journal of Mathematical Physics |
| Volume | 59 |
| Issue number | 12 |
| DOIs | |
| State | Published - Dec 1 2018 |
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All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
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On similarity flows for the compressible Euler system. / Jenssen, Helge Kristian; Tsikkou, Charis.
In: Journal of Mathematical Physics, Vol. 59, No. 12, 121507, 01.12.2018.Research output: Contribution to journal › Article
TY - JOUR
T1 - On similarity flows for the compressible Euler system
AU - Jenssen, Helge Kristian
AU - Tsikkou, Charis
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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UR - http://www.scopus.com/inward/citedby.url?scp=85059614466&partnerID=8YFLogxK
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 -