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On smooth self similar solution to compressible Euler equation with physical vacuum
Abstract
We consider the barotropic Euler equations in dimension d > 1 with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. The existence of smooth decaying solutions through the sonic line has been obtained in the work (F. Merle et al., 2022) in a suitable range of parameters. We work in this paper in a different range of parameters and show the existence of holosphere like solutions which are smooth through the sonic line, but display vanishing density at a sphere.
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BibTeX
@misc{kim2026smooth,
title = {{On smooth self similar solution to compressible Euler equation with physical vacuum}},
author = {Jihoi Kim},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30443},
eprint = {2608.30443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30443},
}
RIS
TY - UNPB
TI - On smooth self similar solution to compressible Euler equation with physical vacuum
AU - Jihoi Kim
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30443
UR - https://arxiv.org/abs/2608.30443
AB - We consider the barotropic Euler equations in dimension d > 1 with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. The existence of smooth decaying solutions through the sonic line has been obtained in the work (F. Merle et al., 2022) in a suitable range of parameters. We work in this paper in a different range of parameters and show the existence of holosphere like solutions which are smooth through the sonic line, but display vanishing density at a sphere.
ER -