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Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains

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Abstract

In this paper, we study the blowup of smooth solutions to the compressible Euler equations with radial symmetry on some fixed bounded domains (\(B_{R}=\{x\in {\mathbb {R}}^{N}:\ |x|\le R\}\), \(N=1,2,\ldots \)) by introducing some new averaged quantities. We consider two types of flows: initially move inward and initially move outward on average. For the flow initially moving inward on average, we show that the smooth solutions will blow up in a finite time if the density vanishes at the origin only (\(\rho (t,0)=0,\ \rho (t,r)>0,\ 0<r\le R\)) for \(N\ge 1\) or the density vanishes at the origin and the velocity field vanishes on the two endpoints (\(\rho (t,0)=0,\ v(t,R)=0\)) for \(N=1\). For the flow initially moving outward, we prove that the smooth solutions will break down in a finite time if the density vanishes on the two endpoints (\(\rho (t,R)=0\)) for \(N=1\). The blowup mechanisms of the compressible Euler equations with constant damping or time-depending damping are obtained as corollaries.

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Acknowledgements

The authors acknowledge the support from the Project of Youth Backbone Teachers of Colleges and Universities in Henan Province (2019GGJS176) and the Dean’s Research Fund 2018-19 (FLASS/DRF/IRS-5) from the Education University of Hong Kong.

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Authors and Affiliations

  1. School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou, 450015, People’s Republic of China

    Jianwei Dong

  2. Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong

    Manwai Yuen

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  1. Jianwei Dong
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  2. Manwai Yuen
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Correspondence to Jianwei Dong.

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Dong, J., Yuen, M. Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains. Z. Angew. Math. Phys. 71, 189 (2020). https://doi.org/10.1007/s00033-020-01392-8

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