Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

Blowup for $C^1$ solutions of compressible Euler equations with time-dependent damping

Pages: 513 – 528

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n2.a9

Authors

Jianli Liu (Department of Mathematics, Shanghai University, Shanghai, China)

Jingjie Wang (Department of Mathematics, Shanghai University, Shanghai, China)

Manwai Yuen (Department of Mathematics and Information Technology, Education University of Hong Kong)

Abstract

In this paper, we will show the blowup phenomenon of solutions to the compressible Euler equations with time-dependent damping. Firstly, under the assumptions that the radially symmetric initial data and initial density contains vacuum states, the singularity of the classical solutions will formed in finite time in $\mathbb{R}^n (n \geq 2)$. Furthermore, we can also find a sufficient condition for the functional of initial data such that smooth solution of the irrotational compressible Euler equations with time-dependent damping breaks down in finite time for all kinds of fractional coefficients in $\mathbb{R}^n (n \geq 2)$.

Keywords

Euler equations, singularity formation, time-dependent damping, vacuum

2010 Mathematics Subject Classification

35B30, 35B44, 35Q31

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This work was partially supported by National Natural Science Foundation of China under Grant No. 11771274, and supported by Seed Fund for General Research Fund/Early Career Scheme of the Dean’s Research Fund 2019-2020 from the Education University of Hong Kong.

Received 5 June 2020

Accepted 14 September 2020

Published 12 April 2021