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

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