#### Vol. 292, No. 2, 2018

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Global existence and blowup of smooth solutions of 3-D potential equations with time-dependent damping

### Fei Hou, Ingo Witt and Huicheng Yin

Vol. 292 (2018), No. 2, 389–426
##### Abstract

In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D irrotational compressible Euler equation with time-dependent damping

 $\left\{\begin{array}{c}\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ {\partial }_{t}\rho +div\left(\rho u\right)=0,\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ {\partial }_{t}\left(\rho u\right)+div\left(\rho u\otimes u+p{I}_{3}\right)=-\alpha \left(t\right)\rho u,\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \rho \left(0,x\right)=\stackrel{̄}{\rho }+\epsilon {\rho }_{0}\left(x\right),\phantom{\rule{1em}{0ex}}u\left(0,x\right)=\epsilon {u}_{0}\left(x\right),\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \end{array}\right\$

where $x\in {ℝ}^{3}$, the frictional coefficient $\alpha \left(t\right)=\mu ∕{\left(1+t\right)}^{\lambda }$ with $\mu >0$ and $\lambda \ge 0$, $\stackrel{̄}{\rho }>0$ is a constant, ${\rho }_{0},{u}_{0}\in {C}_{0}^{\infty }\left({ℝ}^{3}\right)$, $\left({\rho }_{0},{u}_{0}\right)\not\equiv 0$, $\rho \left(0,x\right)>0$, $curl{u}_{0}\equiv 0$, and $\epsilon >0$ is sufficiently small. For $0\le \lambda \le 1$, we show that there exists a global ${C}^{\infty }\left(\left[0,\infty \right)×{ℝ}^{3}\right)$-smooth solution $\left(\rho ,u\right)$ by introducing and establishing some uniform time-weighted energy estimates of $\left(\rho ,u\right)$, while for $\lambda >1$, in general, the smooth solution $\left(\rho ,u\right)$ blows up in finite time. Therefore, $\lambda =1$ appears to be the critical value for the global existence of small amplitude smooth solution $\left(\rho ,u\right)$.

##### Keywords
compressible Euler equations, damping, time-weighted energy inequality, Klainerman–Sobolev inequality, blowup, hypergeometric function
##### Mathematical Subject Classification 2010
Primary: 35L70
Secondary: 35L65, 35L67, 76N15