On the global existence and blowup of smooth solutions of 3-D compressible Euler equations with time-depending damping

In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping $${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0,{\mathrm{\partial }}_t(\rho u)+\mathrm{div}(\rho u\otimes u+p{I}_3)=-\frac{\mu }{(1+t{)}^{\lambda }}\rho u,\rho (0,x)=\bar{\rho }+\epsilon {\rho }_0(x),u(0,x)=\epsilon u_0(x),$$ where $x\in {\mathbb{R}}^3$, $\mu >0$, $\lambda \ge 0$, and $\bar{\rho }>0$ are constants, ${\rho }_0,u_0\in C_0^{\mathrm{\infty }}({\mathbb{R}}^3)$, $(\rho_0, u_0)\not\equiv 0$,$\rho(0,\cdot)>0$,and$\varepsilon>0$ is sufficiently small. For $0\le \lambda \le 1$, we show that there exists a global smooth solution $(\rho ,u)$ when $\mathrm{curl}{u}_0\equiv 0$, while for $\lambda >1$, in general, the solution $(\rho ,u)$ will blow up in finite time. Therefore, $\lambda =1$ appears to be the critical value for the global existence of small amplitude smooth solutions.