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 $$ \partial_t\rho+\operatorname{div}(\rho u)=0, \quad \partial_t(\rho u)+\operatorname{div}\left(\rho u\otimes u+p\,I_{3}\right)=-\,\frac{\mu}{(1+t)^{\lambda}}\,\rho u, \quad \rho(0,x)=\bar \rho+\varepsilon\rho_0(x),\quad u(0,x)=\varepsilon u_0(x), $$ where $x\in\mathbb R^3$, $\mu>0$, $\lambda\geq 0$, and $\bar\rho>0$ are constants, $\rho_0,\, u_0\in C_0^{\infty}(\mathbb R^3)$, $(\rho_0, u_0)\not\equiv 0$, $\rho(0,\cdot)>0$, and $\varepsilon>0$ is sufficiently small. For $0\leq\lambda\leq1$, we show that there exists a global smooth solution $(\rho, u)$ when $\operatorname{curl} u_0\equiv 0$, 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.