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 $$ partial_trho+operatorname{div}(rho u)=0, quad partial_t(rho u)+operatorname{div}left(rho uotimes u+p,I_{3}right)=-,frac{mu}{(1+t)^{lambda}},rho u, quad rho(0,x)=bar rho+varepsilonrho_0(x),quad u(0,x)=varepsilon u_0(x), $$ where $xinmathbb R^3$, $mu>0$, $lambdageq 0$, and $barrho>0$ are constants, $rho_0,, u_0in C_0^{infty}(mathbb R^3)$, $(rho_0, u_0) otequiv 0$, $rho(0,cdot)>0$, and $varepsilon>0$ is sufficiently small. For $0leqlambdaleq1$, we show that there exists a global smooth solution $(rho, u)$ when $operatorname{curl} u_0equiv 0$, 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.