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

Abstract

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 $$ \partial_tρ+\operatorname{div}(ρu)=0, \quad \partial_t(ρu)+\operatorname{div}\left(ρu\otimes u+p\,I_{3}\right)=-\,\fracμ{(1+t)^λ}\,ρu, \quad ρ(0,x)=\bar ρ+\varepsilonρ_0(x),\quad u(0,x)=\varepsilon u_0(x), $$ where $x\in\mathbb R^3$, $μ>0$, $λ\geq 0$, and $\barρ>0$ are constants, $ρ_0,\, u_0\in C_0^{\infty}(\mathbb R^3)$, $(ρ_0, u_0)\not\equiv 0$, $ρ(0,\cdot)>0$, and $\varepsilon>0$ is sufficiently small. For $0\leqλ\leq1$, we show that there exists a global smooth solution $(ρ, u)$ when $\operatorname{curl} u_0\equiv 0$, while for $λ>1$, in general, the solution $(ρ, u)$ will blow up in finite time. Therefore, $λ=1$ appears to be the critical value for the global existence of small amplitude smooth solutions.