On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping

Abstract

In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping \begin{equation*} \partial_tρ+\operatorname{div}(ρu)=0, \quad \partial_t(ρu)+\operatorname{div}\left(ρu\otimes u+p\,I_d\right)=-α(t)ρu, \quad ρ(0,x)=\bar ρ+\varepsilonρ_0(x),\quad u(0,x)=\varepsilon u_0(x), \end{equation*} where $x=(x_1,\cdots ,x_d)\in {\mathbb{R}}^d$ $(d=2,3)$, the frictional coefficient is $\alpha (t)=\frac{\mu }{(1+t{)}^{\lambda }}$ with $\lambda \ge 0$ and $\mu >0$, $\bar{\rho }>0$ is a constant, ${\rho }_0,u_0\in C_0^{\infty }({\mathbb{R}}^d)$, $({\rho }_0,u_0)\not\equiv 0$, $\rho (0,x)>0$, and $\epsilon >0$ is sufficiently small. One can totally divide the range of $\lambda \ge 0$ and $\mu >0$ into the following four cases: Case 1: $0\le \lambda <1$, $\mu >0$ for $d=2,3$; Case 2: $\lambda =1$, $\mu >3-d$ for $d=2,3$; Case 3: $\lambda =1$, $\mu \le 3-d$ for $d=2$; Case 4: $\lambda >1$, $\mu >0$ for $d=2,3$. \noindent We show that there exists a global $C^{\infty }-$smooth solution $(\rho ,u)$ in Case 1, and Case 2 with $\mathrm{curl}{u}_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(\rho ,u)$ blows up in finite time. Therefore, $\lambda =1$ and $\mu =3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(\rho ,u)$ in $d-$dimensional compressible Euler equations with time-depending damping.

32 pages, 2 figures

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