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 $${\partial }_t\rho +\mathrm{div}(\rho u)=0,{\partial }_t(\rho u)+\mathrm{div}\left(\rho u\otimes u+p{I}_3\right)=-\frac{\mu }{(1+t{)}^{\lambda }}\rho u,\rho (0,x)=\bar{\rho }+\epsilon {\rho }_0(x),u(0,x)=\epsilon u_0(x),$$ where $x\in {\mathbb{R}}^3$, $\mu >0$, $\lambda \ge 0$, and $\bar{\rho }>0$ are constants, ${\rho }_0,u_0\in C_0^{\infty }({\mathbb{R}}^3)$, $({\rho }_0,u_0)\not\equiv 0$, $\rho (0,\cdot )>0$, and $\epsilon >0$ 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.

28 pages

References in corpus (2)

• From $p_0(n)$ to $p_0(n+2)$
• Shock Formation in Small-Data Solutions to $3D$ Quasilinear Wave Equations

Cited by in corpus (7)

• Global existence and convergence to the modified Barenblatt solution for the compressible Euler equations with physical vacuum and time-dependent damping
• Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques
• Optimal decay rates of the compressible Euler equations with time-dependent damping in ${\mathbb{R}}^n$: (II) over-damping case
• Global existence and asymptotic behavior of solutions to the Euler equations with time-dependent damping
• Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant
• Singularity formation for compressible Euler equations with time-dependent damping
• Singularity formation for the 1D compressible Euler equation with variable damping coefficient