Global existence and blowup of smooth solutions of 3-D potential equations with time-dependent damping

Fei Hou, Ingo Witt and Huicheng Yin

Vol. 292 (2018), No. 2, 389–426

DOI: 10.2140/pjm.2018.292.389

Abstract

In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D irrotational compressible Euler equation with time-dependent damping

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ {\partial }_t\rho +div\left(\rho u\right)=0,\hfill \\ \hfill \\ \hfill \\ {\partial }_t\left(\rho u\right)+div\left(\rho u\otimes u+p{I}_3\right)=-\alpha \left(t\right)\rho u,\hfill \\ \hfill \\ \hfill \\ \rho \left(0,x\right)=\bar{\rho }+\epsilon {\rho }_0\left(x\right),u\left(0,x\right)=\epsilon u_0\left(x\right),\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

where $x\in {ℝ}^3$, the frictional coefficient $\alpha \left(t\right)=\mu ∕{\left(1+t\right)}^{\lambda }$ with $\mu >0$ and $\lambda \ge 0$, $\bar{\rho }>0$ is a constant, ${\rho }_0,u_0\in C_0^{\infty }\left({ℝ}^3\right)$, $\left({\rho }_0,u_0\right)\not\equiv 0$, $\rho \left(0,x\right)>0$, $curl{u}_0\equiv 0$, and $\epsilon >0$ is sufficiently small. For $0\le \lambda \le 1$, we show that there exists a global $C^{\infty }\left(\left[0,\infty \right)\times {ℝ}^3\right)$-smooth solution $\left(\rho ,u\right)$ by introducing and establishing some uniform time-weighted energy estimates of $\left(\rho ,u\right)$, while for $\lambda >1$, in general, the smooth solution $\left(\rho ,u\right)$ blows up in finite time. Therefore, $\lambda =1$ appears to be the critical value for the global existence of small amplitude smooth solution $\left(\rho ,u\right)$.

Keywords

compressible Euler equations, damping, time-weighted energy inequality, Klainerman–Sobolev inequality, blowup, hypergeometric function

Mathematical Subject Classification 2010

Primary: 35L70

Secondary: 35L65, 35L67, 76N15

Milestones

Received: 31 December 2016

Accepted: 2 May 2017

Published: 17 October 2017