In this talk, we consider the multi-dimensional compressible Euler equations with time-dependent damping of the form $-\frac{\mu}{(1+t)^\lambda}\rho\boldsymbol u$ in $\mathbb R^n$, where $n\ge2$, $\mu>0$, and $\lambda\in[-1,1)$. When $\lambda>0$ ( $\lambda<0$), the damping effect time-asymptotically gets weaker (stronger), which is called under-damping (over damping). We show the optimal decay estimates of the solutions in the under-damping and over-damping cases, respectively, and see how the under-damping effect influences the structure of the Euler system. The time-dependent damping affects essentially the structure of solutions to Euler equations. Different from the traditional view that the stronger damping usually makes the solutions decaying faster, here we recognize that the weaker damping with $0\le\lambda<1$ enhances the faster decay for the solutions, and the effect of the stronger damping with $-1\le \lambda <0$ reduces the decay of the solutions to be slower. The approach adopted for proof is the technical Fourier analysis and Green function method. This is a joint work with Shanming Ji.

会议网址：https://meeting.tencent.com/s/uKzd5zei6Zsm

会议ID：657 608 878