Global existence and asymptotic behavior of solutions to the Euler equations with time-dependent damping

We study the isentropic Euler equations with time-dependent damping, given by $\frac{\mu}{(1+t)^\lambda}\rho u$. Here, $\lambda,\mu$ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when $0<\lambda<1,0<\mu$ in multi-dimensions $n\geq 1$. The asymptotic behavior will coincide with the one that obtained by many authors in the case $\lambda=0$. We will also show that the solution can only decay polynomially in time while in the three dimensions, the vorticity will decay exponentially fast.