Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping

We are concerned with the space-time decay rate of high-order spatial derivatives of solutions for 3D compressible Euler equations with damping. For any integer $\ell \ge 3$, Kim (2022) showed the space-time decay rate of the $k(0\le k\le \ell -2)$th-order spatial derivative of the solution. By making full use of the structure of the system, and employing different weighted energy methods for $0\le k\le \ell -2,k=\ell -1,k=\ell$, it is shown that the space-time decay rate of the $(\ell -1)$th-order and $\ell$th-order spatial derivative of the strong solution in weighted Lebesgue space $L_{\sigma }^2$ are $t^{-\frac{3}{4}-\frac{\ell -1}{2}+\frac{\sigma }{2}}$ and $t^{-\frac{3}{4}-\frac{\ell }{2}+\frac{\sigma }{2}}$ respectively, which are totally new as compared to that of Kim (2022) ^[1].