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

Abstract

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\geq3 $, Kim (2022) showed the space-time decay rate of the $ k(0\leq k\leq \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\leq k \leq \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].

Keywords

• compressible euler equations
• space-time decay rate
• weighted estimate