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

Qin Ye; Qin Ye

doi:10.3934/era.2023197

Electronic Research Archive

2023, Volume 31, Issue 7: 3879-3894. doi: 10.3934/era.2023197

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Research article

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

Qin Ye ^,

School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China

Received: 03 November 2022 Revised: 12 January 2023 Accepted: 28 January 2023 Published: 12 May 2023

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].
- compressible Euler equations,
- space-time decay rate,
- weighted estimate
Citation: Qin Ye. Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping[J]. Electronic Research Archive, 2023, 31(7): 3879-3894. doi: 10.3934/era.2023197

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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].

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