Global existence and convergence rates of solutions for the compressible  Euler equations with damping

Ruiying Wei; Yin Li; Zheng-an Yao

doi:10.3934/dcdsb.2020047

Article Contents

2020, Volume 25, Issue 8: 2949-2967. Doi: 10.3934/dcdsb.2020047

This issue Previous Article Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media Next Article Spectral theory and time asymptotics of size-structured two-phase population models

Global existence and convergence rates of solutions for the compressible Euler equations with damping

1.
School of Mathematics and Statistics, Shaoguan University, 512005, Shaoguan, China
2.
Department of Mathematics, Sun Yat-sen University, 510275, Guangzhou, China

^* Corresponding author: Yin Li
^* Corresponding author: Yin Li

Received: December 31, 2018

Revised: August 31, 2019

Early access: February 2020

Published: August 2020

We would like to express our sincere thanks to Academician Boling Guo of institute of Applied Physics and Computational Mathematics in Beijing for their fruitful help and discussions. This work is partially supported by the National Natural Science Foundation of China(Nos.11926354,11701380 and 11971496),Natural Science Foundation of Guangdong Province (Nos.2019A1515011320,2017A030307022,2016A030310019 and 2016A030307042),the Education Research Platform Project of Guangdong Province(No.2018179).

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Abstract

The Cauchy problem for the 3D compressible Euler equations with damping is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space $ H^3(\mathbb{R}^{3}) $ only, but we don't need the bound of $ L^1 $ norm. Moreover, the optimal $ L^{2} $-$ L^{2} $ convergence rates are also obtained for the solution. Our proof is based on the benefit of the low frequency and high frequency decomposition, here, we just need spectral analysis of the low frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.

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Mathematics Subject Classification: Primary: 35Q35, 35B40; Secondary: 76P05.

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References

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