L2-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

doi:10.1007/s10473-022-0618-6

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (6): 2505-2522.doi: 10.1007/s10473-022-0618-6

• 论文 • 上一篇

L²-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

Shifeng GENG¹, Feimin HUANG², Xiaochun WU³

1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China;
2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
3. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China

收稿日期:2022-08-27 出版日期:2022-12-25 发布日期:2022-12-16
通讯作者: Xiaochun WU, E-mail: xcwu22@csu.edu.cn E-mail:xcwu22@csu.edu.cn
基金资助:
S. Geng’s research was supported in part by the National Natural Science Foundation of China (12071397) and Excellent Youth Project of Hunan Education Department (21B0165). F. Huang’s research was supported in part by the National Key R&D Program of China 2021YFA1000800 and the National Natural Science Foundation of China (12288201).

L²-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

Shifeng GENG¹, Feimin HUANG², Xiaochun WU³

1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China;
2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
3. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China

Received:2022-08-27 Online:2022-12-25 Published:2022-12-16
Contact: Xiaochun WU, E-mail: xcwu22@csu.edu.cn E-mail:xcwu22@csu.edu.cn
Supported by:
S. Geng’s research was supported in part by the National Natural Science Foundation of China (12071397) and Excellent Youth Project of Hunan Education Department (21B0165). F. Huang’s research was supported in part by the National Key R&D Program of China 2021YFA1000800 and the National Natural Science Foundation of China (12288201).

摘要/Abstract

摘要： In this paper, we are concerned with the asymptotic behavior of $L^{\infty}$ weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping $-\frac{m}{(1+t)^{\lambda}}$. As $\lambda \in (0,\frac17]$, we prove that the $L^{\infty}$ weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation (GPME) in $L^2(\mathbb R)$. As $\lambda \in (\frac17,1)$, we prove that the $L^{\infty}$ weak-entropy solution converges to an expansion around the nonlinear diffusion wave in $L^2(\mathbb R)$, which is the best asymptotic profile. The proof is based on intensive entropy analysis and an energy method.

关键词: L²-convergence, compressible Euler Equations, time asymptotic expansion, time-dependent damping, relative entropy inequality

Abstract: In this paper, we are concerned with the asymptotic behavior of $L^{\infty}$ weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping $-\frac{m}{(1+t)^{\lambda}}$. As $\lambda \in (0,\frac17]$, we prove that the $L^{\infty}$ weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation (GPME) in $L^2(\mathbb R)$. As $\lambda \in (\frac17,1)$, we prove that the $L^{\infty}$ weak-entropy solution converges to an expansion around the nonlinear diffusion wave in $L^2(\mathbb R)$, which is the best asymptotic profile. The proof is based on intensive entropy analysis and an energy method.

Key words: L²-convergence, compressible Euler Equations, time asymptotic expansion, time-dependent damping, relative entropy inequality

中图分类号:

35L65

Shifeng GENG, Feimin HUANG, Xiaochun WU. L²-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING[J]. 数学物理学报(英文版), 2022, 42(6): 2505-2522.

Shifeng GENG, Feimin HUANG, Xiaochun WU. L²-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING[J]. Acta mathematica scientia,Series B, 2022, 42(6): 2505-2522.

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L²-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

L²-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

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