Electron. J. Differential Equations, Vol. 2020 (2020), No. 102, pp. 1-25.

Optimal time decay rates for the compressible Navier-Stokes system with and without Yukawa-type potential

Qing Chen, Guochun Wu, Yinghui Zhang, Lan Zou

Abstract:
We consider the time decay rates of smooth solutions to the Cauchy problem for the compressible Navier-Stokes system with and without a Yukawa-type potential. We prove the existence and uniqueness of global solutions by the standard energy method under small initial data assumptions. Furthermore, if the initial data belong to $L^1(\mathbb R^3)$, we establish the optimal time decay rates of the solution as well as its higher-order spatial derivatives. In particular, we obtain the optimal decay rates of the highest-order spatial derivatives of the velocity. Finally, we derive the lower bound time decay rates for the solution and its spacial derivatives.

Submitted February 2, 2020. Published September 29, 2020.
Math Subject Classifications: 35Q30, 76N15
Key Words: Compressible flow; energy method; optimal decay rates.
DOI: 10.58997/ejde.2020.102

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Qing Chen
School of Applied Mathematics
Xiamen University of Technology
Xiamen, Fujian 361024, China
email: chenqing@xmut.edu.cn
Guochun Wu
Fujian Province University Key Laboratory of Computational Science
School of Mathematical Sciences
Huaqiao University
Quanzhou 362021, China
email: guochunwu@126.com
Yinghui Zhang
School of Mathematics and Statistics
Guangxi Normal University
Guilin, Guangxi 541004, China
email: yinghuizhang@mailbox.gxnu.edu.cn
Lan Zou
Fujian Province University Key Laboratory of Computational Science
School of Mathematical Sciences
Huaqiao University
Quanzhou 362021, China
email: zlyoung@163.com

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