Communications in Mathematical Sciences

Volume 21 (2023)

Number 8

Optimal large-time behavior of the compressible Phan–Thein–Tanner model

Pages: 2145 – 2167



Yin Li (Faculty of Education, Shaoguan University, Shaoguan, China)

Ruiying Wei (School of Mathematics and Statistics, Shaoguan University, Shaoguan, China)

Guochun Wu (School of Mathematical Sciences, Huaqiao University, Quanzhou, China)

Zheng-An Yao (School of Mathematics, Sun Yat-sen University, Guangzhou, China)


In this paper, we investigate global existence and optimal decay rates of strong solutions to the three dimensional compressible Phan–Thein–Tanner model. We prove the global existence of the solutions by the standard energy method under the 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 also 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 spatial derivatives. Our method is based on Hodge decomposition, low-frequency and high-frequency decomposition, delicate spectral analysis, and energy methods.


Phan–Thein–Tanner model, optimal large-time behavior, global existence

2010 Mathematics Subject Classification

35Q30, 76N15, 76P05

Received 3 May 2022

Received revised 14 October 2022

Accepted 1 March 2023

Published 15 November 2023