Contents Online
Communications in Mathematical Sciences
Volume 21 (2023)
Number 8
Optimal large-time behavior of the compressible Phan–Thein–Tanner model
Pages: 2145 – 2167
DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n8.a4
Authors
Abstract
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.
Keywords
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