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Vanishing limit for the three-dimensional incompressible Phan-Thien–Tanner system

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  27 March 2023

Yuhui Chen ,
Minling Li Open the ORCID record for Minling Li [Opens in a new window] ,
Qinghe Yao  and
Zheng-an Yao
Yuhui Chen
Affiliation:
School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou 510275, China (chenyh339@mail.sysu.edu.cn)
Minling Li
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China (limling3@mail2.sysu.edu.cn)
Qinghe Yao
Affiliation:
School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou 510275, China (yaoqhe@mail.sysu.edu.cn)
Zheng-an Yao
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China (mcsyao@mail.sysu.edu.cn)
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Abstract

This paper focuses on the vanishing limit problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) system, which is commonly used to describe the dynamic properties of polymeric fluids. Our purpose is to show the relation of the PTT system to the well-known Oldroyd-B system (with or without damping mechanism). The suitable a priori estimates and global existence of strong solutions are established for the PTT system with small initial data. Taking advantage of uniform energy and decay estimates for the PTT system with respect to time $t$ and coefficients $a$ and $b$, then allows us to justify in particular the vanishing limit for all time. More precisely, we prove that the solution $(u,\,\tau )$ of PTT system with $0\leq b\leq Ca$ converges globally in time to some limit $(\widetilde {u},\,\widetilde {\tau })$ in a suitable Sobolev space when $a$ and $b$ go to zero simultaneously (or, only $b$ goes to zero). We may check that $(\widetilde {u},\,\widetilde {\tau })$ is indeed a global solution of the corresponding Oldroyd-B system. In addition, a rate of convergence involving explicit norm will be obtained. As a byproduct, similar results are also true for the local a priori estimates in large norm.

Keywords

Navier–Stokes equationsPhan-Thien–Tanner systemglobal well-posednessvanishing limitconvergence rate

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 35A01: Existence problems 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 3 , June 2024 , pp. 673 - 698
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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