The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien–Tanner system with magnetic field

Article type: Research Article

Authors: Chen, Yuhui^a | Li, Minling^{b; *} | Yao, Qinghe^a | Yao, Zheng-an^b

Affiliations: [a] School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou, 510275, China. E-mails: chenyh339@mail.sysu.edu.cn, yaoqhe@mail.sysu.edu.cn | [b] School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China. E-mails: limling3@mail2.sysu.edu.cn, mcsyao@mail.sysu.edu.cn

Correspondence: [*] Corresponding author. E-mail: limling3@mail2.sysu.edu.cn.

Abstract: In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture L2-convergence result via auxiliary logarithmic time decay estimates with the initial data in Lp(R2)∩L2(R2). Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized.

Keywords: Viscoelastic fluids, sharp time decay rates, global-in-time stability, large solutions

DOI: 10.3233/ASY-211736

Journal: Asymptotic Analysis, vol. 129, no. 3-4, pp. 451-484, 2022