The sharp time decay rates for the incompressible Phan-Thien–Tanner system with magnetic field in [formula omitted]

Abstract

Our focus in this paper is to investigate the sharp time decay rates (upper and lower bounds) for the higher order spatial derivatives of large solutions to the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid. In dimension two, the sharp time decay rates of large solution (u,H) and all its derivatives have been shown by Chen et al. (2022). The L2-decay rate of the stress tensor τ was improved to (1+t)−1, however, the sharp time decay rates of any spatial derivatives of τ cannot be achieved. Based on Fourier time-splitting and semigroup methods, we shall obtain the optimal upper and lower bounds of time decay rates for the higher spatial derivatives of order k (k=1,2) of τ. More precisely, the kth order spatial derivatives of τ decay at (1+t)−1−k2-rate, which will be shown to be sharp. In addition, our result is valid for the incompressible Oldroyd-B system.