Global existence and optimal time decay rates for the three-dimensional incompressible Phan-Thien–Tanner model

Abstract

In this paper, we consider an initial-value problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) model. This model is regard to the dynamic mechanical properties of polymeric fluids. We shall mainly investigate two different cases, one is without damping on the stress tensor (i.e. [Formula: see text]) and the other is [Formula: see text]. Under some suitable assumptions on the initial data, we show that the solutions of the PTT model will blow up in finite time. This result also reveals a special solution. Furthermore, we prove the global existence and the optimal time decay rates of strong solutions to the PTT model when the initial data close enough to this special solution. In particular, it also works for the simpler case of the Oldroyd-B model.