The present paper investigates the global large solutions and long time behavior of the solutions to the incompressible polymeric flow in the whole space. Our main results and innovations can be stated as follows: Employing the weighted Chemin-Lerner technique and a dedicated energy argument, we establish the existence of a unique global solution for the incompressible polymeric Euler flow with some large initial data. Specifically, for a broad class of initial stress tensors satisfying div τ0 = 0, we demonstrate the existence of global solutions with only a mild constraint imposed on the initial velocity field. In contrast to the classical incompressible Euler equations, which have been shown to exhibit finite-time singularities even with small initial data, our findings underscore the stabilizing effect of the stress tensor on the fluid dynamics. We obtain the large time asymptotic behavior of the solutions for the incompressible polymeric Euler flow with small initial data. We establish the global existence and the large time asymptotic behavior of the solutions for the incompressible polymeric flow without dissipation of the stress tensor.
Liang et al. (Thu,) studied this question.