Global well-posedness for the Phan-Thein-Tanner model in critical Besov spaces without damping

In this paper, we mainly investigate the Cauchy problem for the Phan-Thein-Tanner (PTT) model. The PPT model can be viewed as a Navier-Stokes equations couple with a nonlinear transport system. This model is derived from network theory for the polymeric fluid. We study about the global well posedness of the PTT model in critical Besov spaces. When the initial data is a small perturbation over around the equilibrium, we prove that the strong solution in critical Besov spaces exists globally.