  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.
