Blow up and global existence for the periodic Phan-Thein-Tanner model

In this paper, we mainly investigate the Cauchy problem for the periodic Phan-Thein-Tanner (PTT) model. This model is derived from network theory for the polymeric fluid. We prove that the strong solutions of PTT model will blow up in finite time if the trace of initial stress tensor $\tr \tau_0(x)$ is negative. This is a very different with the other viscoelastic model. On the other hand, we obtain the global existence result with small initial data when $\tr \tau_0(x)\geq c_0>0$ for some $c_0$. Moreover, we study about the large time behavior.