Global solutions and large time behavior for some Oldroyd-B type models in $R^2$

In this paper, we are concerned with global solutions to the co-rotation Oldroyd-B type model and large time behavior for the general Oldroyd-B type model. We first establish the energy estimate and B-K-M criterion for the 2-D co-rotation Oldroyd-B type model. Then, we obtain global solutions with large data in Sobolev space by proving the boundedness of vorticity. As a corollary, we prove the global existence of corresponding Hooke model near equilibrium. Furthermore, we present the global existence for the 2-D co-rotation Oldroyd-B type model in critical Besov space by a refined estimate in Besov spaces with index $0$. Finally, we study large time behaviour for the general Oldroyd-B type model. Applying the Fourier spiltting method, we prove the $H^1$ decay rate for global solutions constructed by T. M. Elgindi and F. Rousset.