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 by proving the boundedness of vorticity. In general case, we apply Fourier spiltting method to prove the H^1 decay rate for global solutions constructed by T.M.Elgindi and F.Rousset.