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