  This paper is devoted to global existence and optimal decay rate of weak
solutions to some inviscid Oldroyd-B models with center diffusion. By virtue of
the properties of Calderon-Zygmund operator and the Littlewood-Paley
decomposition theory, we firstly prove that the 2-D co-rotation inviscid
Oldroyd-B model admits global weak solutions with some large data under
different integrability conditions. Furthermore, we prove the energy
conservation of weak solutions for the co-rotation case. These obtained results
generalize and cover the classical results for the Euler equation. Moreover, we
establish global weak solutions with small data for the 2-D noncorotation
inviscid Oldroyd-B model without damping. Finally, we prove optimal decay rate
of global weak solutions for the noncorotation case by the improved Fourier
splitting method.
