  A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by
[T. M. Elgindi, F. Rousset, Commun. Pure Appl. Math. 68 (2015), 2005--2021]
where the global existence and uniqueness of the strong solution were
established for arbitrarily large initial data. As pointed out by [A. V. Bhave,
R. C. Armstrong, R. A. Brown, J. Chem. Phys. 95(1991), 2988--3000], the
diffusion coefficient is significantly smaller than other effects, it is
interesting to study the non-diffusive model. In the present work, we obtain
the global-in-time existence and uniqueness of the strong solution to the
non-diffusive model with small initial data via deriving some uniform
regularity estimates and taking vanishing diffusion limits. In addition, the
large time behavior of the solution is studied and the optimal time-decay rates
for each order of spatial derivatives are obtained. The main challenges focus
on the lack of dissipation and regularity effects of the system and on the
slower decay in the two-dimensional settings. A combination of the spectral
analysis and the Fourier splitting method is adopted.
