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In this paper, we consider the 2-dimensional non-viscous Oldroyd-B model. Inthe case of the ratio equal 1~($\alpha=0$), it is a difficult case since thevelocity field $u(t,x)$ is no longer decay. Fortunately, by {observing theexponential decay} of the stress tensor $\tau(t,x)$, we succeeded in provingthe global existence for this system with some large initial data. Moreover, wegive an unsteady result: when the ratio is close to 1~($a\rightarrow 0$), thesystem is not steady for large time. This implies an interesting physicalphenomenon that the term $a\mathbb{D}u$ is a bridge between the transformationof kinetic energy $u$ and elastic potential energy $\tau$, but this process istransient for large time, which leads the instability.

Global Regularity and instability for the incompressible non-viscous Oldroyd-B model