Global well-posedness, stability and instability for the non-viscous Oldroyd-B model

In this paper we consider the 3-dimensional incompressible Oldroyd-B model. First, we establish two results of the global existence for different kinds of the coupling coefficient $k$. Then, we prove that the solutions $(u,\tau)$ are globally steady when $k^m\rightarrow k>0$, though $(u,\tau)$ corresponds to different decays for different kinds of $k>0~$. Finally, we show that the energy of $u(t,x)$ will have a jump when $k\rightarrow 0$ in large time, which implies a non-steady phenomenon. In a word, we find an interesting physical phenomenon of \eqref{1} such that smaller coupling coefficient $k$ will have a better impact for the energy dissipation of $(u,\tau)$, but $k$ can't be too small to zero, or the dissipation will vanish instantly. While the damping term $\tau$ and $\mathbb{D}u$ always bring the well impact for the energy dissipation.