Stability for the 2-D plane Poiseuille flow in finite channel

In this paper, we study the stability for 2-D plane Poiseuille flow $(1-y^2,0)$ in a channel $\mathbb{T}\times (-1,1)$ with Navier-slip boundary condition. We prove that if the initial perturbation for velocity field $u_0$ satisfies that $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^{\frac{7}{2}+}}\le {\epsilon }_1{\nu }^{2\mathrm{/}3}$ for some suitable small $0<{\epsilon }_1\ll 1$ independent of viscosity coefficient $\nu$, then the solution to the Navier-Stokes equations is global in time and does not transit from the plane Poiseuille flow. This result improves the result of \cite{DL1} from $3\mathrm{/}4$ to $2\mathrm{/}3$.