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

Abstract

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 $\Vert u_0{\Vert }_{H^{\frac{7}{2}+}}\le {\epsilon }_1{\nu }^{2/3}$ 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 $3/4$ to $2/3$.

This version fixes an error in the proof of precious version, and improves the result of [18] form 3/4 to 2/3 for slip boundary value problem. The case of non-slip boundary value problem is not included in this version