A refined uniqueness result of Leray's problem in an infinite-long pipe with the Navier-slip boundary

We consider the generalized Leray's problem with the Navier-slip boundary condition in an infinite pipe $\mathcal{D}=\Sigma\times\mathbb{R}$. We show that if the flux $\Phi$ of the solution is no larger than a critical value that is \emph{independent with the friction ratio} of the Navier-slip boundary condition, the solution to the problem must be the parallel Poiseuille flow with the given flux. Compared with known related 3D results, this seems to be the first conclusion with the size of critical flux being uniform with the friction ratio $\alpha\in]0,\infty]$, and it is surprising since the prescribed uniqueness breaks down immediately when $\alpha=0$, even if $\Phi=0$. Our proof relies primarily on a refined gradient estimate of the Poiseuille flow with the Navier-slip boundary condition. Additionally, we prove the critical flux $\Phi_0\geq\frac{\pi}{16}$ provided that $\Sigma$ is a unit disk.
Comment: 20 pages. All comments are welcome!