Asymptotic stability of symmetric flows with viscous inflow boundary condition

We study the two-dimensional incompressible Navier-Stokes equations in a channel $\mathrm{\Omega }=(0,L)\times (0,H)$ with small viscosity $\epsilon \ll 1$, an $\epsilon$-Navier slip condition on the horizontal walls, and a viscous inflow condition for the perturbation stream function. For a broad class of symmetric base profiles $u_0(y)$ vanishing on the walls, we construct an exact steady solution $(u_s,v_s)$ that is $O({\epsilon }^{1\mathrm{/}3})$-close to the shear $(u_0,0)$. We then develop a new weighted vorticity energy method to prove uniform linear stability and exponential decay: perturbations decay exponentially in a weighted $L^2$ norm on the time scale $O({\epsilon }^{-1\mathrm{/}3})$. In the short-channel regime $L\ll 1$, the method yields nonlinear asymptotic stability with threshold $O({\epsilon }^{2\mathrm{/}3})$. In the long-channel regime, assuming concavity together with a spectral condition, we introduce a quantity $\mathit{\text{Rayleigh}}\mathit{\text{vorticity}}$ to control the non-favorable terms and obtain nonlinear stability with threshold $O({\epsilon }^{5\mathrm{/}6+})$.