Asymptotic stability of symmetric flows with viscous inflow boundary condition

We study the two-dimensional incompressible Navier-Stokes equations in a channel $\Omega=(0,L)\times(0,H)$ with small viscosity $\varepsilon\ll1$, an $\varepsilon$-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 $O(\varepsilon^{1/3})$-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 $O(\varepsilon^{-1/3})$. In the short-channel regime $L\ll1$, the method yields nonlinear asymptotic stability with threshold $O(\varepsilon^{2/3})$. In the long-channel regime, assuming concavity together with a spectral condition, we introduce a quantity \textit{Rayleigh vorticity} to control the non-favorable terms and obtain nonlinear stability with threshold $O(\varepsilon^{5/6+})$.