Asymptotic stability of symmetric flows with viscous inflow boundary condition

TL;DR

This paper establishes the asymptotic stability of symmetric viscous flows in a channel with small viscosity, using a novel energy method to prove exponential decay and nonlinear stability under certain conditions.

Contribution

It introduces a new weighted vorticity energy method and constructs exact steady solutions for symmetric flows with viscous inflow, advancing understanding of stability in low-viscosity regimes.

Findings

Proves uniform linear stability with exponential decay for small viscosity.

Establishes nonlinear asymptotic stability with explicit thresholds in short-channel regimes.

Develops a new concept of Rayleigh vorticity to handle long-channel stability analysis.

Abstract

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 $O(\varepsilon^{5/6+})$.