Uniqueness and uniform structural stability of Poiseuille flows in a two-dimensional strip

Abstract

In this paper, we prove the uniform nonlinear structural stability of Poiseuille flows with arbitrarily large flux for the steady Navier-Stokes system in a two-dimensional periodic strip when the period is not very large. The key point is to establish the a priori estimate for the corresponding linearized problem via the careful analysis for the associated boundary layers. Furthermore, the well-posedness theory for the Navier-Stokes system is also proved even when the L2-norm of the external force is large. These results prove the uniqueness of solutions for the steady Navier-Stokes system even when the flux is large and the flow is not symmetric. In particular, if the vertical velocity is suitably small where the smallness is independent of the flux, then Poiseuille flow is the unique solution of the steady Navier-Stokes system in the periodic strip.