Time periodic solutions of the Navier–Stokes equations with the time periodic Poiseuille flow under (GOC) for a symmetric perturbed channel in $\mathbb{R}^2$

Abstract

Beirao da Veiga [ 5] proves that for a straight channel in $\mathbb{R}^n$ ($n\ge2$) and for a given time periodic flux there exists a unique time periodic Poiseuille flow. As a by product, existence of the time periodic Poiseuille flow in perturbed channels (Leray's problem) is shown for the Stokes problem ($n\ge2$) and for the Navier–Stokes problem ($n\le4$). Concerning the Navier–Stokes case, in [ 5] a quantitative condition required to show the existence of the solutions depends not just on the flux of the time periodic Poiseuille flow but also on the domain itself. Kobayashi [ 16], [ 18] proves that for a perturbed channel in $\mathbb{R}^n$ $(n=2,3)$ there exists a time periodic solution of the Navier–Stokes equations with the Poiseuille flow applying the theory of the steady problem to the time periodic problem. In this paper, applying Fujita [ 8] and Kobayashi [ 18], we succeed in proving the existence of a time periodic solution for a symmetric perturbed channel in $\mathbb{R}^2$.