Time periodic solutions of the Navier–Stokes equations with the time periodic Poiseuille flow under (GOC) for a symmetric perturbed channel in ℝ²

Abstract

Beirão da Veiga [5] proves that for a straight channel in ℝⁿ (n ≥ 2) 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 ≥ 2) and for the Navier–Stokes problem (n ≤ 4). 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 ℝⁿ (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 ℝ².