Time periodic solutions of the Navier-Stokes equations with the time periodic Poiseuille flow in two and three dimensional perturbed channels

Abstract

H. Beirão da Veiga proved that, for a straight channel in $\boldsymbol{R}^n$ ($n$ arbitarily large) and for a given flux with the time periodicity, there exists a unique time periodic Poiseuille flow in a straight channel in $\boldsymbol{R}^n$. Furthermore, the existence of a time periodic solution in a perturbed channel (Leray's problem) is shown for the Stokes problem (arbitary dimension) and for the Navier-Stokes problem ($n\le4$). Concerning the Navier-Stokes case, a quatitative condition requaired to show the existence of a time periodic solution depends not just on the flux of the time periodic Poiseuille flow but also on the domain it self. In this paper, by applying the result of H. Beirão da Veiga and C. J. Amick, we succeed in proving the independence of such a condition on the particular domain.