On the Leray-Hopf Extension Condition for the Steady-State Navier-Stokes Problem in Multiply-Connected Bounded Domains

Abstract

Employing the approach of A. Takeshita [Pacific J. Math., Vol. 157 (1993), 151--158], we give an elementary proof of the invalidity of the Leray-Hopf Extension Condition for certain multiply connected bounded domains of R^n, n=2,3, whenever the flow through the different components of the boundary is non-zero. Our proof is alternative to and, to an extent, more direct than the recent one proposed by J.G. Heywood [J. Math. Fluid Mech. Vol. 13 (2011), 449--457].