A Remark on the Existence of 2-D Steady Navier–Stokes Flow in Bounded Symmetric Domain Under General Outflow Condition

Abstract

Let Ω be a 2-dimensional bounded domain, symmetric with respect to the x2-axis. The boundary has several connected components, intersecting the x2-axis. The boundary value is symmetric with respect to the x2-axis satisfying the general outflow condition. The existence of the symmetric solution to the steady Navier–Stokes equations was established by Amick [2] and Fujita [4]. Fujita [4] proved a key lemma concerning the solenoidal extension of the boundary value by virtual drain method. In this note, we give a different proof via elementary approach by means of the stream function.