Construction of axisymmetric steady states of an inviscid incompressible fluid by spatially discretized equations for pseudo-advected vorticity

Here, p(: Ω→ R) is the pressure, ∂Ω is the boundary, and n is the unit outward normal vector on ∂Ω. In the axisymmetric case, the existence of solutions to (1.2) was discussed as the problem of vortex rings in, for example, [2, 4, 5, 8]. Their methods were based on a variational principle for kinetic energy. By contrast, Vallis et al. [13] proposed a completely different approach to the solvability of (1.2). Assume that the pair (v,q) : Ω×{t > 0} →R3 ×R satisfies the nonstationary system vt +ω× ( v +αvt )=−∇(q+ |v|2 2 ) , ∇· v = 0, v ·n|∂Ω = 0 (1.3)