Making use of a line integral defined without use of the partition of unity, Green's theorem is proved in the case of two-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces W1,p (Ω) ≡ H1,p (Ω) (1 ≤ p < ∞).
In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder (r≤d) where (r, θ, z) denotes the cylindrical co-ordinates in ℝ3 is considered. The motion is with swirl (i.e. the θ-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g.  that for the problem without swirl (f q = 0 in (f)) in the whole space, as the flux constant k tends to ∞1) dist(0z, δA) = O(k 1/2); diam A = O(exp(−c 0 k 3/2));2) k1/2Ψ)k∈ℕ converges to a vortex cylinder U m (see (1.2)).We show that for the problem with swirl, as k ↗ ∞, 1) holds; if m ≤ q + 2 then 2) holds and if m > q + 2 it holds with U q+2 instead of U m. Moreover, these results are independent of f 0, f q and d > 0.