In this article, we study the limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of n k disjoint disks with centers { z i k } and radii ε k . We assume that the initial velocities u 0 k are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, n k → ∞ , and we assume ε k → 0 as k → ∞ .

Let γ i k be the circulation of u 0 k around the circle { | x − z i k | = ε k } . We prove that the limit as k → ∞ retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) ω 0 k = curl u 0 k has a uniform compact support and converges weakly in L p 0 , for some p 0 > 2 , to ω 0 ∈ L c p 0 ( R 2 ) , (2) ∑ i = 1 n k γ i k δ z i k ⇀ μ weak-∗ in BM ( R 2 ) for some bounded Radon measure μ, and (3) the radii ε k are sufficiently small. Then the corresponding solutions u k converge strongly to a weak solution u of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity ω = curl u , with initial data ω 0 , where the transporting velocity field is generated from ω, so that its curl is ω + μ . As a byproduct, we obtain a new existence result for this modified Euler system.

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Asymptotic behavior of 2D incompressible ideal flow around small disks